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Topics in quantum information and the theory of open quantum systems
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Topics in quantum information and the theory of open quantum systems
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TOPICS IN QUANTUM INFORMATION AND THE THEORY OF OPEN QUANTUM SYSTEMS by Ognyan Oreshkov A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) May 2008 Copyright 2008 Ognyan Oreshkov Dedication To my wife Iskra ii Acknowledgements First and foremost, I would like to express my gratitude to Todd A. Brun for his guidance and support throughout the years of our work together. He has been a great advisor and mentor! I am deeply indebted to him for introducing me to the field of quantum information science and helping me advance in it. From him I learned not only how to do research, but also numerous other skills important for the career of a scientist, such as writing, giving presentations, or communicating professionally. I highly appreciate the fact that he was always supportive of any research direction I wanted to undertake, never exerted pressureon my work, and was available to give me advice or encouragement every time I needed them. This provided for me the optimal environment to develop, and made my work with him a wonderful experience. I am also greatly indebted to Daniel A. Lidar who has had an enormous impact on my work. He provided inspiration for many of the studies presented in this thesis. I havelearnedtonsfrommydiscussionswithhimandfromhiscoursesonopenquantum systems and quantum error correction. His interest in what I do, the lengthy email discussions we had, and his invitations to present my research at his group meetings, have been a major stimulus for my work. IamalsothankfultoIgorDevetaktowhomIoweasignificantpartofmyknowledge in quantum error correction and quantum communication. It was a pleasure to have him in our research group. iii I want to thank Paolo Zanardi for encouraging me to complete my work on holo- nomic quantum computation, and for the fruitful discussions we had. Special thanks are due to Stephan Haas for advising me about the course of my Ph.D. studies from their very beginning. Back then, he made me feel that I can rely on him for advice or help of any sort, and he has been corroborating this ever since. I thank Hari Krovi and Mikhail Ryazanov for our collaboration on the project on the non-Markovian evolution of a qubit coupled to an Ising spin bath. I also thank Alireza Shabani for stimulating conversations regarding the measure of fidelity for encoded information. Thanks to Martin Varbanov for sharing my excitement about my research, and the numerous discussions we had. Thanks are due also to all members of the Physics department that I haven’t mentioned explicitly but with whom I have interacted during the course of my Ph.D. program, because I have learned a lot from all of them. Finally, I would like to acknowledge the people whose contribution to my Ph.D. workhasbeenindirectbutofenormoussignificance. IwantthankmyparentsKatyusha and Viktor, who ensured I received the best education when I was a child, gave me confidence in myself, and supported all my endeavors throughout my life. I thank my wifeIskra for her unconditional love, her belief in me, and her constant support, without which this work would have been impossible. iv Table of Contents Dedication ii Acknowledgements iii List of Figures viii Abstract xi Chapter 1: Introduction 1 1.1 Quantum information and open quantum systems . . . . . . . . . . . . 1 1.1.1 Deterministic dynamics of open quantum systems . . . . . . . . . 2 1.1.2 Quantum measurements . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.3 Quantum error correction . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Chapter 2: Generating quantum measurements using weak measure- ments 13 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Decomposing projective measurements . . . . . . . . . . . . . . . . . . . 15 2.3 Decomposing generalized measurements . . . . . . . . . . . . . . . . . . 18 2.4 Measurements with multiple outcomes . . . . . . . . . . . . . . . . . . . 24 2.5 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Chapter 3: Applications of the decomposition into weak measurements to the theory of entanglement 28 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.1 LOCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.2 Entanglement monotones . . . . . . . . . . . . . . . . . . . . . . 32 3.2.3 Infinitesimal operations . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Local operations from infinitesimal local operations . . . . . . . . . . . . 33 3.3.1 Unitary transformations . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.2 Generalized measurements . . . . . . . . . . . . . . . . . . . . . . 34 3.4 Differential conditions for entanglement monotones . . . . . . . . . . . . 35 3.4.1 Local unitary invariance . . . . . . . . . . . . . . . . . . . . . . . 36 3.4.2 Non-increase under infinitesimal local measurements . . . . . . . 37 v 3.4.3 Monotonicity under operations with information loss . . . . . . . 41 3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.5.1 Norm of the state . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.5.2 Local purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.5.3 Entropy of entanglement . . . . . . . . . . . . . . . . . . . . . . . 47 3.6 A new entanglement monotone . . . . . . . . . . . . . . . . . . . . . . . 49 3.7 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.8 Appendix: Proof of sufficiency . . . . . . . . . . . . . . . . . . . . . . . 54 Chapter 4: Non-Markovian dynamics of a qubit coupled to a spin bath via the Ising interaction 59 4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 Exact dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2.2 Exact solution for the system-spin dynamics . . . . . . . . . . . . 65 4.2.3 Limiting cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3 Approximation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.3.1 Born and Born-Markov approximations . . . . . . . . . . . . . . 74 4.3.2 NZ and TCL master equations . . . . . . . . . . . . . . . . . . . 78 4.3.3 Post-Markovian (PM) master equation . . . . . . . . . . . . . . . 82 4.4 Comparison of the analytical solution and the different approximation techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4.1 Exact Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.4.2 NZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.4.3 TCL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.4.4 NZ, TCL, and PM . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.4.5 Coarse-graining approximation . . . . . . . . . . . . . . . . . . . 93 4.5 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.6 Appendix A: Bath correlation functions . . . . . . . . . . . . . . . . . . 97 4.7 Appendix B: Cumulants for the NZ and TCL master equations . . . . . 102 Chapter 5: Continuous quantum error correction for non-Markovian decoherence 104 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.1.1 Continuous quantum error correction . . . . . . . . . . . . . . . . 104 5.1.2 Markovian decoherence . . . . . . . . . . . . . . . . . . . . . . . 107 5.1.3 The Zeno effect. Error correction versus error prevention . . . . . 108 5.1.4 Non-Markovian decoherence . . . . . . . . . . . . . . . . . . . . . 110 5.1.5 Plan of this chapter . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.2 The single-qubit code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.3 The three-qubit bit-flip code . . . . . . . . . . . . . . . . . . . . . . . . 118 5.3.1 A Markovian error model . . . . . . . . . . . . . . . . . . . . . . 118 5.3.2 A non-Markovian error model . . . . . . . . . . . . . . . . . . . . 120 5.4 Relation to the Zeno regime . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.5 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 vi 5.6 Appendix: Implementation of the quantum-jump error-correcting pro- cess via weak measurements and weak unitary operations . . . . . . . . 136 5.6.1 The single-qubit model . . . . . . . . . . . . . . . . . . . . . . . . 137 5.6.2 The bit-flip code . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.6.3 General single-error-correcting stabilizer codes. . . . . . . . . . . 143 Chapter 6: Correctable subsystems under continuous decoherence 151 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.2 Correctable subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.3 Completely positive linear maps . . . . . . . . . . . . . . . . . . . . . . 156 6.4 Markovian dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.5 Conditions on the system-environment Hamiltonian . . . . . . . . . . . 165 6.5.1 Conditions independent of the state of the environment . . . . . 166 6.5.2 Conditions depending on the initial state of the environment . . 171 6.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Chapter 7: Robustness of operator quantum error correction against initialization errors 175 7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7.2 Review of the noiseless-subsystem conditions on the Kraus operators . . 176 7.3 Fidelity between the encoded information in two states . . . . . . . . . . 179 7.3.1 Motivating the definition. . . . . . . . . . . . . . . . . . . . . . . 179 7.3.2 Properties of the measure . . . . . . . . . . . . . . . . . . . . . . 181 7.4 Robustness of OQEC with respect to initialization errors . . . . . . . . 187 7.5 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Chapter 8: A fault-tolerant scheme for holonomic quantum computa- tion 192 8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 8.2 Holonomic Quantum Computation . . . . . . . . . . . . . . . . . . . . . 194 8.3 Stabilizer codes and fault-tolerant computation . . . . . . . . . . . . . . 195 8.4 The scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 8.4.1 Encoded operations in the Clifford group . . . . . . . . . . . . . 199 8.4.2 Encoded operations outside of the Clifford group . . . . . . . . . 209 8.4.3 Using the “cat” state . . . . . . . . . . . . . . . . . . . . . . . . . 212 8.4.4 Fault tolerance of the scheme . . . . . . . . . . . . . . . . . . . . 213 8.5 Effects on the accuracy threshold for environment noise . . . . . . . . . 214 8.6 Fault-tolerant holonomic computation with low-weight Hamiltonians . . 217 8.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 8.8 Appendix: Calculating the holonomy for the Z gate . . . . . . . . . . . 224 8.8.1 Linear interpolation . . . . . . . . . . . . . . . . . . . . . . . . . 224 8.8.2 Unitary interpolation . . . . . . . . . . . . . . . . . . . . . . . . . 229 Chapter 9: Conclusion 232 Bibliography 237 vii List of Figures 1 Comparison of the exact solution at β =1 and β =10 for N =100. . . 87 2 Comparison of the exact solution at β =1 and β =10 for N =4. . . . 88 3 Comparison of the exact solution at β = 1 and β = 10 for N = 100 for randomly generated g n and Ω n . . . . . . . . . . . . . . . . . . . . . 89 4 Comparison of the exact solution at β = 1 and β = 10 for N = 4 for randomly generated g n and Ω n . . . . . . . . . . . . . . . . . . . . . . . 90 5 Comparison of the exact solution, NZ2, NZ3 and NZ4 at β = 1 and β = 10 for N = 100. The exact solution is the solid (blue) line, NZ2 is the dashed (green) line, NZ3 is the dot-dashed (red) line and NZ4 is the dotted (cyan) line. . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6 Comparison of the exact solution, NZ2, NZ3 and NZ4 at β = 1 and β = 10 for N = 4. The exact solution is the solid (blue) line, NZ2 is the dashed (green) line, NZ3 is the dot-dashed (red) line and NZ4 is the dotted (cyan) line. . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 7 Comparison of the exact solution, TCL2, TCL3 and TCL4 at β = 1 and β = 10 for N = 100. The exact solution is the solid (blue) line, TCL2 is the dashed (green) line, TCL3 is the dot-dashed (red) line and TCL4 is the dotted (cyan) line. Note that for β = 1, the curves nearly coincide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 8 Comparison of the exact solution, TCL2, TCL3 and TCL4 at β = 1 and β = 10 for N = 4. The exact solution is the solid (blue) line, TCL2 is the dashed (green) line, TCL3 is the dot-dashed (red) line and TCL4 is the dotted (cyan) line. Note that for β = 1, TCL3, TCL4 and the exact solution nearly coincide. . . . . . . . . . . . . . . 94 viii 9 Comparison of TCL2and the exact solution to demonstrate thevalid- ity of the TCL approximation for N = 4 and β = 1. The solid (blue) line denotes the exact solution and the dashed (green) line is TCL2. Notethat thetimeaxis hereisonalinearscale. TCL2breaksdownat αt≈0.9, whereitremainsflat, whiletheexact solutionhasarecurrence. 95 10 Comparison of the exact solution, NZ4, TCL4 and PM at β = 1 and β = 10 for N = 100. The exact solution is the solid (blue) line, PM is the dashed (green) line, NZ4 is the dot-dashed (red) line and TCL4 is the dotted (cyan) line. Note that for β = 1, TCL4, PM and the exact solution nearly coincide for short and medium times. Only PM captures the recurrences of the exact solution at long times. . . . . . . 96 11 Comparison of the exact solution, NZ4, TCL4 and PM at β = 1 and β = 10 for N = 4. The exact solution is the solid (blue) line, PM is the dashed (green) line, NZ4 is the dot-dashed (red) line and TCL4 is the dotted (cyan) line. Note that for β = 1, TCL4 and the exact solution nearly coincide for short and medium times. . . . . . . . . . . 97 12 Comparison of the exact solution, NZ4, TCL4 and PM atαt=0.1 for N = 100 for different β ∈ [0.01,10]. The exact solution is the solid (blue) line, PM is thedashed(green) line, NZ4is thedot-dashed (red) line and TCL4 is the dotted (cyan) line. . . . . . . . . . . . . . . . . . 98 13 Comparison of the exact solution, NZ4, TCL4 and PM at αt = 0.5 for N = 4 for different β ∈ [0.01,10]. The exact solution is the solid (blue) line, PM is thedashed(green) line, NZ4is thedot-dashed (red) line and TCL4 is the dotted (cyan) line. . . . . . . . . . . . . . . . . . 99 14 Comparison of the exact solution, NZ4, TCL4 and PM at β = 1 and β = 10 for N = 100 for random values of g n and Ω n . The exact solution is the solid (blue) line, PM is the dashed (green) line, NZ4 is the dot-dashed (red) line and TCL4 is the dotted (cyan) line. Note that for β = 1 and β = 10, TCL4, PM and the exact solution nearly coincide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 15 Comparison of the exact solution, NZ4, TCL4 and PM at β = 1 and β =10 forN =4 for random values ofg n and Ω n . The exact solution is the solid (blue) line, PM is the dashed (green) line, NZ4 is the dot- dashed (red) line and TCL4 is the dotted (cyan) line. Note that for β =1, TCL4, PM and theexact solution nearly coincide for shortand medium times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 ix 16 Comparison of the exact solution and the optimal coarse-graining ap- proximation for N = 50 and β = 1. The exact solution is the solid (blue)lineandthecoarse-grainingapproximationisthedashed(green) line. Note the linear scale time axis. . . . . . . . . . . . . . . . . . . . 102 17 Fidelity of the single-qubit code with continuous bit-flip errors and correction, as a function of dimensionless time γt, for three different values of the ratio R =κ/γ. . . . . . . . . . . . . . . . . . . . . . . . . 117 18 These are the allowed transitions between the different components of the system (272) and their rates, arising from both the decoher- ence(bit-flip)process(withrateγ)andthecontinuouserror-correction process (with rate κ). Online, the transitions due to decoherence are black, and the transitions due to error correction are red. . . . . . . . 123 19 Long-time behavior of three-qubit system with bit-flip noise and con- tinuous error correction. The ratio of correction rate to decoherence rate is R =κ/γ =100. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 20 Short-time behavior of three-qubit system with bit-flip noise and con- tinuous error correction. The ratio of correction rate to decoherence rate is R =κ/γ =100. . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 x Abstract This thesis examines seven topics in quantum information and the theory of open quantumsystems. Thefirstoneconcerns weak measurements andtheir universality as ameansofgeneratingquantummeasurements. Itisshownthateverygeneralizedmea- surementcanbedecomposedintoasequenceofweak measurementswhichallows usto think of measurements as resulting form continuous stochastic processes. The second topicconcernsanapplication ofthedecompositionintoweakmeasurementstothethe- ory of entanglement. Necessary and sufficient differential conditions for entanglement monotones are derived, and are used to find a new entanglement monotone for three- qubit states. The third topic examines the performance of different master equations for the description of non-Markovian dynamics. Thesystem studiedis a qubit coupled to a spin bath via the Ising interaction. The fourth topic studies continuous quantum error correction in the case of non-Markovian decoherence. It is shown that dueto the existence of aZenoregime innon-Markovian dynamics, theperformanceof continuous quantum error correction may exhibit a quadratic improvement if the time resolution of the error-correcting operations is sufficiently high. The fifth topic concerns condi- tions for correctability of subsystem codes in the case of continuous decoherence. The obtained conditions on the Lindbladian and the system-environment Hamiltonian can be thought of as generalizations of the previously known conditions for noiseless sub- systems to the case where the subsystem is time-dependent. The sixth topic examines the robustness of quantum error-correcting codes against initialization errors. It is xi shown that operator codes are robust against imperfect initialization without theneed for restriction of the standard error-correction conditions. For this purpose, a new measure of fidelity for encoded information is introduced and its properties are dis- cussed. The last topic concerns holonomic quantum computation and stabilizer codes. A fault-tolerant scheme for holonomic computations is presented, demonstrating the scalability of the holonomic method. The scheme opens the possibility for combining the benefits of error correction with the inherent resilience of the holonomic approach. xii Chapter 1: Introduction 1.1 Quantum information and open quantum systems The field of quantum information and quantum computation has grown rapidly during the last two decades [114]. It has been shown that quantum systems can be used for information processing tasks that cannot be accomplished by classical means. Examples include quantum algorithms that can outperform the best known classical algorithms, such as Shor’s factoring algorithm [144] or Grover’s search algorithm [69], quantum communication protocols which use entanglement for teleportation of quan- tum states [19] or superdense coding [23], or quantum cryptographic protocols which offer provably secure ways of confidential key distribution between distant parties [18]. This has triggered an immense amount of research, leading to advances in many areas of quantum physics. One area that has developed significantly as a result of the new growing field is that of open quantum systems. This development has been stimulated on one hand by the need to understand the full spectrum of operations that can be applied to a quantum state, as well as the information-processing tasks that can be accomplished withthem. Except forunitarytransformations, whichgenerally describethedynamics ofclosedsystems,thetoolsofquantuminformationscienceinvolvealsomeasurements, completely positive (CP) maps [114], and even non-CP operations [139]. These more general operations result from interactions of the system of interest with auxiliary systems, and thus require knowledge of the dynamics of open quantum systems. 1 At the same time, it has been imperative to understand and find means to over- come the effects of noise on quantum information. Quantum superpositions, which are crucial for the workings of most quantum information processing schemes, can be easily destroyed by external interactions. This process, known as decoherence, has presented a major obstacle to the construction of reliable quantum information devices. This has prompted studies on the mechanisms of information loss in open quantum systems and the invention of methods to overcome them, giving rise to one of the pillars of quantum information science—the theory of quantum error correction [142, 148, 22, 88, 55, 174, 104, 102, 89, 51, 83, 172, 93, 94, 24]. Quantumerrorcorrectionstudiestheinformation-preservingstructuresunderopen- system dynamics and the methods for encoding and processing of information using thesestructures. Amajorresultinthetheoryoferrorcorrectionstatesthatiftheerror rateperinformationunitisbelowacertainvalue,bytheuseoffault-toleranttechniques andconcatenation, anarbitrarilylargeinformationprocessingtaskcanbeimplemented reliably with a modest overhead of resources [143, 53, 88, 2, 85, 90, 68, 67, 130]. This result, known as the accuracy threshold theorem, is of fundamental significance for quantum information science. It shows that despite the unavoidable effects of noise, scalable quantum information processing is possible in principle. In this thesis, we examines topics from three intersecting areas in the theory open quantumsystemsandquantuminformation—thedeterministicdynamicsofopenquan- tum systems, quantum measurements, and quantum error correction. 1.1.1 Deterministic dynamics of open quantum systems All transformations in quantum mechanics, except for those that result from mea- surements, are usually thought of as arising from continuous evolution driven by a Hamiltonian that acts on the system of interest and possibly other systems. These transformations are therefore the result of the unitary evolution of a larger system 2 that contains the system in question. Alternative interpretations are also possible— for example some transformation can be thought of as resulting from measurements whose outcomes are discarded. This description, however, can also be understood as originating from unitary evolution of a system which includes the measurement apparatus and all systems on which the outcome has been imprinted. Including the environment in the description is generally difficult due to the large number of environment degrees of freedom. This is why it is useful to have a descrip- tion which involves only the effective evolution of the reduced density operator of the system. When the system and the environment are initially uncorrelated, the effective evolution of the density operator of the system can be described by a completely posi- tivetrace-preserving(CPTP)linearmap[92]. CPTPmapsarewidelyusedinquantum information science for describing noise processes and operations on quantum states [114]. They do not, however, describe the most general form of transformation of the state of an open system, since the initial state of the system and the environment can be correlated in a way which gives rise to non-CP transformations. Furthermore, the effective transformation by itself does not provide direct insights into the process that drivesthetransformation. Forthelatteroneneedsadescriptionintermsofagenerator oftheevolution, similar totheway theSchr¨ odingerequation describestheevolution of a closed system as generated by a Hamiltonian. The main difficulty in obtaining such a description for open systems is that the evolution of the reduced density matrix of the system is subject to non-trivial memory effects arising from the interaction with the environment [30]. In the limit where the memory of the environment is short-lived, the evolution of an open system can be described [30] by a time-local semi-group master equation in the Lindblad form [105]. Such an equation can be thought of as corresponding to a sequence of weak (infinitesimal) CPTP maps. When the memory of the environment cannot be ignored and the effective transformation (which is not necessarily CP) on 3 the initial state is reversible, the evolution can be described by a time-local master equation, known as the time-convolutionless (TCL) master equation [141, 140]. In contrast to the Lindblad equation, this equation does not describe completely positive evolution. Themost general continuous deterministicevolution of anopenquantumsystem is described by the Nakajima-Zwanzig (NZ) equation [111, 179]. This equation involves convolution in time. Both the TCL and NZ equations are quite complicated to obtain from first principles and are usually used for perturbative descriptions. Somewhere in betweentheLindbladequationandtheTCLorNZequationsarethephenomenological post-Markovian master equations such as the one proposed in Ref. [137]. Inthisthesis, wewillexaminethedeterministicevolution ofopenquantumsystems both from the point of view of the full evolution of the system and the environment and from the point of view of the reduced dynamics of the system. We will study the performance of different master equation for the description of the non-Markovian evolutionofaqubitcoupledtoaspinbath[96],compareMarkovianandnon-Markovian models in light of continuous quantum error correction [119], and study the conditions for preservation of encoded information under Markovian evolution of the system and general Hamiltonian evolution of the system and the environment [121]. 1.1.2 Quantum measurements Inadditiontodeterministic transformations, thestate ofan openquantumsystem can alsoundergostochastictransformations. Thesearetransformationsforwhichthestate may change in a number of different ways with non-unit probability. Since according to the postulates of quantum mechanics the only non-deterministic transformations are those that result from measurements [163, 107], stochastic transformations most generally result from measurements applied on the system and its environment. Just like deterministic transformations, stochastic transformations need not be completely 4 positive. If the system of interest is initially entangled with its environment and after some joint unitary evolution of the system and the environment a measurement is performed on the environment, the effective transformation on the system resulting from this measurement need not be CP. The class of completely positive stochastic operations are commonly referred to as generalized measurements [92]. Although this class includes standard projective measurements[163,107]aswellasotheroperationswhoseoutcomesrevealinformation aboutthestate,notalloperationsinthiscategoryrevealinformation. Someoperations simplyconsistofdeterministicoperationsappliedwithprobabilitiesthatdonotdepend on the state, i.e., they amount to trivial measurements. In recent years, a special type of generalized quantum measurements, the so called weak measurements [4, 5, 98, 126, 6], have become of significant interest. A mea- surement is called weak if all of its outcomes result in small (infinitesimal) changes to the state. Weak measurements have been studied both in the abstract, and as a means of understanding systems with continuous monitoring. In the latter case, we can think of the evolution as the limit of a sequence of weak measurements, which gives rise to continuous stochastic evolutions called quantum trajectories (see, e.g., [41, 48, 61, 64, 52, 65, 128]). Such evolutions have been used also as models of de- coherence (see, e.g., [32]). Weak measurements have found applications in feedback quantum control schemes [54] such as state preparation [74, 151, 152, 167, 46] or con- tinuous quantum error correction [7, 134]. In this thesis, we look at weak measurements as a means of generating quantum transformations[117]. Weshowthatanygeneralizedmeasurementcanbeimplemented as a sequence of weak measurements, which allows us to use the tools of differential calculus in studies concerning measurement-driven transformations. We apply this result to the theory of entanglement, deriving necessary and sufficient conditions for a function on quantum state to be an entanglement monotone [118]. We use these 5 conditions to find a new entanglement monotone for three-qubit pure states, a subject ofpreviouslyunsuccessfulinquiries[63]. Wealsodiscusstheuseofweakmeasurements for continuous quantum error correction. 1.1.3 Quantum error correction Whether deterministic or stochastic, the evolution of a system coupled to its envi- ronment is generally irreversible. This is because the environment, by definition, is outside of the experimenter’s control. As irreversible transformations involve loss of information, they could be detrimental to a quantum information scheme unless an error-correcting method is employed. A common form of error correction involves encoding the Hilbert space of a single information unit, say a qubit, in a subspace of the Hilbert space of a larger number of qubits [142, 148, 22, 88]. The encoding is such that if a single qubit in the code undergoes an error, the original state can be recovered by applying an appropriate operation. Clearly, there is a chance that more than one qubit undergoes an error, but according to the theory of fault tolerance [143, 53, 88, 2, 85, 90, 68, 67, 130] this problem can be dealt with by the use of fault-tolerant techniques and concatenation. Error correction encompasses a wide variety of methods, each suitable for different types of noise, different tasks, or using different resources. Examples include passive error-correction methods which protect against correlated errors, such as decoherence- free subspaces [55, 174, 104, 102] and subsystems [89, 51, 83, 172], the standard ac- tive methods [142, 148, 22, 88] which are suitable for fault-tolerant computation [67], entanglement assisted quantum codes [34, 35] useful in quantum communication, or linear quantum error-correction codes [139] that correct non-completely positive er- rors. Recently, a general formalism called operator quantum error correction (OQEC) [93, 94, 24] was introduced, which unified in a common framework all previously pro- posed error-correction methods. This formalism employs the most general encoding 6 of information—encoding in subsystems [87, 161]. OQEC was generalized to include entanglement-assisted error correction resulting in the most general quantum error- correction formalism presently known [73, 62]. In the standard formulation of error correction, noise and the error-correcting op- erations are usually represented by discrete transformations [93, 94, 24]. In practice, however, these transformations result from continuous processes. The more general situation where both the noise and the error-correcting processes are assumed to be continuous, is the subject of continuous quantum error correction [124, 135, 7, 134]. In the paradigm of continuous error correction, error-correcting operations are gen- erated by weak measurements, weak unitary operations or weak completely positive maps. This approach often leads to a better performance in the setting of continuous decoherence than that involving discrete operations. In this thesis, we discuss topics concerning both the discrete formalism and the continuous one. The topics we study include continuous quantum error correction for non-Markovian decoherence[119], conditions forcorrectability ofoperatorcodesunder continuous decoherence [121], the performance of OQEC under imperfect encoding [116], as well as fault-tolerant computation based on holonomic operations [120]. 1.2 Outline This work examines seven topics in the areas of deterministic open-quantum-system dynamics, quantum measurements, and quantum error correction. Some of the topics concern all of these three themes, while others concern only two or only one. As each of the main results has a significance of its own, each topics has been presented as a separate study in one of the following chapters. The topics are ordered in view of the background material they introduce and the logical relation between them. 7 Wefirststudythethemeofweakmeasurementsandtheirapplicationstothetheory of entanglement. In Chapter 2, we show that every generalized quantum measurement canbegeneratedasasequenceofweakmeasurements[117], whichallowsustothinkof measurements in quantum mechanics as generated by continuous stochastic processes. In the case of two-outcome measurements, the measurement procedure has the struc- tureofarandomwalkalongacurveinstatespace, withthemeasurementendingwhen one of the end points is reached. In the continuous limit, this procedure corresponds to a quantum feedback control scheme for which the type of measurement is continu- ously adjusted dependingon the measurement current. This result presents not only a practical prescriptionfortheimplementation ofanygeneralizedmeasurement, butalso reveals a rich mathematical structure, somewhat similar to that of Lie algebras, which allows us to study thetransformations caused by measurements by looking only at the properties of infinitesimal stochastic generators. The result suggests the possibility of constructing a unified theory of quantum measurement protocols. Chapter 3 presents an application of the weak-measurement decomposition to a study of entanglement. Thetheory of entanglement concerns the transformations that are possible to a state under local operations and classical communication (LOCC). The universality of weak measurements allows us to look at LOCC as the class of transformations generated by infinitesimal local operations. We show that a necessary and sufficient condition for a function of the state to be an entanglement monotone under local operations that do not involve information loss is that the function be a monotoneunderinfinitesimal local operations. We thenderivenecessary andsufficient differentialconditionsforafunctionofthestatetobeanentanglementmonotone[118]. We first derive two conditions for local operations without information loss, and then show that they can beextended to more general operations by addingthe requirement of convexity. We then demonstrate that a number of known entanglement monotones 8 satisfy these differential criteria. We use the differential conditions to construct a new polynomial entanglement monotone for three-qubit pure states. In Chapter 4, we extend the scope of our studies to include the deterministic dy- namics of open quantum systems. We study the analytically solvable Ising model of a single-qubit system coupled to a spin bath for a case for which the Markovian approx- imation of short bath correlation times cannot be applied [96]. The purpose of this study is to analyze and elucidate the performance of Markovian and non-Markovian master equations describing the dynamics of the system qubit, in comparison to the exact solution. We find that the time-convolutionless master equation performs par- ticularly well up to fourth order in the system-bath coupling constant, in comparison to the Nakajima-Zwanzig master equation. Markovian approaches fare poorly due to the infinite bath correlation time in this model. A recently proposed post-Markovian master equation performs comparably to the time-convolutionless master equation for a properly chosen memory kernel, and outperforms all the approximation methods considered here at long times. Our findings shed light on the applicability of master equations to thedescription of reducedsystem dynamics in the presence of spin baths. In Chapter 5, we investigate further the difference between Markovian and non- Markovian decoherence—this time, form the point of view of its implications for the performance of continuous quantum error correction. We study the performance of a quantum-jump error correction model in the case where each qubit in a codeword is subject to a general Hamiltonian interaction with an independent bath [119]. We first consider the scheme in the case of a trivial single-qubit code, which provides useful insights into the workings of continuous error correction and the difference between Markovian and non-Markovian decoherence. We then study the model of a bit-flip code with each qubit coupled to an independent bath qubit and subject to continuous correction, and find its solution. We show that for sufficiently large error-correction rates, the encoded state approximately follows an evolution of the type of a single 9 decohering qubit, but with an effectively decreased coupling constant. The factor by which the coupling constant is decreased scales quadratically with the error-correction rate. This is compared to the case of Markovian noise, where the decoherence rate is effectively decreased by a factor which scales only linearly with the rate of error correction. The quadratic enhancement depends on the existence of a Zeno regime in the Hamiltonian evolution which is absent in purely Markovian dynamics. We analyze the range of validity of this result and identify two relevant time scales. Finally, we extend the result to more general codes and argue that there the performance of continuous error correction will exhibit the same qualitative characteristics. In the appendix of Chapter 5, we discuss another application of weak measurements—we show how the quantum-jump error-correction scheme can be implemented using weak measurements and weak unitary operations. In Chapter 6, we study the conditions under which a quantum code is perfectly correctable during a time interval of continuous decoherence for the most general type of encoding—encodingin subsystems. We studythe case of Markovian decoherence as well as the general case of Hamiltonian evolution of the system and the environment, and derive necessary and sufficient conditions on the Lindbladian and the system- environment Hamiltonian [121], respectively. Our approach is based on a result ob- tained in Ref. [95] according to which a subsystem is correctable if and only if it is unitarily recoverable. The conditions we derive can be thought of as generalizations of the previously derived conditions for decoherence-free subsystems to the case where the subsystem is time-dependent. As a special case we consider conditions for uni- tary correctability. In the case of Hamiltonain evolution, the conditions for unitary correctability concern only the effect of the Hamiltonian on the system, whereas the conditions for general correctability concern the entire system-environment Hamilto- nian. We also derive conditions on the Hamiltonian which depend on the initial state 10 of the environment. We discuss possible implications of our results for approximate quantum error correction. Chapter 7 also concerns subsystem codes. Here we study the performance of oper- ator quantum error correction (OQEC) in the case of imperfect encoding [116]. In the OQEC,thenotionofcorrectability isdefinedundertheassumptionthatstatesareper- fectly initialized inside a particular subspace, a factor of which (a subsystem) contains the protected information. It was believed that in the case of imperfect initialization, OQEC codes would require more restrictive than the standard conditions if they are to protect encoded information from subsequent errors. In this chapter, we examine this requirement by looking at the errors on the encoded state. In order to quanti- tatively analyze the errors in an OQEC code, we introduce a measure of the fidelity between the encoded information in two states for the case of subsystem encoding. A major part of the chapter concerns the definition of the measure and the derivation of its properties. In contrast to what was previously believed, we obtain that more restrictive conditions are not necessary neither for DFSs nor for general OQEC codes. Thisisbecausetheeffective noisethatcan ariseinsidethecodeasaresultofimperfect initialization is such that it can only increase the fidelity of an imperfectly encoded state with a perfectly encoded one. In Chapter 8, we present a scheme for fault-tolerant holonomic computation on stabilizer codes [120]. In the holonomic approach, logical states are encoded in the degenerate eigenspace of a Hamiltonian and gates are implemented by adiabatically varyingtheHamiltonianalongloopsinparameterspace. Theresultisatransformation ofpurelygeometric origin, whichisrobustagainst varioustypesoferrorsinthecontrol parameters driving the evolution. In the proposed scheme, single-qubit operations on physical qubits are implemented by varying Hamiltonians that are elements of the stabilizer, or in the case of subsystem codes—elements of the gauge group. By construction, the geometric transformations in each eigenspace of the Hamiltonian are 11 transversalwhichensuresthaterrorsdonotpropagate. Weshowthatforcertaincodes, likethenine-qubitShorcodeoritssubsystemversions,itispossibletorealizeuniversal fault-tolerant computation using Hamiltonians of weight three. The scheme proves that holonomic quantum computation is a scalable method and opens the possibility for bringing together the benefits of error correction and the operational robustness of the holonomic approach. It also presents an alternative to the standard fault-tolerant methods based on dynamical transformations, which have been argued to be in a possible conflict with the assumption of Markovian decoherence that often underlies the derivation of threshold results. Chapter 9 summarizes the results and discusses problems for future research. 12 Chapter 2: Generating quantum measurements using weak measurements 2.1 Preliminaries In the original formulation of measurement in quantum mechanics, measurement out- comesareidentifiedwithasetoforthogonalprojectionoperators,whichcanbethought ofascorrespondingtotheeigenspacesofaHermitianoperator,orobservable[163,107]. Afterameasurement,thestateisprojectedintooneofthesubspaceswithaprobability given by the square of the amplitude of the state’s component in that subspace. In recent years a more general notion of measurement has become common: the generalized or positive-operator valued measurement (POVM) [92]. This formulation can include many phenomena not captured by projective measurements: detectors with non-unit efficiency, measurement outcomes that include additional randomness, measurementsthatgiveincompleteinformation,andmanyothers. POVMshavefound numerous applications in the rapidly-growing field of quantum information processing [114]. Some examples include protocols for unambiguous state discrimination [125] and optimal entanglement manipulation [113, 76]. Upon measurement, a system with density matrix ρ undergoes a random transfor- mation ρ→ρ j =M j ρM † j /p j , X j M † j M j =I, (1) 13 with probability p j = Tr(M j ρM † j ), where the index j labels the possible outcomes of the measurement. Eq. (1) is not the most general stochastic operation that can be applied to a state. For example, one can consider the transformation ρ→ρ j = X i M ij ρM † ij /p j , X i,j M † ij M ij =I, (2) where p j = Tr( P i M ij ρM † ij ) is the probability for the j th outcome (see Chapter 3). The letter can be thought of as resulting from a measurement of the type (1) with measurement operators M ij of which only the information about the indexj labelling the outcome is retained. Therefore, when we talk about generalized measurements, we will refer to the transformation (1). The transformation (1) is commonly comprehended as a spontaneous jump, unlike unitary transformations, for example, which are thought of as resulting from continu- ous unitary evolutions. Any unitary transformation can beimplemented as a sequence of weak (i.e., infinitesimal) unitary transformations. One may ask if a similar decom- position exists for generalized measurements. This would allow us to think of POVMs as resulting from continuous stochastic evolutions and possibly make use of the pow- erful tools of differential calculus in the study of the transformations that a system undergoes upon measurement. In this chapter we show that any generalized measurement can be implemented as a sequence of weak measurements and present an explicit form of the decomposition. The main result was first presented in Ref. [117]. We call a measurement weak if all outcomes result in very small changes to the state. (There are other definitions of weak measurements that include the possibility of large changes to the state with low 14 probability; we will not be considering measurements of this type.) Therefore, a weak measurement is one whose operators can be written as M j =q j (I +ε j ), (3) where q j ∈C, 0≤|q j |≤1, andε is an operator with small normkεk≪1. 2.2 Decomposing projective measurements It has been shown that any projective measurement can be done as a sequence of weak measurements; and by using an additional ancilla system and a joint unitary transformation, it is possible to do any generalized measurement using weak measure- ments[21]. Thisprocedure,however, doesnotdecomposetheoperationontheoriginal systemintoweakoperations, sinceitusesoperationsactingonalargerHilbertspace— that of the system plus the ancilla. If we wish to study the behavior of a function—for instance, an entanglement monotone—defined on a space of a particular dimension, it complicates matters to add and remove ancillas. We will show that an ancilla is not needed, and give an explicit construction of the weak measurement operators for any generalized measurement that we wish to decompose. It is easy to show that a measurement with any number of outcomes can be per- formed as a sequence of measurements with two outcomes. Therefore, for simplicity, we will restrict our considerations to two-outcome measurements. To give the idea of theconstruction, wefirstshow how every projective measurement can beimplemented as a sequence of weak generalized measurements. In this case the measurement oper- ators P 1 and P 2 are orthogonal projectors whose sumP 1 +P 2 =I is the identity. We introduce the operators P(x) = r 1−tanh(x) 2 P 1 + r 1+tanh(x) 2 P 2 , x∈R. (4) 15 NotethatP 2 (x)+P 2 (−x) =I andthereforeP(x)andP(−x)describeameasurement. Ifx=ǫ, where|ǫ|≪ 1, the measurement is weak. Consider the effect of the operators P(x)onapurestate|ψi. Thestatecanbewrittenas|ψi=P 1 |ψi+P 2 |ψi = √ p 1 |ψ 1 i+ √ p 2 |ψ 2 i, where|ψ 1,2 i =P 1,2 |ψi/ √ p 1,2 are the two possible outcomes of the projective measurement andp 1,2 =hψ|P 1,2 |ψi arethe correspondingprobabilities. Ifx is positive (negative), the operator P(x) increases (decreases) the ratio √ p 2 / √ p 1 of the|ψ 2 i and |ψ 1 i components of the state. By applying the same operator P(ǫ) many times in a row for some fixed ǫ, the ratio can be made arbitrarily large or small depending on the sign of ǫ, and hence the state can be transformed arbitrarily close to|ψ 1 i or|ψ 2 i. The ratio of the p 1 and p 2 is the only parameter needed to describe the state, since p 1 +p 2 =1. Also note that P(−x)P(x) = (1−tanh 2 (x)) 1/2 I/2 is proportional to the identity. If we apply the same measurement P(±ǫ) twice and two opposite outcomes occur, the system returns to its previous state. Thus we see that the transformation of the state under many repetitions of the measurement P(±ǫ) follows a random walk along a curve|ψ(x)i in state space. The position on this curve can be parameterized by x = ln p p 1 /p 2 . Then |ψ(x)i can be written as p p 1 (x)|ψ 1 i+ p p 2 (x)|ψ 2 i, where p 1,2 (x) =(1/2)[1±tanh(x)]. The measurement given by the operators P(±ǫ) changes x by x → x±ǫ, with probabilities p ± (x) = (1±tanh(ǫ)(p 1 (x)−p 2 (x)))/2. We continue this random walk until|x|≥X,forsomeX whichissufficientlylargethat|ψ(X)i≈|ψ 1 iand|ψ(−X)i≈ |ψ 2 i to whatever precision we desire. What are the respective probabilities of these two outcomes? 16 Define p(x) to be the probability that the walk will end at X (rather than −X) given that it began at x. This must satisfy p(x) = p + (x)p(x +ǫ) +p − (x)p(x−ǫ). Substituting our expressions for the probabilities, this becomes p(x) =(p(x+ǫ)+p(x−ǫ))/2+tanh(ǫ)tanh(x)(p(x+ǫ)−p(x−ǫ))/2. (5) If we go to the infinitesimal limit ǫ → dx, this becomes a continuous differential equation d 2 p dx 2 +2tanh(x) dp dx =0, (6) with boundary conditions p(X) = 1, p(−X) = 0. The solution to this equation is p(x) = (1/2)[1+tanh(x)/tanh(X)]. In the limit where X is large, tanh(X) → 1, so p(x) =p 1 (x). Theprobabilitiesoftheoutcomesforthesequenceofweakmeasurements are exactly the same as those for a single projective measurement. Note that this is also true for a walk with a step size that is not infinitesimal, since the solution p(x) satisfies (5) for an arbitrarily large ǫ. Alternatively, instead of looking at the state of the system during the process, we could look at an operator that effectively describes the system’s transformation to the current state. This has the advantage that it is state-independent, and will lead the way to decompositions of generalized measurements; it also becomes obvious that the procedure works for mixed states, too. We think of the measurement process as a random walk along a curve P(x) in operator space, given by Eq. (4), which satisfies P(0) = I/ √ 2, lim x→−∞ P(x) = P 1 , lim x→∞ P(x) = P 2 . It can be verified that P(x)P(y) ∝ P(x +y), where the constant of proportionality is (cosh(x+y)/2cosh(x)cosh(y)) 1/2 . Due to normalization of the state, operators which differ by an overall factor are equivalent in their effects on the state. Thus, the random walk driven by weak measurement operators P(±ǫ) has a step size|ǫ|. 17 2.3 Decomposing generalized measurements Next we consider measurements where the measurement operators M 1 and M 2 are positivebutnotprojectors. Weusethewellknownfactthatageneralizedmeasurement can be implemented as joint unitary operation on the system and an ancilla, followed by a projective measurement on the ancilla [114]. (One can think of this as an indirect measurement; one lets the system interact with the ancilla, and then measures the ancilla.) Later we will show that the ancilla is not needed. We consider two-outcome measurements andtwo-level ancillas. Inthis caseM 1 andM 2 commute, andhencecan be simultaneously diagonalized. Let the system and ancilla initially be in a state ρ⊗|0ih0|. Consider the unitary operation U(0) =M 1 ⊗Z +M 2 ⊗X, (7) where X =σ x and Z =σ z are Pauli matrices acting on the ancilla bit. By applying U(0) to the extended system we transform it to: U(0)(ρ⊗|0ih0|)U † (0) = M 1 ρM 1 ⊗|0ih0|+M 1 ρM 2 ⊗|0ih1| + M 2 ρM 1 ⊗|1ih0|+M 2 ρM 2 ⊗|1ih1|. (8) Then a projective measurement on the ancilla in the computational basis would yield oneof thepossiblegeneralized measurement outcomes forthesystem. Wecanperform the projective measurement on the ancilla as a sequence of weak measurements by the procedure we described earlier. We will then prove that for this process, there exists a correspondingsequenceof generalized measurements withthesameeffect acting solely on the system. To prove this, we first show that at any stage of the measurement process,thestateoftheextendedsystemcanbetransformedintotheformρ(x)⊗|0ih0| by a unitary operation which does not depend on the state. 18 The net effect of the joint unitary operation U(0), followed by the effective mea- surement operator on the ancilla, can be written in a block form in the computational basis of the ancilla: ¯ M(x) ≡ (I⊗P(x))U(0) = q 1−tanh(x) 2 M 1 q 1−tanh(x) 2 M 2 q 1+tanh(x) 2 M 2 − q 1+tanh(x) 2 M 1 . (9) Ifthecurrentstate ¯ M(x)(ρ⊗|0ih0|) ¯ M † canbetransformedtoρ(x)⊗|0ih0|byaunitary operatorU(x)which isindependentofρ, thenthelower left block ofU(x) ¯ M(x) should vanish. We look for such a unitary operator in block form, with each block being Hermitian and diagonal in the same basis as M 1 andM 2 . One solution is: U(x) = A(x) B(x) B(x) −A(x) , (10) where A(x) = p 1−tanh(x)M 1 (I +tanh(x)(M 2 2 −M 2 1 )) − 1 2 , (11) B(x)= p 1+tanh(x)M 2 (I +tanh(x)(M 2 2 −M 2 1 )) − 1 2 . (12) (Since M 2 1 +M 2 2 = I, the operator (I +tanh(x)(M 2 2 −M 2 1 )) − 1 2 always exists.) Note thatU(x) is Hermitian, soU(x) =U † (x) is its own inverse, and atx=0 it reduces to the operator (7). After every measurement on the ancilla, dependingon the value ofx, we apply the operation U(x). Then, before the next measurement, we apply its inverse U † (x) = U(x). By doing this, we can think of the procedure as a sequence of generalized measurements on the extended system that transform it between states of the form ρ(x)⊗|0ih0| (a generalized measurement preceded by aunitary operation andfollowed byaunitaryoperationdependentontheoutcomeisagainageneralized measurement). 19 The measurement operators are now ˜ M(x,±ǫ)≡U(x±ǫ)(I⊗P(±ǫ))U(x), and have the form ˜ M(x,±ǫ) = M(x,±ǫ) N(x,±ǫ) 0 O(x,±ǫ) . (13) HereM,N,O areoperators acting on thesystem. Uponmeasurement, the state of the extended system is transformed ρ(x)⊗|0ih0|→ M(x,±ǫ)ρ(x)M † (x,±ǫ) p(x,±ǫ) ⊗|0ih0|, (14) with probability p(x,±ǫ)=Tr n M(x,±ǫ)ρ(x)M † (x,±ǫ) o . (15) By imposing ˜ M † (x,ǫ) ˜ M(x,ǫ)+ ˜ M(x,−ǫ) † ˜ M(x,−ǫ) =I, we obtain that M † (x,ǫ)M(x,ǫ)+M † (x,−ǫ)M(x,−ǫ) =I, (16) where the operators in the last equation acts on the system space alone. Therefore, the same transformations that the system undergoes during this procedure can be achieved by the measurements M(x,±ǫ) acting solely on the system. Depending on the current value ofx, we perform the measurement M(x,±ǫ). Due to the one-to-one correspondence with the random walk for the projective measurement on the ancilla, this procedure also follows a random walk with a step size |ǫ|. It is easy to see that if the measurements on the ancilla are weak, the corresponding measurements on the system are also weak. Therefore we have shown that every measurement with positive operatorsM 1 andM 2 , can beimplemented as a sequence of weak measurements. This 20 is the main result of this chapter. From the construction above, one can find the explicit form of the weak measurement operators: M(x,ǫ) = r 1−tanh(ǫ) 2 A(x)A(x+ǫ)+ r 1+tanh(ǫ) 2 B(x)B(x+ǫ). (17) Theseexpressionscan besimplifiedfurther. Thecurrentstate ofthesystem at any point during the procedure can be written as M(x)ρM(x)/Tr(M 2 (x)ρ), (18) where M(x) = r I +tanh(x)(M 2 2 −M 2 1 ) 2 , x∈R. (19) The weak measurement operators can be written as M(x,±ǫ) = s C ± I +tanh(x±ǫ)(M 2 2 −M 2 1 ) I +tanh(x)(M 2 2 −M 2 1 ) , (20) where the weights C ± are chosen to ensure that these operators form a generalized measurement: C ± =(1±tanh(ǫ)tanh(x))/2. (21) Note that this procedure works even if the step of the random walk is not small, sinceP(x)P(y)∝P(x+y) for arbitrary values ofx andy. So it is not surprisingthat the effective operator which gives the state of the system at the point x is M(x) ≡ M(0,x). In the limit when ǫ → 0, the evolution under the described procedure can be described by a continuous stochastic equation. We can introduce a time stepδt and a rate γ =ǫ 2 /δt. (22) 21 Then we can define a mean-zero Wiener process δW as follows: δW =(δx−M[δx] / √ γ), (23) where M[δx] is the mean of δx, M[δx] =ǫ(p + (x)−p − (x)). (24) The probabilities p ± (x) can be written in the form p ± (x) = 1 2 (1±2hQ(x)iǫ), (25) wherehQ(x)i denotes the expectation value of the operator Q(x) = 1 2 (M 2 2 −M 2 1 )+tanh(x)I I +tanh(x)(M 2 2 −M 2 1 ) . (26) NotethatM[(δW) 2 ]=δt+O(δt 2 ). Expandingthechangeofastate|ψiuponthemea- surement M(x,±ǫ) up to second order inδW and taking the limit δW →0 averaging over many steps, we obtain the following coupled stochastic differential equations: |dψi =− γ 2 (Q(x)−hQ(x)i) 2 |ψidt+ √ γ(Q(x)−hQ(x)i)|ψidW, (27) dx=2γhQ(x)idt+ √ γdW. (28) This process corresponds to a continuous measurement of an observable Q which is continuously changed depending on the measurement current x. In other words, it is a feedback-control scheme where depending on the measurement current, the type of measurement is continuously adjusted. 22 Finally, consider the most general type of two-outcome generalized measurement, with the only restriction being M † 1 M 1 +M † 2 M 2 = I. By polar decomposition the measurement operators can be written M 1,2 =V 1,2 q M † 1,2 M 1,2 , (29) whereV 1,2 areappropriateunitaryoperators. Onecan thinkof theseunitariesascaus- ing an additional disturbance to the state of the system, in addition to the reduction due to the measurement. The operators (M † 1,2 M 1,2 ) 1/2 are positive, and they form a measurement. We could then measure M 1 and M 2 by first measuring these positive operators by a sequence of weak measurements, and then performing either V 1 or V 2 , depending on the outcome. However, wecanalsodecomposethismeasurementdirectlyintoasequenceofweak measurements. Let the weak measurement operators for (M † 1,2 M 1,2 ) 1/2 be M p (x,±ǫ). LetV(x)beanycontinuousunitaryoperatorfunctionsatisfyingV(0) =I andV(±x)→ V 1,2 as x→∞. We then define M(x,y)≡V(x+y)M p (x,y)V † (x). (30) By construction M(x,±y) are measurement operators. Since V(x) is continuous, if y = ǫ, where ǫ ≪ 1, the measurements are weak. The measurement procedure is analogous to the previous cases and follows a random walk along the curveM(0,x) = V(x)M p (0,x). In summary, we have shown that for every two-outcome measurement described by operators M 1 and M 2 acting on a Hilbert space of dimension d, there exists a 23 continuous two-parameter family of operators M(x,y) over the same Hilbert space with the following properties: M(x,0) =I/ √ 2, (31) M(0,x)→M 1 as x→−∞, (32) M(0,x)→M 2 as x→+∞, (33) M(x+y,z)M(x,y)∝M(x,z +y), (34) M † (x,y)M(x,y)+M † (x,−y)M(x,−y) =I. (35) We have presented an explicit solution for M(x,y) in terms of M 1 and M 2 . The measurement is implemented as a random walk on the curve M(0,x) by consecutive application of the measurements M(x,±ǫ), which depend on the current value of the parameterx. Inthecasewhere|ǫ|≪1,themeasurementsdrivingtherandomwalkare weak. Since any measurement can be decomposed into two-outcome measurements, weak measurements are universal. 2.4 Measurements with multiple outcomes Even though two-outcome measurements can be used to construct any multi-outcome measurement,itisinterestingwhetheradirectdecompositionsimilartotheonewepre- sented can beobtained formeasurements with multipleoutcomes as well. InRef. [157] it was shown that such a decomposition exists. For a measurement withn positive op- eratorsL j ,j =1,...,n, n P j=1 L 2 j =I,theeffectivemeasurementoperatorM(x)describing the state during the procedure is given by [157] M(s)= p f(s) v u u t ( n X j=1 s j L 2 j ), (36) 24 where f(s)=1+n n X j=1 s j (1−s j ). (37) Here the parameter s is chosen such that n P j=1 s j = 1, s ∈ [0,1], i.e., it describes a simplex. The system of stochastic equations describing the process in this case can be written as |dψi =− γ 8 g jk (s)(Q j (s)−hQ j (s)i)(Q k (s)−hQ k (s)i)|ψidt+ 1 2 √ γ(Q i (s)−hQ i (s)i)|ψia i α (s)dW α , (38) ds=γg ij (s)hQ j (s)idt+ √ γa i α (s)dW α , (39) where Q i (s) = L 2 i s m L 2 m , (40) g ij (s) = n X α,β=1 s i (δ i α −s α )(δ α β − 1 n )s j (δ j β −s β ), (41) a(s) is the square root of g(s), g ij (s) = n X k=1 a i k (s)a j k (s), (42) and we have assumed Einstein’s summation convention. The decomposition can be easily generalized to the case of non-positive measure- ment operators in a way similar to the one we described for the two-outcome case—by inserting suitable weak unitaries between the weak measurements. 2.5 Summary and outlook The result presented in this chapter may have important implications for quantum control and the theory of quantum measurements in general. It provides a practical 25 prescription for the implementation of any generalized measurement using weak mea- surementswhichmaybeusefulinexperimentswherestrongmeasurementsaredifficult to implement. The decomposition might be experimentally feasible for some quantum optical or atomic systems. The result also reveals an interesting mathematical structure, somewhat similar to that of Lie algebras, which allows us to think of measurements as generated by infinitesimalstochasticgenerators. Oneapplicationofthisispresentedinthefollowing chapter, wherewederivenecessary andsufficientconditions forafunctiononquantum statestobeanentanglement monotone. Anentanglement monotone[159]isafunction which does not increasing on average under local operations. For pure states the operations are unitaries and generalized measurements. Since all unitaries can be broken into a series of infinitesimal steps and all measurements can be decomposed into weak measurements, it suffices to look at the behavior of a prospective monotone under small changes in the state. Thus we can use this result to derive differential conditions on the function. These observations suggest that it may be possible to find a unified description of quantum operations where every quantum operations can be continuously generated. Clearly, measurements do not form a group since they do not have inverse elements, but it may be possible to describe them in terms of a semi-group. The problem with using measurements as the elements of the semigroup is that a strong measurement is not equal to a composition of weak measurements, since the sequence of weak mea- surements that builds up a particular strong measurement is not pre-determined—the measurements depend on a stochastic parameter. It may be possible, however, to use more general objects—measurement protocols—which describe measurements applied conditioned on a parameter in some underlying manifold. If such a manifold exists for the most general possible notion of a protocol, the basic objects could be describable 26 by stochastic matrices on this manifold. Such a possibility is appealing since stochas- tic processes are well understood and this may have important implications for the study of quantum control protocols. In addition, such a description could be useful for describing general open-system dynamics. These questions are left open for future investigation. 27 Chapter 3: Applications of the decomposition into weak measurements to the theory of entanglement 3.1 Preliminaries In this chapter we apply the result on the universality of weak measurements to the theoryofentanglement. Thetheoryofentanglement concernsthetransformationsthat are possible to a state under local operations with classical communication (LOCC). The paradigmatic experiment is a quantum system comprising several subsystems, each in a separate laboratory under control of a different experimenter: Alice, Bob, Cara, etc. Each experimenter can perform any physically allowed operation on his or her subsystem—unitarytransformations, generalized measurements, indeed any trace- preserving completely positive operation–and communicate their results to each other without restriction. Theyarenot, however, allowed tobringtheir subsystemstogether and manipulate them jointly. An LOCC protocol consists of any number of local operations, interspersed with any amount of classical communication; the choice of operations at later times may depend on the outcomes of measurements at any earlier time. The results of Bennett et al. [17, 20, 22] and Nielsen [113], among many others [158, 76, 72, 77, 160], have given us a nearly complete theory of entanglement for bipartitesystemsinpurestates. Unfortunately,greatdifficultieshavebeenencountered in trying to extend these results both to mixed states and to states with more than 28 two subsystems (multipartite systems). The reasons for this are many; but one reason is that the set LOCC is complicated and difficult to describe mathematically [21]. One mathematical tool which has proven very useful is that of the entanglement monotone: a function of the state which is invariant under local unitary transforma- tions and always decreases (or increases) on average after any local operation. These functions were described by Vidal [159], and large classes of them have been enumer- ated since then. We will consider those protocols in LOCC that preserve pure states as the set of operations generated by infinitesimal local operations: operations which can be performed locally and which leave the state little changed including infinitesimal local unitaries and weak generalized measurements. In Bennett et al. [21] it was shown that infinitesimal local operations can beused to performany local operation with the additionaluseoflocalancillarysystems–extrasystemsresidinginthelocallaboratories, whichcanbecoupledtothesubsystemsforatimeandlaterdiscarded. Aswesawinthe previoussection, anylocal generalized measurementcanbeimplementedasasequence of weak measurements without the use of ancillas. This implies that a necessary and sufficient condition for a function of the state to be a monotone underlocal operations that preserve pure states is the function to be a monotone under infinitesimal local operations. In this chapter we derive differential conditions for a function of the state to be an entanglement monotone by considering the change of the function on average under infinitesimal local operations up to the lowest order in the infinitesimal parameter. We thus obtain conditions that involve at most second derivatives of the function. We then prove that these conditions are both necessary and sufficient. We show that the conditions are satisfied by a number of known entanglement monotones and we use them to construct a new polynomial entanglement monotone for three-qubit pure states. 29 WehopethatthisapproachwillprovideanewwindowwithwhichtostudyLOCC, andperhapsavoid someofthedifficultiesinthetheoryofmultipartiteandmixed-state entanglement. Bylookingonlyatthedifferentialbehaviorofentanglement monotones, we avoid concerns about the global structure of LOCC or the class of separable oper- ations. In Section 3.2, we define the basic concepts of this chapter: LOCC operations, entanglement monotones, and infinitesimal operations. In Section 3.3, we show how all local operations that preserve pure states can be generated by a sequence of in- finitesimal local operations. In Section 3.4, we derive differential conditions for a function of the state to be an entanglement monotone. There are two such conditions for pure-state entanglement monotones: the first guarantees invariance under local unitary transformations (LU invariance), and involves only the first derivatives of the function, whilethe second guarantees monotonicity underlocal measurements, and in- volves second derivatives. For mixed-state entanglement monotones we add a further condition, convexity, which ensures that a function remains monotonic under opera- tions that lose information (and can therefore transform pure states to mixed states). In Section 3.5, we look at some known monotones–the norm of the state, the local purity, and the entropy of entanglement–and show that they obey the differential cri- teria. In Section 3.6, we use the differential conditions to construct a new polynomial entanglement monotone for three-qubit pure states which depends on the invariant identified by Kempe [82]. In Section 3.7 we conclude. In the Appendix (Section 3.8), we show that higher derivatives of the function are not needed to prove monotonicity. 30 3.2 Basic definitions 3.2.1 LOCC An operation (or protocol) in LOCC consists of a sequence of local operations with classical communication between them. Initially, we will consider only those local operations that preserve pure states: unitaries, in which the state is transformed ρ→UρU † , U † U =UU † =I, (43) and generalized measurements, in which the state randomly changes as in Eq. (1), ρ→ρ j =M j ρM † j /p j , X j M † j M j =I, with probability p j = Tr n M † j M j ρ o , where the index j labels the possible outcomes of the measurement. Note that we can think of a unitary as being a special case of a generalized measurement with only one possible outcome. One can think of this class of operations as being limited to those which do not discard information. Later, we will relax this assumption to consider general operations, which can take pure states to mixed states. Such operations do involve loss of information. Examples include performing a measurement without retaining the result, performing an unknown uni- tary chosen at random, or entangling the system with an ancilla which is subsequently discarded. TherequirementthatanoperationbelocalmeansthattheoperatorsU orM j must haveatensor-productstructureU ≡U⊗I,M j ≡M j ⊗I,wheretheyactastheidentity onall exceptoneofthesubsystems. Theability touseclassical communication implies that the choice of later local operations can depend arbitrarily on the outcomes of all earlier measurements. One can think of an LOCC operation as consisting of a series of “rounds.” In each round, a single local operation is performed by one of the local 31 parties; if it is a measurement, the outcome is communicated to all parties, who then agree on the next local operation. 3.2.2 Entanglement monotones For the purposes of this study, we define an entanglement monotone to be a real- valued function of the state with the following properties: if we start with the system in a state ρ and perform a local operation which leaves the system in one of the states ρ 1 ,··· ,ρ n withprobabilitiesp 1 ,...,p n ,thenthevalueofthefunctionmustnotincrease on average: f(ρ)≥ X j p j f(ρ j ). (44a) Furthermore,wecanstartwithastateselectedrandomlyfromanensemble{ρ k ,p k }. If we dismiss the information about which particular state we are given (which can be done locally), the function of the resultant state must not exceed the average of the function we would have if we keep this information: X k p k f(ρ k )≥f X k p k ρ k ! . (44b) Some functions may obey a stronger form of monotonicity, in which the function cannot increase for any outcome: f(ρ)≥f(ρ j ), ∀j, (45) but this is not the most common situation. Some monotones may be defined only for pure states, or may only be monotonic for pure states. In the latter case, monotonic- ity is defined as non-increase on average under local operations that do not involve information loss. 32 3.2.3 Infinitesimal operations We call an operation infinitesimal if all outcomes result in only very small changes to the state. That is, if after an operation the system can be left in states ρ 1 ,··· ,ρ n , we must have ||ρ−ρ j ||≪1, ∀j. (46) For a unitary, this means that U =exp(iε)≈I +iε, (47) where ε is a Hermitian operator with small norm,||ε||≪ 1, ε =ε † . For a generalized measurement, every measurement operator M j can be written as in Eq. (3), M j =q j (I +ε j ), where 0≤q j ≤1 andε j is an operator with small norm||ε j ||≪1. 3.3 Local operations from infinitesimal local operations In this section we show how any local operation that preserves pure states can be performedasasequenceofinfinitesimallocaloperations. Theoperationsthatpreserve pure states are unitary transformations and generalized measurements. 3.3.1 Unitary transformations Every local unitary operator has the representation U =e iH , (48) 33 where H is a local hermitian operator. We can write U = lim n→∞ (I +iH/n) n , (49) and define ε=H/n (50) for a suitably large value of n. Thus, in the limit n→∞, any local unitary operation canbethoughtofasaninfinitesequenceofinfinitesimallocalunitaryoperationsdriven by operators of the form U ε ≈I +iε, (51) where ε is a small (kεk≪1) local hermitian operator. 3.3.2 Generalized measurements As was shown in Chapter 2, any measurement can begenerated by a sequence of weak measurements. Since a measurement with any number of outcomes can be imple- mented as a sequenceof two-outcome measurements, it suffices to consider generalized measurements with two outcomes. The form of the weak operators needed to generate any measurement (Eq. (20)) is M(x,±ǫ) = s C ± I +tanh(x±ǫ)(M 2 2 −M 2 1 ) I +tanh(x)(M 2 2 −M 2 1 ) , where C ± =(1±tanh(ǫ)tanh(x))/2. From these expressions it is easy to see that if |ǫ|≪ 1, we have M(x,ǫ) = p 1/2(I + O(ǫ)), i.e., the coefficients q j in Eq. (3) areq 1 =q 2 = 1 √ 2 . Furthermore, if the original measurement is local, the weak measurements are also local. 34 Clearly, the fact that infinitesimal local operations are part of the set of LO means that an entanglement monotone must be a monotone under infinitesimal local oper- ations. The discussion in this section implies that if a function is a monotone under infinitesimal local unitaries and generalized measurements, it is a monotone under all local unitaries and generalized measurements (the operations that do not involve in- formation loss and preserve pure states). Based on this result, in the next section we derive necessary and sufficient conditions for a function to be an entanglement monotone. 3.4 Differential conditions for entanglement monotones Let us now consider the change in the state under an infinitesimal local operation. Without loss of generality, we assume that the operation is performed on Alice’s sub- system. In this case, it is convenient to write the density matrix of the system as ρ = X i,j,l,m ρ ijlm |i A ihl A |⊗|j BC... ihm BC... |, (52) where the set{|i A i} and the set{|j BC... i} are arbitrary orthonormal bases for subsys- tem A and the rest of the system, respectively. Any function of the state f(ρ) can be thought of as a function of the coefficients in the above decomposition: f(ρ) =f(ρ ijlm ). (53) 35 3.4.1 Local unitary invariance Unitary operations are invertible, and therefore the monotonicity condition reduces to an invariance condition for LU transformations. Under local unitary operations on subsystemA the components of ρ transform as follows: ρ ijlm → X k,p U ik ρ kjpm U ∗ lp , (54) where U ik are the components of the local unitary operator in the basis {|i A i}. We consider infinitesimal local unitary operations: U lk = e iε lk , (55) where ε is a local hermitian operator acting on subsystem A, and kεk≪1. (56) Up to first order in ε the coefficients ρ ijlm transform as ρ ijlm →ρ ijlm +i[ε,ρ] ijlm . (57) Requiring LU-invariance of f(ρ), we obtain that the function must satisfy X i,j,l,m ∂f ∂ρ ijlm [ε,ρ] ijlm =0. (58) Analogous equations must be satisfied for arbitrary hermitian operators ε acting on the other parties’ subsystems. In a more compact form, the condition can be written as Tr ∂f ∂ρ [ε,ρ] =0, (59) 36 where ε is an arbitrary local hermitian operator. 3.4.2 Non-increase under infinitesimal local measurements As mentioned earlier, a measurement with any number of outcomes can be imple- mentedasasequenceofmeasurementswithtwooutcomes, andageneral measurement can be done as a measurement with positive operators, followed by a unitary condi- tioned on the outcome; therefore, it suffices to impose the monotonicity condition for two-outcome measurements with positive measurement operators. Consider local measurements on subsystem A with two measurement outcomes, given by operators M 2 1 +M 2 2 =I. Without loss of generality, we assume M 1 = p (I +ε)/2, M 2 = p (I−ε)/2, (60) whereε is again a small local hermitian operator acting onA (in the previous section we saw that any two-outcome measurement with positive operators can be generated by weak measurements of this type). Upon measurement, the state undergoes one of two possible transformations ρ → M 1,2 ρM 1,2 p 1,2 , (61) with probabilities p 1,2 =Tr M 1,2 2 ρ . Sinceε is small, we can expand M 1 = 1 √ 2 (I +ε/2−ε 2 /8−···), (62) M 2 = 1 √ 2 (I−ε/2−ε 2 /8−···). (63) 37 The condition for non-increase on average of the function f under infinitesimal local measurements is p 1 f(M 1 ρM 1 /p 1 )+p 2 f(M 2 ρM 2 /p 2 )≤f(ρ). (64) Expanding (64) in powers of ε up to second order, we obtain 1 4 Tr ∂f ∂ρ [[ε,ρ],ε] +Tr ( ∂ 2 f ∂ρ ⊗2 Tr(ερ)ρ− 1 2 {ε,ρ} ⊗2 ) ≤0, (65) where {ε,ρ} is the anti-commutator of ε and ρ. The inequality must be satisfied for an arbitrary local hermitian operator ε. So long as (65) is satisfied by a strict inequality, it is obvious that we need not consider higher-order terms in ε. But what about the case when the condition is sat- isfied by equality? In the appendix we will show that even in the case of equality, (65) is still the necessary and sufficient condition for monotonicity under local generalized measurements. TherewealsoprovethesufficiencyoftheLU-invariancecondition(59). This allows us to state the following Theorem 1: A twice-differentiable functionf(ρ) of the density matrix is a monotone under local unitary operations and generalized measurements, if and only if it satisfies (59) and (65). We point out that from the condition of LU invariance applied up to second-order in ε, one obtains Tr ∂f ∂ρ [[ε,ρ],ε] =−Tr ∂ 2 f ∂ρ ⊗2 (i[ε,ρ]) ⊗2 . (66) Therefore, in the case when both Eq. (59) and Eq. (65) are satisfied, condition (65) can be written equivalntly in the form Tr ( ∂ 2 f ∂ρ ⊗2 " Tr(ερ)ρ− 1 2 {ε,ρ} ⊗2 − i 2 [ε,ρ] ⊗2 #) ≤0. (67) 38 Unitary operations and generalized measurements are the operations that preserve pure states. Other operations (which involve loss of information), such as positive maps, would in general cause pure states to evolve into mixed states. A measure of pure-state entanglement need not be defined over the entire set of density matrices, butonlyover purestates. Thusameasureof pure-stateentanglement, whenexpressed as a function of the density matrix, may have a significantly simpler form than its generalizations tomixedstates. Forexample, theentropyofentanglement forbipartite pure states can be written in the well-known form S A (ρ) = −Tr(ρ A logρ A ), where ρ A is the reduced density matrix of one of the parties’ subsystems. When directly extended over mixed states, this function is not well justified, since S A (ρ) may have a differentvaluefromS B (ρ). Moreover,S A (ρ)byitselfisnotamixed-stateentanglement monotone, since it may increase under local positive maps on subsystem A (these properties of the entropy of entanglement will be discussed further in Section 3.5). Onegeneralization of the entropy of entanglement to mixed states is the entanglement of formation [22], which is defined as the minimum of P i p i S A (ρ i ) over all ensembles of bipartite pure states {ρ i ,p i } realizing the mixed state: ρ = P i p i ρ i . This quantity is a mixed-state entanglement monotone. As a function of ρ, it has a much more complicated form than the above expression for the entropy of entanglement. In fact, there is no known analytic expression for the entanglement of formation in general. The problem of extending pure-state entanglement monotones to mixed states is an important one, since every mixed-state entanglement monotone can be thought of as an extension of a pure-state entanglement monotone. Note, however, that a pure- state entanglement monotone may have many different mixed-state generalizations. The relation between the entanglement of formation and the entropy of entanglement presents one way to perform such an extension (convex-roof extension). For every pure-state entanglement monotonem(ρ), onecan defineamixed-state extensionM(ρ) as the minimum of P i p i m(ρ i ) over all ensembles of pure states {ρ i ,p i } realizing the 39 mixed state: ρ= P i p i ρ i . It is easy to verify thatM(ρ) is an entanglement monotone for mixed states. On the set of pure states the function M(ρ) reduces to m(ρ). As the example with the entropy of entanglement suggests, not every form of a pure- state entanglement monotone corresponds to a mixed-state entanglement monotone when trivially extended to all states – there are additional conditions that a mixed- state entanglement monotone must satisfy. On the basis of the above considerations, it makes sense to consider separate sets of differential conditions for pure-state and mixed-state entanglement monotones. Corollary 1: Atwice-differentiable functionf(ρ)ofthedensitymatrix is apure-state entanglement monotone, if and only if it satisfies (59) and (65) for pureρ. For pure states ρ = |ψihψ|, the elements of ρ are ρ ijℓm = α ij α ∗ ℓm , where the {α ij } are the state amplitudes: |ψi = P i,j α ij |i A i|j BC... i. Any function on pure states f(ρ)≡ f(|ψi) is therefore a function of the state amplitudes and their complex conjugates: f(|ψi) =f({α ij },{α ∗ ij }). (68) By making the substitutionρ ijℓm =α ij α ∗ ℓm into (59) and (65), we can (after consider- able algebra) derive alternative forms of the differential conditions for functions of the state vector: X i,j,k ∂f ∂α ij ε ik α kj = X i,j,k ∂f ∂α ∗ ij ε ∗ ik α ∗ kj , (69) X i,j,k,l,m,n ∂ 2 f ∂α ij ∂α mn (ε ik α kj −hεiα ij )(ε mℓ α ℓn −hεiα mn )+c.c.≤0. (70) HereεisalocalhermitianoperatoractingonsubsystemA.Analogousconditionsmust be satisfied for ε acting on the other parties’ subsystems. 40 3.4.3 Monotonicity under operations with information loss Besidesmonotonicityunderlocalunitariesandgeneralizedmeasurements,anentangle- mentmonotoneformixedstatesshouldalsosatisfymonotonicityunderlocaloperations which involve loss of information. The most general transformation that involves loss of information has the form ρ→ρ k = 1 p k X j M k,j ρM † k,j , (71) where p k =Tr X j M k,j ρM † k,j (72) is the probability for outcome k. The operators{M k,j } must satisfy X k,j M † k,j M k,j =I. (73) We can see that this includes unitary transformations, generalized measurements, and completely positive trace-preserving maps as special cases. It occasionally makes sense to consider even more general transformations, where the operators need not sum to the identity: X k,j M † k,j M k,j ≤I. (74) This corresponds to a situation where only certain outcomes are retained, and others are discarded; the probabilities add up to less than 1 dueto these discarded outcomes. We say such a transformation involves postselection. With or without postselection, weareconcerned withthecase whereall operations aredonelocally, sothatall theoperators{M k,j }act onasinglesubsystem. Everysuch transformation can be implemented as a sequence of local generalized measurements 41 (possibly discarding some of the outcomes) and local completely positive maps. In operator-sum representation [92], a completely positive map can be written ρ→ X k M k ρM † k , (75) where X k M † k M k ≤I. (76) Therefore, in addition to (59) and (65) we must impose the condition f(ρ)≥f X k M k ρM † k ! . (77) for all sets of local operators{M k } satisfying (76). Suppose the parties are supplied with a state ρ k taken from an ensemble{ρ k ,p k }. Discarding the information of the actual state amounts to the transformation {ρ k ,p k }→ρ ′ = X k p k ρ k . (78) As pointed out in [159], discarding information should not increase the entanglement of the system on average. Therefore, for any ensemble {ρ k ,p k }, an entanglement monotone on mixed states should be convex: X k p k f(ρ k )≥f X k p k ρ k ! . (79) Condition (79), together with condition (65) for monotonicity under local generalized measurements, implies monotonicity under local completely positive maps: f X k M k ρM † k ! ≤ X k p k f M k ρM † k p k ! ≤f(ρ). (80) 42 It is easy to see that if this inequality holds without postselection, it must also hold with postselection. It follows that a function of the density matrix is an entanglement monotone for mixed states if and only if it is (1) a convex function on the set of density matrices and (2) a monotone under local unitaries and generalized measurements. Fortunately, there are also simple differential conditions for convexity. A necessary and sufficient condition for a twice-differentiable function of multiple variables to be convex on a convex set is that its Hessian matrix be positive on the interior of the convex set (in this case, the set of density matrices). Therefore, in addition to (59) and (65) we add the differential condition Tr ∂ 2 f(ρ) ∂ρ ⊗2 σ ⊗2 ≥0, (81) which must be satisfied at every ρ on the interior of the set of density matrices for an arbitrary traceless hermitian matrix σ. Corollary 2: A twice-differentiable function f(ρ) of the density matrix is a mixed- state entanglement monotone, if and only if it satisfies (59), (65) and (81). 3.5 Examples In this section we demonstrate how conditions (59), (65) and (81) can be used to verify if a function is an entanglement monotone. We show this for three well known entanglement monotones: the norm of the state of the system, the trace of the square of the reduced density matrix of any subsystem, and the entropy of entanglement. In the next section we will use some of the observations made here to construct a new polynomial entanglement monotone for three-qubit pure states. 43 3.5.1 Norm of the state The most trivial example is the norm or the trace of the density matrix of the system: I 1 =Tr{ρ}. (82) Clearly I 1 is a monotone under LOCC, since all operations that we consider either preserveor decrease thetrace. But for thepurposeof demonstration, let usverify that I 1 satisfies the differential conditions. The LU-invariance condition (59) reads Tr ∂I 1 ∂ρ [ε,ρ] =Tr{[ε,ρ]} =0. (83) The second equality follows from the cyclic invariance of the trace. Sincethetraceislinear,thesecondtermincondition(65)vanishes,andweconsider only the first term: Tr ∂I 1 ∂ρ [[ε,ρ],ε] =Tr{[[ε,ρ],ε]} =0. (84) The condition is satisfied with equality, again due to the cyclic invariance of the trace, implying that the norm remains invariant under local measurements. The convexity condition (81) is also satisfied by equality. 3.5.2 Local purity The second example is the purity of the reduced density matrix: I 2 =Tr ρ 2 A , (85) 44 whereρ A is the reduced density matrix of subsystemA (which in general need not be a one-party subsystem). Note that this is an increasing entanglement monotone for pure states—the purity of the local reduced density matrix can only increase under LOCC. It has been shown in [33] that every m-th degree polynomial of the components of the density matrix ρ can be written as an expectation value of an observable O on m copies of ρ: f(ρ)=Tr Oρ ⊗m . (86) Here we have Tr ρ 2 A =Tr Cρ ⊗2 , (87) where the components of C are C lpsnkjqm =δ jp δ mn δ lq δ ks . (88) Therefore Tr ∂I 2 ∂ρ [ε,ρ] = Tr{C([ε,ρ]⊗ρ+ρ⊗[ε,ρ])} = Tr A {[ε,ρ] A ρ A +ρ A [ε,ρ] A } = 2Tr A {ρ A [ε,ρ] A }, (89) wherebyO A wedenotethepartialtraceofanoperatorOoverallsubsystemsexceptA. If ε does not act on subsystem A, then [ε,ρ] A = 0 and the above expression vanishes. If it acts on subsystemA, then [ε,ρ] A =[ε,ρ A ] and the expression vanishes due to the cyclic invariance of the trace. Now consider condition (65). If ε does not act on subsystemA, then [[ε,ρ],ε] A =0. (90) 45 From (65) we get 0 ≤ 1 4 Tr ∂I 2 ∂ρ [[ε,ρ],ε] +Tr ( ∂ 2 I 2 ∂ρ ⊗2 Tr{ερ}ρ− 1 2 {ε,ρ} ⊗2 ) =2Tr ( Tr{ερ}ρ− 1 2 {ε,ρ} 2 A ) . (91) The inequality follows from the fact that (Tr{ερ}ρ−(1/2){ε,ρ}) 2 A is a positive oper- ator. If ε acts on A, we can use the fact that for pure states Tr ρ 2 A =Tr ρ 2 B , (92) where B denotes the subsystem complementary to A. Then we can apply the same argumentasbeforeforthefunctionTr ρ 2 B . ThereforeI 2 doesnotdecrease onaverage under local generalized measurements, and is an entanglement monotone for pure states. What about mixed states? For increasing entanglement monotones the convex- ity condition (81) becomes a concavity condition—the direction of the inequality is inverted. In the case of I 2 , however, we have Tr ∂ 2 I 2 (ρ) ∂ρ ⊗2 σ ⊗2 =2Tr σ 2 A ≥0, (93) i.e., thefunctionisconvex. ThismeansthatTr{ρ 2 A }isnot agoodmeasureofentangle- ment for mixed states. Indeed, when extended to mixed states, I 2 cannot distinguish between entanglement and classical disorder. 46 3.5.3 Entropy of entanglement Finally consider the von Neumann entropy of entanglement: S A =−Tr(ρ A logρ A ). (94) Expanding around ρ A =I, we get S A =−Tr[(ρ A −I)+ 1 2 (ρ A −I) 2 − 1 6 (ρ A −I) 3 +...]. (95) The LU-invariance follows from the fact that every term in this expansion satisfies (59). If we substitute the n-th term in the condition, we obtain Tr([ε,ρ] A (ρ A −I) n−1 )=0. (96) This is true either because [ε,ρ] A =0 whenε does not act onA, or because otherwise [ε,ρ] A =[ε,ρ A ] and the equation follows from the cyclic invariance of the trace. Now to prove that S A satisfies (65), we will first assume that ρ −1 A exists. Then we can formally write ∂ ∂ρ logρ A = ∂ρ A ∂ρ ∂ ∂ρ A logρ A = ∂ρ A ∂ρ ρ −1 A . (97) 47 Consider the case whenε does not act on A. SubstitutingS A in (65), we get 1 4 Tr ∂S A ∂ρ [[ε,ρ],ε] +Tr ( ∂ 2 S A ∂ρ ⊗2 Tr{ερ}ρ− 1 2 {ε,ρ} ⊗2 ) =0+Tr ( ∂ ∂ρ ⊗ −logρ A ∂ρ A ∂ρ − ∂ρ A ∂ρ Tr{ερ}ρ− 1 2 {ε,ρ} ⊗2 ) =−Tr ( ρ −1 A ∂ρ A ∂ρ ∂ρ A ∂ρ Tr{ερ}ρ− 1 2 {ε,ρ} ⊗2 ) =−Tr A ρ −1 A Tr{ερ}ρ− 1 2 {ε,ρ} A Tr{ερ}ρ− 1 2 {ε,ρ} A =−Tr A ( ρ −1/2 A Tr{ερ}ρ− 1 2 {ε,ρ} A 2 ) ≤0. (98) Ifρ −1 A does not exist, it is only on a subset of measure zero – where one or more of the eigenvalues ofρ A vanish. Therefore, we can always find an arbitrarily close vicinity in the parameters describing ρ A , where ρ −1 A is regular and where (65) is satisfied. Since the condition is continuous, it cannot be violated on this special subset. Ifε acts on A, we can use an equivalent definition of the entropy of entanglement: S A =S B =−Tr{ρ B logρ B }, (99) and apply the same arguments. Therefore S A is an entanglement monotone for pure states. The convexity condition is not satisfied, since Tr ∂ 2 S A ∂ρ ⊗2 σ ⊗2 =−Tr{ρ −1 A σ 2 A }≤0. (100) This reflects the fact that the entropy of entanglement, like I 2 , does not distinguish between entanglement and classical randomness. 48 3.6 A new entanglement monotone It has been shown [63] that the set of all entanglement monotones for a multipartite pure state uniquely determine the orbit of the state under the action of the group of local unitary transformations. For three-qubit pure states the orbit is uniquely determined by 5 independent continuous invariants (not counting the norm) and one discrete invariant [1, 44]. Therefore, for pure states of three qubits there must exist five independent continuous entanglement monotones that are functions of the five independent continuous invariants. Any polynomial invariant in the amplitudes of a state |ψi = X i,j,k... α ijk... |i A i|j B i|k C i··· is a sum of homogenous polynomials of the form [153] P στ··· (|ψi) =α i 1 j 1 k 1 ... α ∗ i 1 j σ(1) k τ(1) ... ···α injnkn... α ∗ inj σ(n) k τ(n) ... , (101) where σ,τ,... are permutations of (1,2,...,n), and repeated indices indicate summa- tion. A set of five independent polynomial invariants for three-qubit pure states is [153] I 1 = P e,(12) (102) I 2 = P (12),e (103) I 3 = P (12),(12) (104) I 4 = P (123),(132) (105) I 5 = |α i 1 j 1 k 1 α i 2 j 2 k 2 α i 3 j 3 k 3 α i 4 j 4 k 4 ǫ i 1 i 2 ǫ i 3 i 4 ǫ j 1 j 2 ǫ j 3 j 4 ǫ k 1 k 3 ǫ k 2 k 4 | 2 . (106) 49 In the last expression ǫ ij is the antisymmetric tensor in 2 dimensions. The first three invariants arethelocal puritiesof subsystemsC, BandA,I 4 is theinvariant identified byKempe[82]andI 5 is(uptoafactor)thesquareofthe3-tangleidentifiedbyCoffman, Kundu and Wootters [45]. According to [63] the four known independent continuous entanglement monotones that do not require maximization over a multi-dimensional space are τ (AB)C =2(1−I 1 ) (107) τ (AC)B =2(1−I 2 ) (108) τ (BC)A =2(1−I 3 ) (109) τ ABC =2 p I 5 , (110) and any fifth independent entanglement monotone must depend on I 4 . Numerical evidence suggested that the tenth order polynomialσ ABC =3−(I 1 +I 2 +I 3 )I 4 might be such an entanglement monotone. However, no rigorous proof of monotonicity was given. Here, we will use conditions (59) and (65) to construct a different independent entanglement monotone, which is of sixth order in the amplitudes of the state and their complex conjugates. Observe that in (101) the amplitudes have been combined in such a way that subsystem A is manifestly traced out. By appropriate rearrangement, one can write the same expression in a form where an arbitrary subsystem is manifestly traced out. Therefore,anypolynomialinvariantcanbewrittenentirelyintermsofthecomponents of Tr A {ρ} or Tr B {ρ}, etc. This immediately implies that the LU-invariance condition (59) issatisfied, sinceifεacts onsubsystemA,wecanconsidertheexpressioninterms of ρ BC... , which, when substituted in (59), would yield zero because [ε,ρ] BC... = 0. It also implies that in order to prove monotonicity under local measurements we can only consider thesecond term in (65), since whenε acts on subsystem A, we can again 50 considertheexpressionforthefunctiononlyintermsofρ BC... andthefirsttermwould vanish according to (90). We will aim at constructing a polynomial function of three-qubit pure states ρ which has the same form when expressed in terms of ρ AB , ρ AC , or ρ BC , in order to avoid the necessity for separate proofs of monotonicity under measurements on the different subsystems. It has been shown in [153] that I 4 = 3Tr{ρ AB (ρ A ⊗ρ B )}−Tr ρ 3 A −Tr ρ 3 B = 3Tr{ρ AC (ρ A ⊗ρ C )}−Tr ρ 3 A −Tr ρ 3 C = 3Tr{ρ BC (ρ B ⊗ρ C )}−Tr ρ 3 B −Tr ρ 3 C . (111) For local measurements on subsystem C it is convenient to use the first of the above expressions for I 4 . The terms Tr ρ 3 A and Tr ρ 3 B are entanglement monotones by themselves. This can be easily seen by plugging them in condition (65): 1 4 Tr ∂Tr n ρ 3 A,B o ∂ρ [[ε,ρ],ε] +Tr ∂ 2 Tr n ρ 3 A,B o ∂ρ ⊗2 Tr(ερ)ρ− 1 2 {ε,ρ} ⊗2 =0+6Tr ( ρ A,B Tr{ερ}ρ− 1 2 {ε,ρ} 2 A,B ) ≥0. (112) These terms, however, are not independent of the invariants I 2 and I 3 . The term which is independent of the other polynomial invariants is Tr{ρ AB (ρ A ⊗ρ B )}. When we plug this term into condition (65) we obtain an expression which is not manifestly positive or negative. Is it possible to construct a function dependent on this term, which similarly to Tr n ρ 3 A,B o would yield a trace of a manifestly positive operator when substituted in (65)? 51 It is easy to see that if the function has the form Tr X 3 , where the operator X(ρ AB ) is a positive operator linearly dependent on ρ AB , it will be an increasing monotone under local measurements on C (for simplicity we assume X(0) =0): 1 4 Tr ( ∂Tr X 3 (ρ AB ) ∂ρ [[ε,ρ],ε] ) +Tr ( ∂ 2 Tr X 3 (ρ AB ) ∂ρ ⊗2 Tr(ερ)ρ− 1 2 {ε,ρ} ⊗2 ) =0+6Tr X(ρ AB )X 2 ((Tr{ερ}ρ− 1 2 {ε,ρ}) AB ) ≥0. (113) Since we want the function to depend on Tr{ρ AB (ρ A ⊗ρ B )}, we choose X(ρ AB ) = 2ρ AB +ρ A ⊗I B +I A ⊗ρ B . This is clearly positive for positive ρ AB . Expanding the trace, we obtain: Tr X 3 (ρ AB ) =12Tr{ρ AB (ρ A ⊗ρ B )}+12Tr ρ 2 AB (I A ⊗ρ B ) +12Tr ρ 2 AB (ρ A ⊗I B ) +6Tr ρ AB (I A ⊗ρ B ) 2 +6Tr ρ AB (ρ A ⊗I B ) 2 +3Tr ρ A ⊗ρ 2 B +3Tr ρ 2 A ⊗ρ B +Tr I A ⊗ρ 3 B +Tr ρ 3 A ⊗I B +8Tr ρ 3 AB =12Tr{ρ AB (ρ A ⊗ρ B )}+12Tr ρ 2 AB (I A ⊗ρ B ) +12Tr ρ 2 AB (ρ A ⊗I B ) +8Tr ρ 3 A +8Tr ρ 3 B +8Tr ρ 3 AB +3Tr ρ 2 A +3Tr ρ 2 B . (114) One can show that Tr ρ 2 AB (I A ⊗ρ B ) =Tr{ρ BC (ρ B ⊗ρ C )}, (115) Tr ρ 2 AB (ρ A ⊗I B ) =Tr{ρ AC (ρ A ⊗ρ C )}. (116) We also have that Tr ρ 3 AB =Tr ρ 3 C . Using this and (111), we obtain Tr X 3 (ρ AB ) =12I 4 +16 Tr ρ 3 A +Tr ρ 3 B +Tr ρ 3 C +3Tr ρ 2 A +3Tr ρ 2 B . (117) 52 ThisexpressionisanincreasingmonotoneunderlocalmeasurementsonC.Ifweaddto it 3Tr ρ 2 AB =3Tr ρ 2 C , it becomes invariant under permutations of the subsystems. Since Tr ρ 2 C is an increasing entanglement monotone, the whole expression will be a monotone under operations on any subsystem. We can define the closely related quantity φ ABC =69−Tr (2ρ AB +ρ A ⊗I B +I A ⊗ρ B ) 3 −3Tr ρ 2 AB . (118) This is a decreasing entanglement monotone that vanishes for product states, which is more standard for a measure of entanglement. It depends on the invariant identified by Kempe and is therefore independent of the other known monotones for three-qubit pure states. 3.7 Summary and outlook We have derived differential conditions for a twice-differentiable function on quantum states to be an entanglement monotone. There are two such conditions for pure- stateentanglementmonotones—invarianceunderlocalunitariesanddiminishingunder local measurements—plus a third condition (overall convexity of the function) for mixed-state entanglement monotones. We have shown that these conditions are both necessaryandsufficient. Wethenverifiedthattheconditionsaresatisfiedbyanumber of known entanglement monotones and we used them to construct a new polynomial entanglement monotone for three-qubit pure states. Itisourhopethatthisapproachtothestudyofentanglementmaycircumventsome of the difficulties that arise due the mathematically complicated nature of LOCC. It may be possible to find new classes of entanglement monotones, for both pure and mixed states, and to look for functions with particularly desirable properties (such as additivity). 53 There may also be other areas of quantum information theory where it will prove advantageous to consider general quantum operations as continuous processes. This seems a very promising new direction for research. 3.8 Appendix: Proof of sufficiency The LU-invariance condition can be written as F(ρ,ε) =0, (119) where we define F(ρ,ε) =f(e iε ρe −iε )−f(ρ) (120) withε being a local hermitian operator. This condition has to be satisfied for every ρ and every ε. By expanding up to first order in ε we obtained condition (59), which is equivalent to Tr ∂F(ρ,ε) ∂ε ε=0 ε =0. (121) This is a linear form of the components of ε and the requirement that it vanishes for every ε implies that ∂F(ρ,ε) ∂ε ij ε=0 =0. (122) Thishastobesatisfiedforeveryρ. Considerthefirstderivative ofF(ρ,ε) withrespect to ε ij , taken at an arbitrary pointε 0 . We have ∂F(ρ,ε) ∂ε ij ε=ε 0 = ∂F(ρ,ε 0 +ε) ∂ε ij ε=0 . (123) But from the form of F(ρ,ε) one can see that F(ρ,ε 0 +ε) = F(ρ ′ ,ε), where ρ ′ = e iε 0 ρe −iε 0 . Therefore 54 ∂F(ρ,ε) ∂ε ij ε=ε 0 = ∂F(ρ ′ ,ε) ∂ε ij ε=0 =0, (124) i.e., the first derivatives of F(ρ,ε) with respect to the components of ε vanish identi- cally. ThismeansthatF(ρ,ε) =F(ρ,0) =0foreveryεandcondition(59)issufficient. The condition for non-increase on average under local generalized measurements (64) can be written as G(ρ,ε)≤0, (125) where G(ρ,ε) =p 1 f(M 1 ρM 1 /p 1 )+p 2 f(M 2 ρM 2 /p 2 )−f(ρ). (126) The operators M 1 and M 2 in terms of ε are given by (60), and the probabilities p 1 and p 2 are defined as before. As we have argued in Section 3.3, it is sufficient that this condition is satisfied for infinitesimalε. By expanding the condition up to second order in ε we obtained condition (65), which is equivalent to Tr ∂ 2 G(ρ,ε) ∂ε ⊗2 ε=0 ε ⊗2 ≤0. (127) Clearly, if this condition is satisfied by a strict inequality, it is sufficient, since correc- tions of higher order in ε can be made arbitrarily smaller in magnitude by taking ε small enough. Concerns about the contribution of higher-order corrections may arise only if the second-order correction to G(ρ,ε) vanishes in some open vicinity of ρ and some open vicinity of ε (we have assumed that the function f(ρ) is continuous). But the second-order correction is a real quadratic form of the components of ε and it can vanish in an open vicinity of ε, only if it vanishes for every ε, i.e., if ∂ 2 G(ρ,ε) ∂ε ij ∂ε kl ε=0 =0. (128) 55 We will now show that if (128) is satisfied in an open vicinity of ρ, there exists an open vicinity of ε = 0 in which all second derivatives of G(ρ,ε) with respect to ε vanish identically. This means that all higher-order corrections to G(ρ,ε) vanish in this vicinity and (125) is satisfied with equality. Consider the two terms of G(ρ,ε) that depend on ε: G 1 (ρ,ε) =p 1 f(M 1 ρM 1 /p 1 ), (129) G 2 (ρ,ε) =p 2 f(M 2 ρM 2 /p 2 ). (130) They differ only by the sign of ε, i.e. G 1 (ρ,ε) =G 2 (ρ,−ε), and therefore ∂ 2 G 1 (ρ,ε) ∂ε ij ∂ε kl ε=0 = ∂ 2 G 2 (ρ,ε) ∂ε ij ∂ε kl ε=0 = 1 2 ∂ 2 G(ρ,ε) ∂ε ij ∂ε kl ε=0 . (131) If (128) is satisfied in an open vicinity of ρ, we have ∂ 2 G 1 (ρ,ε) ∂ε ij ∂ε kl ε=0 = ∂ 2 G 2 (ρ,ε) ∂ε ij ∂ε kl ε=0 =0 (132) in this vicinity. Consider the second derivatives of G(ρ,ε) with respect to the compo- nents of ε, taken at a point ε 0 : ∂ 2 G(ρ,ε) ∂ε ij ∂ε kl ε=ε 0 = ∂ 2 G 1 (ρ,ε) ∂ε ij ∂ε kl ε=ε 0 + ∂ 2 G 2 (ρ,ε) ∂ε ij ∂ε kl ε=ε 0 = ∂ 2 G 1 (ρ,ε 0 +ε) ∂ε ij ∂ε kl ε=0 + ∂ 2 G 2 (ρ,ε 0 +ε) ∂ε ij ∂ε kl ε=0 . (133) From the expression for G 1 (ρ,ε) one can see that ε occurs in G 1 (ρ,ε) only in the combination q I−ε 2 ρ q I−ε 2 . In G 1 (ρ,ε 0 +ε) it will appear only in q I−ε 0 −ε 2 ρ q I−ε 0 −ε 2 . But r I−ε 0 −ε 2 = r I−ε ′ 2 p I−ε 0 , (134) 56 where ε ′ =ε(I−ε 0 ) −1 . (135) So we can write r I−ε 0 −ε 2 ρ r I−ε 0 −ε 2 =p ′ r I−ε ′ 2 ρ ′ r I−ε ′ 2 , (136) where ρ ′ = p I−ε 0 ρ p I−ε 0 /p ′ (137) and p ′ =Tr n p I−ε 0 ρ p I−ε 0 o . (138) Then one can verify that G 1 (ρ,ε 0 +ε) =p ′ G 1 (ρ ′ ,ε ′ ). (139) Similarly G 2 (ρ,ε 0 +ε) =p ′′ G 2 (ρ ′′ ,ε ′′ ), (140) where ε ′′ =ε(I +ε 0 ) −1 , (141) ρ ′′ = p I +ε 0 ρ p I +ε 0 /p ′′ , (142) p ′′ =Tr n p I +ε 0 ρ p I +ε 0 o . (143) 57 Note that∂ε ′ pq /∂ε ij and∂ε ′′ pq /∂ε ij have nodependenceonε. Nor dop ′ andp ′′ . There- fore we obtain ∂ 2 G(ρ,ε) ∂ε ij ∂ε kl ε=ε 0 =p ′ ∂ 2 G 1 (ρ ′ ,ε ′ ) ∂ε ij ∂ε kl ε=0 +p ′′ ∂ 2 G 2 (ρ ′′ ,ε ′′ ) ∂ε ij ∂ε kl ε=0 = X p,q,r,s ∂ε ′ pq ∂ε ij ∂ε ′ rs ∂ε kl p ′ ∂ 2 G 1 (ρ ′ ,ε ′ ) ∂ε ′ pq ∂ε ′ rs ε ′ =0 + X p,q,r,s ∂ε ′′ pq ∂ε ij ∂ε ′′ rs ∂ε kl p ′′ ∂ 2 G 2 (ρ ′′ ,ε ′′ ) ∂ε ′′ pq ∂ε ′′ rs ε ′′ =0 . (144) We assumed that (132) is satisfied in an open vicinity of ρ. If ρ ′ and ρ ′′ are within this vicinity, the above expression will vanish. But from (137) and (142) we see that askε 0 k tends to zero, the quantitieskρ ′ −ρk andkρ ′′ −ρk also tend to zero. Therefore there exists an open vicinity of ε 0 = 0, such that for every ε 0 in this vicinity, the correspondingρ ′ andρ ′′ will be within the vicinity ofρ for which (132) is satisfied and ∂ 2 G(ρ,ε) ∂ε ij ∂ε kl ε=ε 0 =0. (145) This means that higher derivatives of G(ρ,ε) with respect to the components of ε taken at points in this vicinity will vanish, in particular derivatives taken atε=0. So higher order corrections in ε to G(ρ,ε) will also vanish. Therefore G(ρ,ε) = 0 in the vicinity of ρ for which we assumed that (65) is satisfied with equality, which implies that condition (65) is sufficient. 58 Chapter 4: Non-Markovian dynamics of a qubit coupled to a spin bath via the Ising interaction In this chapter, we turn our attention to the deterministic dynamics of open quantum systems. The chapter is based on a study made in collaboration with Hari Krovi, Mikhail Ryazanov and Daniel Lidar [96]. 4.1 Preliminaries As we pointed out in Chapter 1, a major conceptual as well as technical difficulty in the practical implementation of quantum information processing schemes is the unavoidable interaction of quantum systems with their environment. This interaction can destroy quantum superpositions and lead to an irreversible loss of information, a processknown as decoherence. Understandingthedynamicsof open quantumsystems is therefore of considerable importance. The Schr¨ odinger equation, which describes the evolution of closed systems, is generally inapplicable to open systems, unless one includes the environment in the description. This is, however, generally difficult, due to the large number of environment degrees of freedom. An alternative is to develop a descriptionfortheevolution of onlythesubsystemofinterest. A multitudeofdifferent approaches have been developed in this direction, exact as well as approximate [8, 30]. Typically the exact approaches are of limited practical usefulness as they are either phenomenological or involve complicated integro-differential equations. The various 59 approximations lead to regions of validity that have some overlap. Such techniques have been studied for many different models, but their performance in general, is not fully understood. In this work we consider an exactly solvable model of a single qubit coupled to an environment of qubits. We are motivated by the physical importance of such spin bathmodels [131] inthedescriptionof decoherence insolid state quantuminformation processors, such as systems based on the nuclear spin of donors in semiconductors [81, 164], or on the electron spin in quantum dots [106]. Rather than trying to accurately model decoherence due to the spin bath in such systems (as in, e.g., Refs. [147, 168]), ourgoalinthisworkistocomparetheperformanceofdifferentmasterequationswhich havebeenproposedintheliterature. Becausethemodelweconsiderisexactlysolvable, we are able to accurately assess the performance of the approximation techniques that we study. In particular, we study the Born-Markov and Born master equations, and the perturbation expansions of the Nakajima-Zwanzig (NZ) [111, 179] and the time- convolutionless (TCL) master equations [141, 140] up to fourth order in the coupling constant. We also study the post-Markovian (PM) master equation proposed in [137]. The dynamics of the system qubit in the model we study is highly non-Markovian and hence we do not expect the traditional Markovian master equations commonly used,e.g., inquantumoptics[42]andnuclearmagneticresonance[145],tobeaccurate. Thisistypicalofspinbaths,andwasnoted,e.g., byBreueretal. [29]. Aswewillseein Chapter5, thenon-Markovian character ofthedynamicscan beusedtoouradvantage inerror-correctionschemes,henceunderstandingthesemodelsisofspecialsignificance. The work by Breuer et al. (as well as by other authors in a number of subsequent publications [123, 36, 71, 173, 40, 75]) is conceptually close to ours in that in both cases ananalytically solvablespin-bathmodelisconsideredandtheanalytical solution for the open system dynamics is compared to approximations. However, there are also important differences, namely, in Ref. [29] a so-called spin-star system was studied, 60 wherethesystemspinhasequalcouplingstoallthebathspins,andtheseareoftheXY exchange-type. In contrast, in our model the system spin interacts via Ising couplings with the bath spins, and we allow for arbitrary coupling constants. As a result there are also important differences in the dynamics. For example, unlike the model in Ref. [29], for our model we find that the odd order terms in the perturbation expansions of Nakajima-Zwanzig and time-convolutionless master equations are non-vanishing. This reflects the fact that there is a coupling between thex andy components of the Bloch vector which is absent in [29]. In view of the non-Markovian behavior of our model, we also discuss the relation between a representation of the analytical solution of our model in terms of completely positive maps, and the Markovian limit obtained via a coarse-grainingmethodintroducedin[103],andtheperformanceofthepost-Markovian master equation [137]. Thischapterisorganizedasfollows. InSection4.2,wepresentthemodel,derivethe exact solution and discuss its behavior in the limit of small times and large number of bathspins,andinthecasesofdiscontinuousspectraldensityco-domainandalternating sign of the system-bath coupling constants. In Section 4.3, we consider second order approximation methods such as the Born-Markov and Born master equations, and a coarse-graining approach to the Markovian semigroup master equation. Then we derive solutions to higher order corrections obtained from the Nakajima-Zwanzig and time-convolutionless projection techniques aswell asderivetheoptimal approximation achievable through the post-Markovian master equation. In Section 4.4, we compare these solutions for various parameter values in the model and plot the results. Finally in Section 4.5, we present our conclusions. 61 4.2 Exact dynamics 4.2.1 The model We consider a single spin- 1 2 system (i.e., a qubit with a two-dimensional Hilbert space H S ) interacting with a bath of N spin- 1 2 particles (described by an N-fold tensor product of two-dimensional Hilbert spaces denoted H B ). The observables describing the spin of a spin- 1 2 particle in each of the three spatial directions are described by the Pauli operators σ x = 0 1 1 0 ,σ y = 0 −i i 0 ,σ z = 1 0 0 −1 . (146) We model the interaction between the system qubit and the bath by the Ising Hamil- tonian H ′ I =ασ z ⊗ N X n=1 g n σ z n , (147) where g n are dimensionless real-valued coupling constants in the interval [−1,1] (n labels thedifferent qubitsin thebath), andα>0is aparameter having thedimension of frequency (we work in units in which ¯ h=1), which describes the coupling strength and will be used below in conjunction with time (αt) for perturbation expansions. The system and bath Hamiltonians are H S = 1 2 ω 0 σ z (148) and H B = N X n=1 1 2 Ω n σ z n . (149) For definiteness, werestrict thefrequenciesω 0 andΩ n totheinterval [−1,1], ininverse time units. Even though the units of time can be arbitrary, by doing so we do not lose 62 generality, sincewewillbeworkingintheinteractionpicturewhereonlythefrequencies Ω n appear in relation to the state of the bath [Eq. (158)]. Since the ratios of these frequencies and the temperature of the bath occur in the equations, only their values relativetothetemperatureareofinterest. Therefore,henceforthwewillomittheunits of frequency and temperature and will treat these quantities as dimensionless. The interaction picture is defined as the transformation of any operator A7→A(t) =exp(iH 0 t)Aexp(−iH 0 t), (150) whereH 0 =H S +H B . The interaction HamiltonianH I chosen here is invariant under this transformation since it commutes with H 0 . [Note that in the next subsection, to simplify our calculations we redefine H S and H ′ I (whence H ′ I becomes H I ), but this does not alter the present analysis.] All the quantities discussed in the rest of this article are assumed to be in the interaction picture. The dynamics can be described using the superoperator notation for the Liouville operator Lρ(t)≡−i[H ′ I ,ρ(t)], (151) where ρ(t) is the density matrix for the total system in the Hilbert space H S ⊗H B . The dynamics is governed by the von Neumann equation d dt ρ(t) =αLρ(t) (152) and the formal solution of this equation can be written as follows: ρ(t) =exp(αLt)ρ(0). (153) 63 The state of the system is given by the reduced density operator ρ S (t) =Tr B {ρ(t)}, (154) whereTr B denotes apartial trace taken over thebath HilbertspaceH B . Thiscan also be written in terms of the Bloch sphere vector ~ v(t) = v x (t) v y (t) v z (t) =Tr{~ σρ S (t)}, (155) where ~ σ ≡ (σ x ,σ y ,σ z ) is the vector of Pauli matrices. In the basis of σ z eigenstates this is equivalent to ρ S (t) = 1 2 (I +~ v·~ σ) = 1 2 1+v z (t) v x (t)−iv y (t) v x (t)+iv y (t) 1−v z (t) . (156) We assume that the initial state is a product state, i.e., ρ(0) =ρ S (0)⊗ρ B , (157) and that the bath is initially in the Gibbs thermal state at a temperature T ρ B =exp(−H B /kT)/Tr[exp(−H B /kT)], (158) wherek is the Boltzmann constant. Sinceρ B commutes with the interaction Hamilto- nian H I , the bath state is stationary throughout the dynamics: ρ B (t) = ρ B . Finally, the bath spectral density function is defined as usual as J(Ω)= X n |g n | 2 δ(Ω−Ω n ). (159) 64 4.2.2 Exact solution for the system-spin dynamics We first shift the system Hamiltonian in the following way: H S 7→H S +θI, θ≡Tr{ X n g n σ z n ρ B }. (160) As a consequence the interaction Hamiltonian is modified from Eq. (147) to H ′ I 7→H I =ασ z ⊗B, (161) where B≡ X n g n σ z n −θI B . (162) This shift is performed because now Tr B [H I ,ρ(0)] =0, or equivalently Tr B {Bρ B }=0. (163) This property will simplify our calculations later when we consider approximation techniques in Section 4.3. Now, we derive the exact solution for the reduced density operator ρ S corresponding to the system. We do this in two different ways. The Kraus operator sum representation is a standard description of the dynamics of a sys- tem initially decoupled from its environment and it will also be helpful in studying the coarse-graining approach to the quantum semigroup master equation. The sec- ond method is computationally more effective and is helpful in obtaining analytical expressions for N ≫ 1. 65 4.2.2.1 Exact Solution in the Kraus Representation In the Kraus representation the system state at any given time can be written as ρ S (t)= X i,j K ij (t)ρ S (0)K ij (t) † , (164) wherethe Krausoperators satisfy P ij K ij (t) † K ij (t)=I S [92]. Theseoperators can be expressed easily in the eigenbasis of the initial state of the bath density operator as K ij (t)= p λ i hj|exp(−iH I t)|ii, (165) where the bath density operator at the initial time is ρ B (0) = P i λ i |iihi|. For the Gibbs thermal state chosen here, the eigenbasis is theN-fold tensor product of theσ z basis. In this basis ρ B = X l exp(−βE l ) Z |lihl|, (166) where β =1/kT. Here E l = N X n=1 1 2 ¯ hΩ n (−1) ln , (167) is the energy of each eigenstate |li, where l = l 1 l 2 ...l n is the binary expansion of the integer l, and the partition function is Z = P l exp(−βE l ). Therefore, the Kraus operators become K ij (t) = p λ i exp(−itα ˜ E i σ z )δ ij , (168) where ˜ E i =hi|B|ii= N X n=1 g n (−1) in −Tr{ X n g n σ z n ρ B }, (169) 66 andλ i =exp(−βE i )/Z. SubstitutingthisexpressionforK ij intoEq. (164)andwriting the system state in the Bloch vector form given in Eq. (156), we obtain v x (t) = v x (0)C(t)−v y (0)S(t), v y (t) = v x (0)S(t)+v y (0)C(t), (170) v z (t) = v z (0), where C(t) = X i λ i cos2α ˜ E i t, S(t) = X i λ i sin2α ˜ E i t. (171) The equations (170) are the exact solution to the system dynamics of the above spin bath model. We see that the evolution of the Bloch vector is a linear combination of rotations around thez axis. This evolution reflects the symmetry of the interaction Hamiltonian which is diagonal in the z basis. By inverting Eqs. (170) for v x (0) or v y (0), we see that the Kraus map is irreversible when C(t) 2 +S(t) 2 = 0. This will become important below, when we discuss the validity of the time-convolutionless approximation. 4.2.2.2 Alternative Exact Solution Another way to derive the exact solution which is computationally more useful is the following. Since all σ z n commute, the initial bath density matrix factors and can be written as ρ B = N O n=1 exp − Ωn 2kT σ z n Tr exp − Ωn 2kT σ z n = N O n=1 1 2 (I +β n σ z n )≡ N Y n=1 ρ n , (172) 67 where β n =tanh − Ω n 2kT , (173) and−1≤β n ≤1. Using this, we obtain an expression for θ defined in Eq. (160) θ = Tr{ N X n=1 g n σ z n N O m=1 1 2 (I +β m σ z m )} = N X n=1 g n Tr{ 1 2 (σ z n +β n I)} Y m6=n Tr{ 1 2 (I +β m σ z m )} = N X n=1 g n β n . (174) The evolution of the system density matrix in the interaction picture is ρ S (t) =Tr B {e −iH I t ρ(0)e iH I t }. (175) In terms of the system density matrix elements in the computational basis {|0i,|1i} (which is an eigenbasis of σ z in H I =ασ z ⊗B), we have hj|ρ S (t)|ki = hj|Tr B {e −iH I t ρ S (0) N O m=1 ρ m e iH I t }|ki = Tr B {e −iαhj|σ z |jiBt hj|ρ S (0)|ki N O m=1 ρ m e +iαhk|σ z |kiBt }. Let us substitutehj|σ z |ji =(−1) j and rewrite e −iαhj|σ z |jiBt =e −iα(−1) j ( P N l=1 g l σ z l −θI)t = N O l=1 e −i(−1) j α(g l σ z l − θ N I)t . Since all the matrices are diagonal, they commute and we can collect the terms by qubits: hj|ρ S (t)|ki =hj|ρ S (0)|kiTr{ N O m=1 e −i[(−1) j −(−1) k ]α(g l σ z l − θ N I)t ρ n }. 68 Let us denote (−1) j −(−1) k =2ǫ jk . The trace can be easily computed to be Q N n=1 Tr{e −i2ǫ jk α(gnσ z n − θ N I)t1 2 (I +β n σ z n )} = N Y n=1 e i2ǫ jk α θ N t [cos(2ǫ jk αg n t)−iβ n sin(2ǫ jk αg n t)]. Thus the final expression for the system density matrix elements is hj|ρ S (t)|ki = hj|ρ S (0)|kie i2ǫ jk αθt N Y n=1 [cos(2ǫ jk αg n t)−iβ n sin(2ǫ jk αg n t)]. Notice that ǫ 00 =ǫ 11 = 0, hence the diagonal matrix elements do not depend on time as before: h0|ρ S (t)|0i =h0|ρ S (0)|0i, h1|ρ S (t)|1i =h1|ρ S (0)|1i. For the off-diagonal matrix elements ǫ 01 =1, ǫ 10 =−1, and the evolution is described by h0|ρ S (t)|1i = h0|ρ S (0)|1if(t), h1|ρ S (t)|0i = h1|ρ S (0)|0if ∗ (t), (176) where f(t)=e i2αθt N Y n=1 [cos(2αg n t)−iβ n sin(2αg n t)]. (177) In terms of theBloch vector components, this can bewritten in the form of Eq. (170), where C(t) = (f(t)+f ∗ (t))/2, S(t) = (f(t)−f ∗ (t))/2i. (178) 69 4.2.3 Limiting cases 4.2.3.1 Short Times Consider the evolution for short times whereαt≪1. Then N Y n=1 [cos(2αg n t)±iβ n sin(2αg n t)] = N Y n=1 q 1−(1−β 2 n )sin 2 (2αg n t) ≈ N Y n=1 [1−2(1−β 2 n )(αg n t) 2 ] ≈ 1−2 " α 2 N X n=1 g 2 n (1−β 2 n ) # t 2 ≈ exp[−2(αt) 2 Q 2 ], (179) where (see Appendix A at the end of this chapter) Q 2 ≡Tr{B 2 ρ B }= N X n=1 g 2 n (1−β 2 n )= Z ∞ −∞ 2J(Ω) 1+cosh( Ω kT ) dΩ. (180) Note that for the above approximation to be valid, we need 2(αt) 2 Q 2 ≪ 1. The total phase of f(t) in Eq. (177) is φ≈2θαt+ N X n=1 (−β n 2αg n t)=2θαt−2α N X n=1 g n β n ! t =0, (181) where we have used Eq. (174). Thus, the off-diagonal elements of the system density matrix become ρ 01 S (t) ≈ ρ 01 S (0)e −2(αt) 2 Q 2 , ρ 10 S (t) ≈ ρ 10 S (0)e −2(αt) 2 Q 2 . (182) 70 Finally, the dynamics of the Bloch vector components are: v x,y (t) ≈ v x,y (0)e −2(αt) 2 Q 2 , v z (t) = v z (t). (183) This represents the well known behavior [112] of the evolution of an open quantum system in the Zeno regime. In this regime coherence does not decay exponentially but is initially flat, as is the case here due to the vanishing time derivative of ρ 01 S (t) at t = 0. As we will see in Section 4.3, the dynamics in the Born approximation (which is also the second order time-convolutionless approximation) exactly matches the last result. 4.2.3.2 Large N WhenN ≫1 and the values ofg n are random, then the different terms in the product of Eq. (177) are smaller than 1 most of the time and have recurrences at different times. Therefore, we expect the function f(t) to be close to zero in magnitude for most of the time and full recurrences, if they exist, to be extremely rare. Wheng n are equal and so are Ω n , then partial recurrences occur periodically, independently of N. Full recurrences occur with a period which grows at least as fast as N. This can be argued from Eq. (170) by imposing thecondition that the arguments of all the cosines and sines are simultaneously equal to an integer multiple of 2π. When J(Ω) has a narrow high peak, e.g., oneg n is much larger than the others, then the corresponding terms in the products in Eq. (177) oscillate faster than the rate at which the whole product decays. This is effectively a modulation of the decay. 71 4.2.3.3 Discontinuous spectral density co-domain As can be seen from Eq. (177), the coupling constants g n determine the oscillation periods of the product terms, while the temperature factors β n determine their mod- ulation depths. If the codomain of spectral density is not continuous, i.e. it can be split into nonoverlapping intervals G j , j = 1,...,J, then Eq. (177) can be represented in the following form: f(t)=e i2αθt P 1 (t)P 2 (t)...P J (t), (184) where P j (t)= Y gn∈G j cos(2αg n t)−iβ n sin(2αg n t) . (185) In this case, if G j are separated by large enough gaps, the evolution rates of different P j (t)canbesignificantlydifferent. ThisisparticularlynoticeableifoneP j (t)undergoes partial recurrences while another P j ′(t) slowly decays. For example, one can envision a situation with two intervals such that one term shows frequent partial recurrences that slowly decay with time, while the other term decays faster, but at times larger than the recurrence time. Theoverall evolution then consists in a small number of fast partial recurrences. In an extreme case, when one g n is much larger then the others, this results in an infinite harmonic modulation of the decay with depth dependent on β n , i.e., on temperature. 72 4.2.3.4 Alternating signs If the bath has the property that every bath qubit m has a pair −m with the same frequency Ω −m =Ω m , but opposite coupling constant g −m =−g m , the exact solution can be simplified. First, β −m =β m , and θ =0. Next, Eq. (177) becomes f(t) = N/2 Y m=1 cos(2αg m t)−iβ m sin(2αg m t) cos(2αg −m t)−iβ −m sin(2αg −m t) = N/2 Y m=1 cos 2 (2αg m t)+β 2 m sin 2 (2αg m t) . (186) This function is real, thus Eq. (178) becomes C(t) = f(t),S(t) = 0, so that v x (t) = v x (0)f(t) and v y (t) = v y (0)f(t). The exact solution is then symmetric under the interchange v x ↔v y , a property shared by all the second order approximate solutions considered below, as well as the post-Markovian master equation. The limiting case Eq.(179)remainsunchanged,andsinceQ 2 dependsong 2 n ,butnotg n ,itandallsecond order approximations also remain unchanged. In the special case |g m | =g, the exact solution exhibits full recurrences with periodT =π/αg. 4.3 Approximation methods In this section we discuss the performance of different approximation methods de- veloped in the open quantum systems literature [8, 30]. The corresponding master equations for the system density matrix can be derived explicitly and since the model considered here is exactly solvable, we can compare the approximations to the exact dynamics. We use the Bloch vector representation and since the z component has no dynamics, a fact which is reflected in all the master equations, we omit it from our comparisons. 73 4.3.1 Born and Born-Markov approximations Both the Born and Born-Markov approximations are second order in the coupling strength α. 4.3.1.1 Born approximation The Born approximation is equivalent to a truncation of the Nakajima-Zwanzig pro- jection operator method at the second order, which is discussed in detail in Section 4.3.2. The Born approximation is given by the following integro-differential master equation: ˙ ρ S (t) =− Z t 0 Tr B {[H I (t),[H I (s),ρ S (s)⊗ρ B ]]}ds. (187) SinceinourcasetheinteractionHamiltonianistime-independent,theintegralbecomes easy to solve. We obtain ˙ ρ S (t) =−2α 2 Q 2 Z t 0 (ρ S (s)−σ z ρ S (s)σ z )ds, (188) where Q 2 is the second order bath correlation function in Eq. (180). Writing ρ S (t) in terms of Bloch vectors as (I +~ v·~ σ)/2 [Eq. (156)], we obtain the following integro- differential equations: ˙ v x,y (t) = −4α 2 Q 2 Z t 0 v x,y (s)ds. (189) These equations can be solved by taking the Laplace transform of the variables. The equations become sV x,y (s)−v x,y (0) =−4α 2 Q 2 V x,y (s) s , (190) 74 where V x,y (s) is the Laplace transform of v x,y (t). This gives V x,y (s) = v x,y (0)s s 2 +4Q 2 α 2 , (191) which can be readily solved by taking the inverse Laplace transform. Doing so, we obtain the solution of the Born master equation for our model: v x,y (t) = v x,y (0)cos(2α p Q 2 t). (192) Note that this solution is symmetric under the interchange v x ↔ v y , but the exact dynamics in Eq. (170) does not have this symmetry. The exact dynamics respects the symmetry: v x → v y and v y → −v x , which is a symmetry of the Hamiltonian. This means that higher order corrections are required to break the symmetry v x ↔ v y in order to approximate the exact solution more closely. One often makes the substitution v x,y (t) for v x,y (s) in Eq. (189) since the integro- differential equation obtained in other models may not be as easily solvable. This approximation, which is valid for short times, yields ˙ v x,y (t) = −4α 2 Q 2 tv x,y (t), (193) which gives v x,y (t) = v x,y (0)exp(−2Q 2 α 2 t 2 ), (194) i.e., we recover Eq. (183). This is the same solution obtained in the second order approximation using the time-convolutionless (TCL) projection method discussed in Section 4.3.2. 75 4.3.1.2 Born-Markov approximation In order to obtain the Born-Markov approximation, we use the following quantities [30][Ch.3]: R(ω) = X E 2 −E 1 =ω P E 1 σ z P E 2 , Γ(ω) = α 2 Z ∞ 0 e iωs Q 2 ds, H L = X ω T(ω)R(ω) † R(ω), (195) where T(ω) = (Γ(ω)−Γ(ω) ∗ )/2i, E i is an eigenvalue of the system Hamiltonian H S , and P E i is the projector onto the eigenspace corresponding to this eigenvalue. In our case H S is diagonal in the eigenbasis of σ z , and only ω = 0 is relevant. This leads to R(0) = σ z and Γ(0) = α 2 R ∞ 0 Q 2 dt. Since Γ(0) is real, we have T(0) = 0. Hence the Lamb shift Hamiltonian H L = 0, and the Lindblad form of the Born-Markov approximation is ˙ ρ S (t) =γ(σ z ρ S σ z −ρ S ), (196) where γ = Γ(0)+Γ(0) ∗ = 2α 2 R ∞ 0 Q 2 dt. But note that Q 2 = Tr B {B 2 ρ B } does not depend on time. This means that Γ and hence γ are both infinite. Thus the Born- Markov approximation is not valid for this model and the main reason for this is the time independence of the bath correlation functions. The dynamics is inherently non-Markovian. A different approach to the derivation of a Markovian semigroup master equation was proposed in [103]. In this approach, a Lindblad equation is derived from the Kraus operator-sum representation by a coarse-graining procedure defined in terms of a phenomenological coarse-graining time scale τ. The general form of the equation is: ∂ρ(t) ∂t =−i[h ˙ Qi τ ,ρ(t)]+ 1 2 M X α,β=1 h˙ χ α,β i τ ([A α ,ρ(t)A † β ]+[A α ρ(t),A † β ]), (197) 76 where the operators A 0 = I and A α ,α = 1,...,M form an arbitrary fixed operator basis in which the Kraus operators (164) can be expanded as K i = M X α=0 b iα A α . (198) The quantities χ α,β (t) and Q(t) are defined through χ α,β (t) = X i b iα (t)b ∗ iβ (t), (199) Q(t) = i 2 M X α=1 (χ α0 (t)K α −χ 0α (t)K † α ), (200) and hXi τ = 1 τ Z τ 0 X(s)ds. (201) For our problem we find ∂ρ(t) ∂t =−i˜ ω[σ Z ,ρ(t)]+˜ γ(σ Z ρ(t)σ Z −ρ(t)), (202) where ˜ ω = 1 2τ S(τ) (203) and ˜ γ = 1 2τ (1−C(τ)) (204) withC(t)andS(t)definedinEq. (171). Inorderforthisapproximationtobejustified, it is required that the coarse-graining time scale τ be much larger than any charac- teristic time scale of the bath [103]. However, in our case the bath correlation time is infinite which, once again, shows the inapplicability of the Markovian approximation. This is further supported by the performance of the optimal solution that one can achieve by varyingτ, which is discussed in Section 4.4. There we numerically examine 77 the average trace-distance between the solution to Eq. (202) and the exact solution as a function of τ. The average is taken over a time T, which is greater than the decay time of the exact solution. We determine an optimal τ for which the average trace distance is minimum and then determine the approximate solution. The solution of Eq. (202) for a particular τ in terms of the Bloch vector components is v x (t) = v x (0) ˜ C τ (t)+v y (0) ˜ S τ (t) v y (t) = v y (0) ˜ C τ (t)−v x (0) ˜ S τ (t), (205) where ˜ C τ (t) = e −˜ γ(τ)t cos(˜ ω(τ)t) and ˜ S τ (t) = e −˜ γ(τ)t sin(˜ ω(τ)t). The average trace distance as a function of τ is given by, ¯ D (ρ exact ,ρ CG )≡ 1 2 Tr|ρ exact −ρ CG | = 1 2T T X t=0 q (C(t)− ˜ C(t)) 2 +(S(t)− ˜ S(t)) 2 q v x (0) 2 +v y (0) 2 , (206) whereρ CG representsthecoarse-grained solutionandwhere|X| = √ X † X. Theresults are presented in Section 4.4. Next we consider the Nakajima-Zwanzig (NZ) and the time-convolutionless (TCL) master equations for higher order approximations. 4.3.2 NZ and TCL master equations Using projection operators one can obtain approximate non-Markovian master equa- tions to higher orders in αt. A projection is defined as follows, Pρ =Tr B {ρ}⊗ρ B , (207) andservestofocusonthe“relevantdynamics”(ofthesystem)byremovingthebath(a recent generalization isdiscussedinRef. [28]). Thechoice ofρ B issomewhat arbitrary 78 and can be taken to beρ B (0) which significantly simplifies the calculations. Using the notation introduced in [29], define hSi≡PSP (208) for any superoperatorS. ThushS n i denote the moments of the superoperator. Note thatfortheLiouvilliansuperoperator,hLi=0byvirtueofthefactthatTr B {Bρ B (0)} = 0 (see [30]). Since we assume that the initial state is a product state, both the NZ and TCL equations are homogeneous equations. The NZ master equation is an integro- differential equation with a memory kernelN(t,s) and is given by ˙ ρ S (t)⊗ρ B = Z t 0 N(t,s)ρ S (s)⊗ρ B ds. (209) The TCL master equation is a time-local equation given by ˙ ρ S (t)⊗ρ B =K(t)ρ S (t)⊗ρ B . (210) When these equations are expanded in αt and solved we obtain the higher order cor- rections. When the interaction Hamiltonian is time independent (as in our case), the above equations simplify to Z t 0 N(t,s)ρ S (s)⊗ρ B ds= ∞ X n=1 α n I n (t,s)hL n i pc ρ S (s) (211) and K(t) = ∞ X n=1 α n t n−1 (n−1)! hL n i oc (212) 79 for the NZ and TCL equations, respectively, where the time-ordered integral operator I n (t,s) is defined as I n (t,s)≡ Z t 0 dt 1 Z t 1 0 dt 2 ··· Z t n−2 0 ds. (213) The definitions of the partial cumulants hLi pc and the ordered cumulants hLi oc are given in Refs. [140, 133, 80]. For our model we have hLi pc =hLi oc =0, (214) and L 2 pc = L 2 L 2 oc = L 2 L 3 pc = L 3 L 3 oc = L 3 L 4 pc = L 4 − L 2 2 L 4 oc = L 4 −3 L 2 2 . (215) Explicit expressions for these quantities are given in Appendix at the end of this chapter. SubstitutingtheseintotheNZandTCLequations(211) and(212), weobtain whatwerefertobelowastheNZnandTCLnmasterequations, withn=2,3,4. These approximate master equations are, respectively, second, third and fourth order in the coupling constant α, and they can be solved analytically. The second order solution 80 of the NZ equation (NZ2) is exactly the Born approximation and the solution is given in Eq. (192). The third order NZ master equation is given by ˙ ρ S (t) = −2α 2 Q 2 I 2 (t,s)(ρ S (s)−σ z ρ S (s)σ z ) + i4α 3 Q 3 I 3 (t,s)(σ z ρ S (s)−ρ S (s)σ z ), (216) and the fourth order is ˙ ρ S (t) = −2α 2 Q 2 I 2 (t,s)(ρ S (s)−σ z ρ S (s)σ z ) + i4α 3 Q 3 I 3 (t,s)(σ z ρ S (s)−ρ S (s)σ z ) + 8α 4 (Q 4 −Q 2 2 )I 4 (t,s)(ρ S (s)−σ z ρ S (s)σ z ). (217) Theseequationsareequivalentto, respectively, 6thand8thorderdifferentialequations (withconstantcoefficients)andaredifficulttosolveanalytically. Theresultswepresent in the next section were therefore obtained numerically. The situation is simpler in the TCL approach. The second order TCL equation is given by ˙ ρ S (t) = −α 2 tTr B {[H I ,[H I ,ρ S (t)⊗ρ B (0)]]} = −2α 2 tQ 2 (ρ S (t)−σ z ρ S (t)σ z ), (218) whose solution is as given in Eq. (194) in terms of Bloch vector components. For TCL3 we find ˙ ρ S (t) =−2α 2 tQ 2 (ρ S (t)−σ z ρ S (t)σ z )+4iQ 3 α 3 t 2 2 (σ z ρ S (t)−ρ S (t)σ z ), (219) 81 and for TCL4 we find ˙ ρ S (t) = [−2α 2 tQ 2 +(8Q 4 −24Q 2 2 )α 4 t 3 6 ](ρ S (t)−σ z ρ S (t)σ z ) + 4iQ 3 α 3 t 2 2 (σ z ρ S (t)−ρ S (t)σ z ). (220) These equation can be solved analytically, and the solutions to the third and fourth order TCL equations are given by v x (t) = f n (αt)[v x (0)cos(g(t))+v y (0)sin(g(t))], v y (t) = f n (αt)[v y (0)cos(g(t))−v x (0)sin(g(t))]. (221) whereg(t) =4Q 3 α 3 t 3 /3,f 3 (αt) =exp(−2Q 2 α 2 t 2 )(TCL3)andf 4 (αt) =exp(−2Q 2 α 2 t 2 +(2Q 4 −6Q 2 2 )α 4 t 4 /3) (TCL4). It is interesting to note that the second order expan- sions of the TCL and NZ master equations exhibit a v x ↔v y symmetry between the components of the Bloch vector, and only the third order correction breaks this sym- metry. Notice that the coefficient ofα 3 does not vanish in this model unlike in the one considered in [29] because both L 3 pc 6= 0 and L 3 oc 6= 0 and hence the third order (and other odd order) approximations exist. 4.3.3 Post-Markovian (PM) master equation In this section we study the performance of the post-Markovian master equation re- cently proposed in [137]: ∂ρ(t) ∂t =D Z t 0 dt ′ k(t ′ )exp(Dt ′ )ρ(t−t ′ ). (222) This equation was constructed via an interpolation between the exact dynamics and thedynamicsin theMarkovian limit. TheoperatorD is thedissipator intheLindblad 82 equation (353), andk(t)isaphenomenological memorykernelwhichmustbefoundby fittingtodataorguessedonphysicalgrounds. Aswasdiscussedearlier, theMarkovian approximation fails for our model, nevertheless, one can use the form of the dissipator we obtained in Eq. (353) Dρ=σ z ρσ z −ρ. (223) It is interesting to examine to what extent Eq. (222) can approximate the exact dynamics. As a measure of the performance of the post-Markovian equation, we will take the trace-distance between the exact solution ρ exact (t) and the solution to the post-Markovian equation ρ 1 (t). The general solution of Eq. (222) can be found by expressingρ(t) in the damping basis [31] and applying a Laplace transform [137]. The solution is ρ(t) = X i μ i (t)R i = X i Tr(L i ρ(t))R i , (224) where μ i (t) =Lap −1 1 s−λ i ˜ k(s−λ i ) μ i (0)≡ξ i (t)μ i (0), (225) (Lap −1 is the inverse Laplace transform) with ˜ k being the Laplace transform of the kernel k, {L i } and {R i } being the left and right eigenvectors of the superoperator D, and λ i the corresponding eigenvalues. For our dissipator the damping basis is {L i }={R i }={ I √ 2 , σ x √ 2 , σ y √ 2 , σ z √ 2 } and the eigenvalues are{0,−2,−2,0}. Therefore, we can immediately write the formal solution in terms of the Bloch vector components: v x,y (t) =Lap −1 1 s+2 ˜ k(s+2) v x,y (0)≡ξ(t)v x,y (0). (226) We see that v x (t) has no dependence on v y (0), and neither does v y (t) on v x (0), in contrast to the exact solution. The difference comes from the fact that the dissipator D does not couple v x (t) and v y (t). This reveals an inherent limitation of the post- Markovian master equation: it inherits the symmetries of the Markovian dissipator 83 D, which may differ from those of the generator of the exact dynamics. In order to rigorously determine the optimal performance, we use the trace distance between the exact solution and a solution to the post-Markovian equation: D(ρ exact (t),ρ 1 (t)) = 1 2 p (C(t)−ξ(t)) 2 +S(t) 2 q v x (0) 2 +v y (0) 2 . (227) Obviously this quantity reaches its minimum for ξ(t) =C(t),∀t independently of the initialconditions. Thekernelforwhichtheoptimalperformanceofthepost-Markovian master equation is achieved, can thus be formally expressed, using Eq. (226), as: k opt (t) = 1 2 e 2t Lap −1 1 Lap(C(t)) −s . (228) It should be noted that the condition for complete positivity of the map generated by Eq. (222), P i ξ i (t)L T i ⊗R i ≥0 [137], amounts here to|ξ(t)| =|C(t)|≤1, which holds for all t. Thus the minimum achievable trace-distance between the two solutions is given by D min (ρ exact (t),ρ 1 (t)) = 1 2 S(t) q v x (0) 2 +v y (0) 2 . (229) The optimal fit is plotted in Section 4.4. Finding a simple analytical expression for the optimal kernel Eq. (228) seems difficult due to the complicated form of C(t). One way to approach this problem is to expand C(t) in powers of αt and consider terms which give a valid approximation for small times αt≪ 1. For example, Eq. (179) yields the lowest non-trivial order as: C 2 (t) =1−2Q 2 α 2 t 2 +O(α 4 t 4 ). (230) 84 Note that this solution violates the complete positivity condition for times larger than t=1/α √ 2Q 2 . The corresponding kernel is: k 2 (t) =2α 2 Q 2 e 2t cosh(2 p Q 2 αt). (231) Alternatively we could try finding a kernel that matches some of the approximate solutions discussed so far. For example, it turns out that the kernel k NZ2 (t) =2α 2 Q 2 e 2t (232) leads to an exact match of the NZ2 solution. Finding a kernel which gives a good description of the evolution of an open system is an important but in general, difficult question which remains open for further investigation. We note that this question was also taken up in the context of the PM in the recent study [108], where the PM was applied to an exactly solvable model describing a qubit undergoing spontaneous emission and stimulated absorption. No attempt was made to optimize the memory kernel andhencetheagreement with theexact solution was not asimpressiveas might be possible with optimization. 4.4 Comparisonoftheanalyticalsolutionandthedifferent approximation techniques In the results shown below, all figures express the evolution in terms of the dimen- sionless parameterαt (plotted on a logarithmic scale). We choose the initial condition v x (0) =v y (0) =1/ √ 2 and plot onlyv x (t) since the structure of the equations forv x (t) and v y (t) is similar. In order to compare the different methods of approximation, we consider various choices of parameter values in our model. Among these choices we consider both low and high temperaturecases. We note that in a spin bath model it is 85 assumed that the environment degrees of freedom are localized and this is usually the case at low temperatures. At higher temperatures one may need to consider delocal- ized environment degrees of freedom in order to account for such environment modes such as phonons, magnons etc. A class of models known as oscillator bath models (e.g., Ref. [165]) consider such effects. In this study, we restrict attention to the spin bath model described here for both low and high temperatures. 4.4.1 Exact Solution We first assume that the frequencies of the qubits in the bath are equal (Ω n =1,∀n), and so are the coupling constants (g n = 1,∀n). In this regime, we consider large and small numbers of bath spins N = 100 and N = 4, and two different temperatures β = 1 and β = 10. Figs. 1 and 2 show the exact solution for N = 100 and N = 4 spins, respectively, up to the second recurrence time. For each N, we plot the exact solution for β =1 and β =10. We also consider the case where the frequencies Ω n and the coupling constants g n can take different values. We generated uniformly distributed random values in the interval [−1,1] for both Ω n andg n . In Figs. (3) and (4) we plot the ensemble average of the solution over 50 random ensembles. The main difference from the solution with equal Ω n and g n is that the partial recurrences decrease in size, especially as N increases. Weattributethisdampingpartially tothefact that welookat theensemble average, which amounts to averaging out the positive and negative oscillations that arise for different values of the parameters. The main reason, however, is that for a generic ensemble of random Ω n and g n the positive and negative oscillations in the sums (171) tend to average out. This is particularly true for large N, as reflected in Fig. 3. We looked at a few individual random cases forN =100 and recurrences were not present there. For N = 20 (not shown here), some small recurrences were still visible. 86 We also looked at the case where one of the coupling constants, sayg i , has a much larger magnitude than the other ones (which were made equal). The behavior was similar to that for a bath consisting of only a single spin. 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 α t v x (t) β=1 β=10 Figure 1: Comparison of the exact solution at β =1 and β =10 for N =100. In the following, we plot the solutions of different orders of the NZ, TCL and PM master equations and compare them for the same parameter values. 4.4.2 NZ In this subsection, we compare the solutions of different orders of the NZ master equationforΩ n =g n =1. Fig.(5)showsthesolutionstoNZ2,NZ3,NZ4andtheexact solution forβ =1 andβ =10 up to the firstrecurrence time of the exact solution. For short times NZ4 is the better approximation. It can be seen that while NZ2 and NZ3 are bounded, NZ4 leaves the Bloch sphere. But note that the approximations under whichthesesolutionshavebeenobtainedarevalidforαt≪1. TheNZ4solutionleaves the Bloch sphere in a regime where the approximation is not valid. For β = 10, NZ2 87 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 10 5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 α t v x (t) β=1 β=10 Figure 2: Comparison of the exact solution at β =1 and β =10 for N =4. again has a periodic behavior (which is consistent with the solution), while the NZ3 and NZ4 solutions leave the Bloch sphere after small times. Fig. (6) shows the same graphs for N = 4. In this case both NZ3 and NZ4 leave the Bloch sphere for β = 1 andβ =10, while NZ2 has a periodic behavior. A clear conclusion from these plots is that the NZ approximation is truly a short-time one: it becomes completely unreliable for times longer than αt≪1. 4.4.3 TCL Fig. (7) plots the exact solution, TCL2, TCL3 and TCL4 at β = 1 and β = 10 for N = 100 spins and Ω n = g n = 1. It can be seen that for β = 1, the TCL solution approximates the exact solution well even for long times. However, the TCL solution cannot reproduce the recurrence behavior of the exact solution (also shown in the figure.) Fig. (8) shows the same graphs for N = 4. In this case, while TCL2 and TCL3 decay, TCL4 increases exponentially and leaves the Bloch sphere after a short 88 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 α t v x (t) β=1 β=10 Figure 3: Comparison of the exact solution at β = 1 and β = 10 for N = 100 for randomly generated g n and Ω n . time. This is because the exponent in the solution of TCL4 in Eq. (221) is positive. Here again the approximations underwhich the solutions have been obtained are valid only for small time scales and the graphs demonstrate the complete breakdown of the perturbation expansion for large values of αt. Moreover, the graphs reveal the sensitivity of the approximation to temperature: the TCL fares much better at high temperatures. Inordertodeterminethevalidity oftheTCLapproximation, welookat theinvert- ibility of the Kraus map derived in Eq. (164) or equivalently Eq. (171). As mentioned earlier, this map is non-invertible if C(t) 2 +S(t) 2 = 0 for some t (or equivalently v x (t) =0 andv y (t) =0). This will happen if and only if at least one of theβ n is zero. Thiscanoccurwhenthebathdensitymatricesofsomeofthebathspinsaremaximally mixed or in the limit of a very high bath temperature. Clearly, when the Kraus map is non-invertible, the TCL approach becomes invalid since it relies on the assumption 89 10 −3 10 −2 10 −1 10 0 10 1 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 α t v x (t) β=1 β=10 Figure 4: Comparison of the exact solution at β = 1 and β = 10 for N = 4 for randomly generated g n and Ω n . that the information about the initial state is contained in the current state. This fact has also been observed for the spin-boson model with a damped Jaynes-Cummings Hamiltonian [30]. At thepointwheretheKrausmapbecomes non-invertible, theTCL solution deviates from the exact solution (see Fig. 9). We verified that both v x and v y vanish at this point. 4.4.4 NZ, TCL, and PM Inthissubsection,wecomparetheexactsolutiontoTCL4,NZ4andthesolutionofthe optimal PM master equation. Fig. (10) shows these solutions for N = 100 and β = 1 and β = 10 when Ω n = g n = 1. Here we observe that while the short-time behavior of the exact solution is approximated well by all the approximations we consider, the long-time behavior is approximated well only by PM. 90 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 −1 −0.5 0 0.5 1 1.5 α t v x (t) β=1 β=10 Figure 5: Comparison of the exact solution, NZ2, NZ3 and NZ4 at β = 1 and β = 10 for N = 100. The exact solution is the solid (blue) line, NZ2 is the dashed (green) line, NZ3 is the dot-dashed (red) line and NZ4 is the dotted (cyan) line. Forβ =1, NZ4 leaves the Bloch sphere after a short time while TCL4 decays with the exact solution. But as before, the TCL solution cannot reproduce the recurrences seen in the exact solution. The optimal PM solution, by contrast, is capable of repro- ducing both the decay and the recurrences. TCL4 and NZ4 leave the Bloch sphere after a short time forβ =10, while PM again reproduces the recurrences in the exact solution. Fig. 11 shows the corresponding graphs for N = 4 and it can be seen that again PM can outperform both TCL and NZ for long times. Figs. 12 and 13 show the performance of TCL4, NZ4 and PM compared to the exact solution at a fixed time (for which the approximations are valid) for different temperatures (β ∈ [0.01,10]). It can be seen that both TCL4 and the optimal PM solution perform better than NZ4 at medium and high temperatures, with TCL4 outperforming PM at medium temperatures. The performance of NZ4 is enhanced at low temperatures, where it performs similarly to TCL4 (see also Figs. 10 and 11). This can be understood from 91 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 −1 −0.5 0 0.5 1 1.5 α t v x (t) β=1 β=10 Figure 6: Comparison of the exact solution, NZ2, NZ3 and NZ4 at β = 1 and β = 10 for N = 4. The exact solution is the solid (blue) line, NZ2 is the dashed (green) line, NZ3 is the dot-dashed (red) line and NZ4 is the dotted (cyan) line. the short-time approximation to the exact solution given in Eq. (183), which up to the precision for which it was derived is also an approximation of NZ2 [Eq. (192)]. As discussed above, this approximation (which also coincides with TCL2) is valid when 2Q 2 (αt) 2 ≪ 1. As temperaturedecreases, so does the magnitudeofQ 2 , which leads to a better approximation at fixed αt. Since NZ2 gives the lowest-order correction, this improvement is reflected in NZ4 as well. In Figs. 14 and 15 we plot the averaged solutions over 50 ensembles of random values for Ω n and g n in the interval [−1,1]. We see that on average TCL4, NZ4 and the optimal PM solution behave similarly to the case when Ω n =g n = 1. Due to the dampingof therecurrences, especially whenN =100, theTCL4andthePM solutions match the exact solution closely for much longer times than in the deterministic case. Again, the PM solution is capable of qualitatively matching the behavior of the exact solution at long times. 92 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 −1 −0.5 0 0.5 1 1.5 α t v x (t) β=1 β=10 Figure 7: Comparison of the exact solution, TCL2, TCL3 and TCL4 at β = 1 and β = 10 for N = 100. The exact solution is the solid (blue) line, TCL2 is the dashed (green) line, TCL3 is the dot-dashed (red) line and TCL4 is the dotted (cyan) line. Note that for β =1, the curves nearly coincide. 4.4.5 Coarse-graining approximation Finally, we examine the coarse-graining approximation discussed in Section 4.3.1. We choose the time over which the average trace distance is calculated to be the time where the exact solution dies down. In Fig. 16 we plot the coarse-grained solution for the value of τ for which the trace distance to the exact solution is minimum. As can be seen, the coarse-graining approximation does not help since the Markovian assumption is not valid for this model. In deriving the coarse-graining approximation [103] one makes the assumption that thecoarse-graining time scale is greater than any characteristic bath time scale. But the characteristic time scale of the bath is infinite in this case. 93 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 α t v x (t) β=1 β=10 Figure 8: Comparison of the exact solution, TCL2, TCL3 and TCL4 at β = 1 and β = 10 for N = 4. The exact solution is the solid (blue) line, TCL2 is the dashed (green) line, TCL3 is the dot-dashed (red) line and TCL4 is the dotted (cyan) line. Note that for β =1, TCL3, TCL4 and the exact solution nearly coincide. 4.5 Summary and conclusions We studied the performance of various methods for approximating the evolution of an Ising model of an open quantum system for a qubit system coupled to a bath bath consisting of N qubits. The high symmetry of the model allowed us to derive the exact dynamics of the system as well as find analytical solutions for the different master equations. We saw that the Markovian approximation fails for this model due to the time independence of the bath correlation functions. This is also reflected in thefact that thecoarse-graining method[103]doesnot approximatetheexact solution well. We discussed the performance of these solutions for various parameter regimes. Unlike other spin bath models discussed in literature (e.g., Ref. [29]), the odd-order bathcorrelation functionsdonotvanish, leadingtotheexistenceofodd-ordertermsin thesolutionofTCLandNZequations. Thesetermsdescribetherotation aroundthez 94 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 α t v x (t) Figure 9: Comparison of TCL2 and the exact solution to demonstrate the validity of the TCL approximation for N =4 and β =1. The solid (blue) line denotes the exact solution and the dashed (green) line is TCL2. Note that the time axis here is on a linear scale. TCL2 breaks down at αt ≈ 0.9, where it remains flat, while the exact solution has a recurrence. axisoftheBlochsphere,afactwhichisreflectedintheexactsolution. Weshowedthat up to fourth order TCL performs better than NZ at medium and high temperatures. For low temperatures we demonstrated an enhancement in the performanceof NZ and showed that NZ and TCL perform equally well. We showed that the TCL approach breaks down for certain parameter choices and related this to the non-invertibility of the Kraus map describing the system dynamics. We also studied the performance of thepost-Markovian master equation obtained in[137] withanoptimal memorykernel. We discussed possible ways of approximating the optimal kernel for short times and derived the kernel which leads to an exact fit to the NZ2 solution. It turns out that PM master equation performs as well as the TCL2 for a large number of spins and outperforms all orders of NZ and TCL considered here at long times, as it captures the recurrences of the exact solution. 95 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 −1 −0.5 0 0.5 1 1.5 α t v x (t) β=1 β=10 Figure10: Comparisonoftheexact solution, NZ4, TCL4andPMatβ =1andβ =10 forN =100. The exact solution is the solid (blue) line, PM is the dashed (green) line, NZ4 is the dot-dashed (red) line and TCL4 is the dotted (cyan) line. Note that for β =1, TCL4, PM and the exact solution nearly coincide for short and medium times. Only PM captures the recurrences of the exact solution at long times. Our study reveals the limitations of some of the best known master equations available in the literature, in the context of a spin bath. In general, perturbative ap- proaches such as low-order NZ and TCL do well at short times (on a time scale set by the system-bath coupling constant) and fare very poorly at long times. These approx- imations are also very sensitive to temperature and do better in the high temperature limit. The PM does not do as well as TCL4 at short times but has the distinct ad- vantage of retaining a qualitatively correct character for long times. This conclusion depends heavily on the proper choice of the memory kernel; indeed, when the memory kernel is not optimally chosen thePM can yield solutions which are not as satisfactory [108]. 96 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 10 5 −1 −0.5 0 0.5 1 1.5 α t v x (t) β=1 β=10 Figure11: Comparisonoftheexact solution, NZ4, TCL4andPMatβ =1andβ =10 for N = 4. The exact solution is the solid (blue) line, PM is the dashed (green) line, NZ4 is the dot-dashed (red) line and TCL4 is the dotted (cyan) line. Note that for β =1, TCL4 and the exact solution nearly coincide for short and medium times. 4.6 Appendix A: Bath correlation functions Here we show how to calculate the bath correlation functions used in our simulations. The k th order bath correlation function is defined as Q k =Tr{B k ρ B }, 97 10 −2 10 −1 10 0 10 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 β v x (α t=0.1) Figure 12: Comparison of the exact solution, NZ4, TCL4 and PM at αt = 0.1 for N = 100 for different β ∈ [0.01,10]. The exact solution is the solid (blue) line, PM is the dashed (green) line, NZ4 is the dot-dashed (red) line and TCL4 is the dotted (cyan) line. where B andρ B were given in Eqs. (162) and (158), respectively. This yields: Q k = Tr{( X n g n σ z n −θI B ) k X l exp(−βE l ) Z |lihl|} = X l exp(−βE l ) Z hl|( X n g n σ z n −θI B ) k |li = X l,l ′ ,...,l ′′′ exp(−βE l ) Z hl|( X n g n σ z n −θI B )|l ′ ihl ′ |( X n ′ g n ′σ z n ′−θI B )|l ′′ ihl ′′ |··· ···|l ′′′ ihl ′′′ |( X n ′′′ g n ′′′σ z n ′′′−θI B )|li = X l,l ′ ,...,l ′′′ exp(−βE l ) Z ( X n g n hl|σ z n |l ′ i−θ)δ ll ′( X n ′ g n ′hl ′ |σ z n ′|l ′′ i−θ)δ l ′ l ′′··· ···( X n ′′′ g n ′′′hl ′′′ |σ z n ′′′|li−θ)δ l ′′′ l = X l exp(−βE l ) Z ( X n g n hl|σ z n |li−θ)( X n ′ g n ′hl|σ z n ′|li−θ)··· ···( X n ′′′ g n ′′′hl|σ z n ′′′|li−θ) = X l exp(−βE l ) Z ( X n g n hl|σ z n |li−θ) k , 98 10 −2 10 −1 10 0 10 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 β v x (α t=0.5) Figure 13: Comparison of the exact solution, NZ4, TCL4 and PM at αt = 0.5 for N =4 for differentβ∈[0.01,10]. The exact solution is the solid (blue) line, PM is the dashed (green) line, NZ4 is the dot-dashed (red) line and TCL4 is the dotted (cyan) line. or Q k = 1 Z X l ( ˜ E l ) k exp(−βE l ), (233) where Z = P l exp(−βE l ) and the expressions for E l and ˜ E l were given in Eqs. (167) and (169), respectively. The above formulas are useful when the energy levels E l and ˜ E l are highly degen- erate, which is the case for example when g n ≡g and Ω n ≡ Ω for all n. For a general choice of these parameters, it is computationally more efficient to consider θ in the 99 10 −3 10 −2 10 −1 10 0 10 1 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 α t v x (t) β=1 β=10 Figure14: Comparisonoftheexact solution, NZ4, TCL4andPMatβ =1andβ =10 for N = 100 for random values of g n and Ω n . The exact solution is the solid (blue) line, PM is the dashed (green) line, NZ4 is the dot-dashed (red) line and TCL4 is the dotted (cyan) line. Note that forβ =1 andβ =10, TCL4, PM and the exact solution nearly coincide. form (174) and the initial bath density matrix in the form (172). For example, the second order bath correlation function is Q 2 = Tr{( N X m=1 g m σ z m −θI)( N X n=1 g n σ z n −θI)ρ B } = Tr{ N X n,m=1 g n g m σ z n σ z m ρ B }−2θTr{ N X n=1 g n σ z n ρ B } | {z } θ +θ 2 = Tr{ N X n,m=1 g n g m σ z n σ z m N O n=1 1 2 (I +β n σ z n )}−θ 2 = N X n6=m Tr{g m 1 2 (σ z m +β m I)}Tr{g n 1 2 (σ z n +β n I)} Y j6=m,n Tr{ 1 2 (I +β j σ z j )} +Tr{ N X n=1 g 2 n ρ B }−θ 2 = N X n,m=1 g m β m g n β n | {z } θ 2 − N X n=1 g 2 n β 2 n + N X n=1 g 2 n −θ 2 = N X n=1 g 2 n (1−β 2 n ). (234) 100 10 −3 10 −2 10 −1 10 0 10 1 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 α t v x (t) β=1 β=10 Figure15: Comparisonoftheexact solution, NZ4, TCL4andPMatβ =1andβ =10 for N =4 for random values of g n and Ω n . The exact solution is the solid (blue) line, PMisthedashed(green) line, NZ4isthedot-dashed(red)lineandTCL4isthedotted (cyan) line. Note that forβ =1, TCL4, PM and the exact solution nearly coincide for short and medium times. Using the identity 1−tanh 2 (−x/2) = 2/(1+coshx), this correlation function can be expressed in terms of the bath spectral density function [Eq. (159)] as follows: Q 2 = N X n=1 g 2 n (1−β 2 n ) = Z ∞ −∞ δ(Ω−Ω n )|g n | 2 (1−tanh 2 (− Ω 2kT ))dΩ = Z ∞ −∞ 2J(Ω)dΩ 1+cosh( Ω kT ) . Higher order correlation functions are computed analogously. 101 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 α t v x (t) Figure 16: Comparison of the exact solution and the optimal coarse-graining approx- imation for N = 50 and β = 1. The exact solution is the solid (blue) line and the coarse-graining approximation is the dashed (green) line. Note the linear scale time axis. 4.7 Appendix B: Cumulants for the NZ and TCL master equations We calculate theexplicit expressionsfor thecumulants appearingin Eq. (215), needed to find the NZ and TCL perturbation expansions up to fourth order. Second order: hL 2 iρ = −Tr B {[H I ,[H I ,ρ]]}⊗ρ B = −Tr B {H 2 I ρ−2H I ρH I +ρH 2 I }⊗ρ B = −2Q 2 (ρ S −σ z ρ S σ z )⊗ρ B ≡ ρ ′ , (235) 102 hL 2 i 2 ρ = PL 2 PPL 2 Pρ = PL 2 Pρ ′ = −2Q 2 (ρ ′ S −σ z ρ ′ S σ z )⊗ρ B , where ρ ′ S =Tr B ρ ′ =−2Q 2 (ρ S −σ z ρ S σ z ). Therefore hL 2 i 2 ρ = −2Q 2 {(−2Q 2 (ρ S −σ z ρ S σ z ))−σ z (−2Q 2 (ρ S −σ z ρ S σ z ))σ z }⊗ρ B = 8Q 2 2 (ρ S −σ z ρ S σ z )⊗ρ B . (236) Third order: hL 3 iρ = iTr B {[H I ,[H I ,[H I ,ρ]]]}⊗ρ B = iTr B {H 3 I ρ−3H 2 I ρH I +3H I ρH 2 I −ρH 3 I }⊗ρ B = 4iQ 3 (σ z ρ S −ρ S σ z )⊗ρ B . (237) Fourth order: hL 4 iρ = Tr B {[H I ,[H I ,[H I ,[H I ,ρ]]]]}⊗ρ B = Tr B {H 4 I ρ−4H 3 I ρH I +6H 2 I ρH 2 I −4H I ρH 3 I +ρH 4 I }⊗ρ B = 8Q 4 (ρ S −σ z ρ S σ z )⊗ρ B . (238) 103 Chapter 5: Continuous quantum error correction for non-Markovian decoherence In this chapter we continue our exploration of non-Markovian decoherence. This time, wecompare Markovian andnon-Markovian error modelsin light of theperformanceof continuous quantum error correction. We consider again an Ising decoherence model of the type we studied in the previous chapter, but in a much simpler version—when the environment consists of only a single qubit. This allows us to solve exactly the evolution of a multi-qubit code in which each qubit is coupled to an independent bath when the code is subject to continuous error correction. The conclusions we obtain, however, extend beyond this model and apply for general non-Markovian decoherence. 5.1 Preliminaries 5.1.1 Continuous quantum error correction Ingeneral, errorprobabilitiesincreasewithtime. Nomatterhowcomplicatedacodeor how many levels of concatenation are involved, the probability of uncorrectable errors isnevertrulyzero, andifthesystemisexposedtonoiseforasufficientlylongtime, the weight of uncorrectable errors can accumulate. To combat this, error correction must be applied repeatedly and sufficiently often. If one assumes that the time for an error- correcting operation is small compared to other relevant time scales of the system, error-correcting operations can be considered instantaneous. Then the scenario of 104 repeated error correction leads to a discrete evolution which often may be difficult to describe. To study the evolution of a system in the limit of frequently applied instantaneous error correction, Paz and Zurek proposed to describe error correction as a continuous quantum jump process [124]. In this model, the infinitesimal error- correcting transformation that the density matrix of the encoded system undergoes during a time step dt is ρ→(1−κdt)ρ+κdtΦ(ρ), (239) where Φ(ρ) is the completely positive trace-preserving (CPTP) map describing a full error-correcting operation, andκ is the error-correction rate. The full error-correcting operation Φ(ρ) consists of a syndrome detection, followed (if necessary) by a unitary correction operation conditioned on the syndrome. Consider, for example, the three-qubit bit-flip code whose purpose is to protect an unknown qubit state from bit-flip (Pauli X) errors. The code space is spanned by |0i=|000iand|1i=|111i,andthestabilizergeneratorsareZZI andIZZ (seeSection 8.3). Here byX =σ x ,Y =σ y ,Z =σ z andI we denote the usual Pauli operators and the identity, respectively, and a string of three operators represents the tensor product of operators on each of the three qubits. The standard error-correction procedure involves a measurement of the stabilizer generators, which projects the state onto one of the subspaces spanned by|000i and|111i,|100i and|011i,|010i and|101i, or|001i and|110i; the outcome of these measurements is the error syndrome. Assuming that the probability for two- or three-qubit errors is negligible, then with high probability the result of this measurement is either the original state with no errors, or with a singleX error on the first, the second, or the third qubit. Depending on the outcome, 105 one then applies an X gate to the erroneous qubit and transforms the state back to the original one. The CPTP map Φ(ρ) for this code can be written explicitly as Φ(ρ) =(|000ih000|+|111ih111|)ρ(|000ih000|+|111ih111|) +(|000ih100|+|111ih011|)ρ(|100ih000|+|011ih111|) +(|000ih010|+|111ih101|)ρ(|010ih000|+|101ih111|) +(|000ih001|+|111ih110|)ρ(|001ih000|+|110ih111|). (240) The quantum-jump process (239) can be viewed as a smoothed version of the dis- crete scenario of repeated error correction, in which instantaneous full error-correcting operations are applied at random times with rate κ. It can also be looked upon as arising from a continuous sequence of infinitesimal CPTP maps of the type (239). In practice, such a weak map is never truly infinitesimal, but rather has the form ρ→(1−ǫ 2 )ρ+ǫ 2 Φ(ρ), (241) whereǫ≪1 is a small but finite parameter, and the weak operation takes a small but nonzero time τ c . For times t much greater than τ c (τ c ≪t), the weak error-correcting map(241) iswell approximated bytheinfinitesimalform(239), wheretherateoferror correction is κ=ǫ 2 /τ c . (242) A weak map of the form (241) could beimplemented, for example, by a weak coupling between the system and an ancilla via an appropriate Hamiltonian, followed by dis- cardingthe ancilla. A closely related scenario, wherethe ancilla is continuously cooled in order to reset it to its initial state, was studied in [135]. Another way of implementing the weak map (241) is via weak measurements fol- lowed by weak unitaries dependent on the outcome. In the appendix at the end of 106 this chapter, we give an example of such an implementation for the case of the bit- flip code—when Φ(ρ) is given by (240). We also present a scheme in terms of weak measurements for codes that correct arbitrary single-qubit errors. In the latter case, the resulting weak map is not the same as (241), but also yields the strong error- correcting map Φ(ρ) when exponentiated. We point out that the weak measurements used in these schemes are not weak versions of the strong measurements for syndrome detection—they are in a different basis. They can be regarded as weak versions of a different set of strong measurements which, when followed by an appropriate unitary, yield the same map Φ(ρ) on average. Thus, the workings of continuous error correc- tion, whenitisdrivenbyweakmeasurements, doesnottranslatedirectlyintotheerror syndromedetection andcorrection ofthestandardparadigm. Inthissense, thecontin- uousapproach can beregardedas a different paradigm forerror correction—one based on weak measurements and weak unitary operations. The idea of using continuous weak measurements and unitary operations for error correction has been explored in the context of different heuristic schemes [7, 134], some of which are based on a direct “continuization” of the syndromemeasurements. In this study we consider continuous error correction of the type given by Eq. (239). 5.1.2 Markovian decoherence So far, continuous quantum error correction has been studied only for Markovian er- ror models. As we discussed in the previous chapter, the Markovian approximation describes situations where the bath correlation times are much shorter than any char- acteristic time scale of the system [30]. In this limit, the dynamics can be described by a semi-group master equation in the Lindblad form [105]: dρ dt =L(ρ)≡−i[H,ρ]+ 1 2 X j λ j (2L j ρL † j −L † j L j ρ−ρL † j L j ). (243) 107 Here H is the system Hamiltonian and the {L j } are suitably normalized Lindblad operators describing different error channels with decoherence rates λ j . For example, the Liouvillian L(ρ) = X j λ j (X j ρX j −ρ), (244) whereX j denotes a local bit-flip operator acting on thej-th qubit, describes indepen- dent Markovian bit-flip errors. For a system undergoing Markovian decoherence and error correction of the type (239), the evolution is given by the equation dρ dt =L(ρ)+κΓ(ρ), (245) where Γ(ρ) = Φ(ρ)−ρ. In [124], Paz and Zurek showed that if the set of errors{L j } are correctable by the code, in the limit of infinite error-correction rate (strong error- correcting operations applied continuously often) the state of thesystem freezes and is protected fromerrorsat all times. Theeffect offreezingcan beunderstoodbynoticing that the transformation arising from decoherence during a short time step Δt, is ρ→ρ+L(ρ)Δt+O(Δt 2 ), (246) i.e., the weight of correctable errors emerging during this time interval is proportional to Δt, whereas uncorrectable errors (e.g., multi-qubit bit flips in the case of the three- qubit bit-flip code) are of order O(Δt 2 ). Thus, if errors are constantly corrected, in the limit Δt→0 uncorrectable errors cannot accumulate, and the evolution stops. 5.1.3 The Zeno effect. Error correction versus error prevention The effect of “freezing” in continuous error correction strongly resembles the quantum Zenoeffect[110], inwhichfrequentmeasurementsslowdowntheevolution ofasystem, 108 freezing the state in the limit where they are applied continuously. The Zeno effect arises when the system and its environment are initially decoupled and they undergo a Hamiltonian-driven evolution, which leads to a quadratic change with time of the state duringtheinitial moments [112] (the socalled Zenoregime). Let the initial state of the system plus the bath beρ SB (0) =|0ih0| S ⊗ρ B (0). For small times, the fidelity of the system’s density matrix with the initial state α(t) = Tr{(|0ih0| S ⊗I B )ρ SB (t)} can be approximated as α(t) =1−Ct 2 +O(t 3 ). (247) In terms of the Hamiltonian H SB acting on the entire system, the coefficient C is C =Tr H 2 SB (|0ih0| S ⊗ρ B (0)) −Tr{H SB (|0ih0| S ⊗I B )H SB (|0ih0| S ⊗ρ B (0))}. (248) According to Eq. (247), if after a short time step Δt the system is measured in an orthogonal basis which includes the initial state |0i, the probability to find the system in a state other than the initial state is of order O(Δt 2 ). Thus if the state is continuously measured (Δt→0), this prevents the system from evolving. IthasbeenproposedtoutilizethequantumZenoeffectinschemesforerrorpreven- tion [178, 15, 156], in whichan unknownencoded state is prevented fromerrorssimply byfrequentmeasurementswhichkeepit insidethecodespace. Theapproachissimilar to errorcorrection inthat theerrors for whichthe codeis designed sendacodeword to a space orthogonal to the code space. The difference is that different errors need not be distinguishable, since the procedure does not involve correction of errors, but their prevention. In [156] it was shown that with this approach it is possible to use codes of smaller redundancy than those needed for error correction and a four-qubit encoding of a qubit was proposed, which is capable of preventing arbitrary independent errors arising from Hamiltonian interactions. The possibility of this approach implicitly as- sumes the existence of a Zeno regime, and fails if we assume Markovian decoherence 109 for all times. This is because the probability of errors emerging during a time step dt in a Markovian model is proportional to dt (rather than dt 2 ), and hence errors will accumulate with time if not corrected. From the above observations we see that error correction is capable of achieving resultsinnoiseregimeswhereerrorprevention fails. Ofcourse, thisadvantage isat the expense of a more complicated procedure—in addition to the measurements used in errorprevention, errorcorrection involves unitarycorrectionoperations, andingeneral requires codes with higher redundancy. At the same time, we see that in the Zeno regime it is possible to reduce decoherence using weaker resources than those needed in the case of Markovian noise. This suggests that in this regime error correction may exhibit higher performance than it does for Markovian decoherence. 5.1.4 Non-Markovian decoherence Markovian decoherenceisanapproximationvalidfortimesmuchlargerthanthemem- oryoftheenvironment. Aswesawinthepreviouschapter,however,inmanysituations of practical significance the memory of the environment cannot be neglected and the evolution is highly non-Markovian. Furthermore, no evolution is strictly Markovian, and for a system initially decoupled from its environment a Zeno regime is always present, short though it may be [112]. If the time resolution of error-correcting opera- tionsishighenoughsothatthey“see”theZenoregime, thiscouldgiverisetodifferent behavior. Theexistence ofaZenoregimeisnottheonlyinterestingfeatureofnon-Markovian decoherence. Themechanism by which errors accumulate in a general Hamiltonian in- teraction with the environment may differ significantly from the Markovian case, since the system may develop nontrivial correlations with the environment. For example, imagine that sometimeafter theinitial encodingof asystem, astrongerror-correcting 110 operation is applied. This brings the state inside the code space, but the state con- tainsanonzeroportionoferrorsnon-distinguishablebythecode. Thusthenewstateis mixedandisgenerally correlated withtheenvironment. Asubsequenterror-correcting operation can only aim at correcting errors arising after this point, since the errors al- ready present inside the code space are in principle uncorrectable. Subsequent errors onthedensity matrix, however, maynot becompletely positive duetothecorrelations with the environment. Nevertheless, it follows from a result in [139] that an error-correction procedure which is capable of correcting a certain class of completely positive (CP) maps, can also correct any linear noise map whose operator elements can be expressed as linear combinations of the operator elements in a correctable CP map. This implies, in particular, that an error-correction procedure that can correct arbitrary single-qubit CP maps can correct arbitrary single-qubit linear maps. In this context, we note that theeffectsofsystem-environmentcorrelationsinnon-Markovianerrormodelshavealso been studied from the perspective of fault tolerance, and it has been shown that the thresholdtheoremcanbeextendedtovarioustypesofnon-Markoviannoise[154,12,3]. Another important difference from the Markovian case is that error correction and the effective noise on the reduced density matrix of the system cannot be treated as independent processes. One could derive an equation for the effective evolution of the system alone subject to interaction with the environment, like the Nakajima- Zwanzig [111, 179] or the time-convolutionless (TCL) [141, 140] master equations, but thegeneratoroftransformationsatagivenmomentingeneralwilldepend(implicitlyor explicitly) on the entire history up to this moment. Therefore, adding error correction can nontrivially affect the effective error model. This means that in studying the performanceof continuous error correction one either has to derivean equation for the effective evolution of the encoded system, taking into account error correction from the very beginning, or one has to look at the evolution of the entire system—including 111 the bath—where the error generator and the generator of error correction can be considered independent. In the latter case, for sufficiently small τ c , the evolution of the entire system including the bath can be described by dρ dt =−i[H,ρ]+κΓ(ρ), (249) where ρ is the density matrix of the system plus bath, H is the total Hamiltonian, and the error-correction generator Γ acts locally on the encoded system. In this study, we take this approach for a sufficiently simple bath model which allows us to find a solution for the evolution of the entire system. 5.1.5 Plan of this chapter The rest of the chapter is organized as follows. To develop understandingof the work- ings of continuous error correction, in Section 5.2 we look at a simple example: an error-correction code consisting of only one qubit which aims at protecting a known state. We discuss the difference in performance for Markovian and non-Markovian decoherence, and argue the implications it has for the case of multi-qubit codes. In Section 5.3, we study the three-qubit bit-flip code. We first review the performance of continuous error correction in the case of Markovian bit-flip decoherence, which was first studied in [124]. We then consider a non-Markovian model, where each qubit in the code is coupled to an independent bath qubit. This model is a simple version of the one studied in the previous chapter, and it allows us to solve for its evolution ana- lytically. In the limit of large error-correction rates, the effective evolution approaches the evolution of a single qubit without error correction, but the coupling strength is now decreased by a factor which scales quadratically with the error-correction rate. This is opposed to the case of Markovian decoherence, where the same factor scales linearly with the rate of error-correction. In Section 5.4, we show that the quadratic 112 enhancement in the performance over the case of Markovian noise can be attributed to the presence of a Zeno regime and argue that for general stabilizer codes and inde- pendent errors, the performanceof continuous error correction would exhibit the same qualitative characteristics. InSection 5.5, weconclude. IntheAppendix(Section 5.6), we present an implementation of the quantum-jump error correcting model that uses weak measurements and weak unitary operations. 5.2 The single-qubit code Consider the problem of protecting a qubit in state |0i from bit-flip errors. This problem can be regarded as a trivial example of a stabilizer code, where the code space is spanned by|0i and its stabilizer is Z. Let us consider the Markovian bit-flip model first. The evolution of the state subject to bit-flip errors and error correction is described by Eq. (245) with L(ρ) =λ(XρX−ρ), (250) and Γ(ρ)=|0ih0|ρ|0ih0|+|0ih1|ρ|1ih0|−ρ. (251) If the state lies on the z-axis of the Bloch sphere, it will never leave it, since both the noise generator (250) and the error-correction generator (251) keep it on the axis. We will take the qubit to be initially in the desired state |0i, and therefore at any later moment it will have the form ρ(t) = α(t)|0ih0| +(1−α(t))|1ih1|, α(t) ∈ [0,1]. The coefficient α(t) has the interpretation of a fidelity with the trivial code space spanned by|0i. For an infinitesimal time step dt, the effect of the noise is to decrease α(t) by 113 the amount λ(2α(t)− 1)dt and that of the correcting operation is to increase it by κ(1−α(t))dt. The net evolution is then described by dα(t) dt =−(κ+2λ)α(t)+(κ+λ). (252) The solution is α(t) =(1−α M ∗ )e −(κ+2λ)t +α M ∗ , (253) where α M ∗ =1− 1 2+r , (254) and r =κ/λ is the ratio between the rate of error correction and the rate of decoher- ence. We see that thefidelity decays, butit is confinedabove its asymptotic valueα M ∗ , which can be made arbitrarily close to 1 for a sufficiently large r. Now let us consider a non-Markovian error model. We choose the simple scenario where the system is coupled to a single bath qubit via the Hamiltonian H =γX⊗X, (255) whereγ is the coupling strength. This is the Ising Hamiltonian (147) for the case of a single bath qubit, but in the basis|+i= |0i+|1i √ 2 ,|−i= |0i−|1i √ 2 . As we noted in Chapter 4 (Section 4.1), the model of a single bath qubit can be a good approximation for situations in which the coupling to a single spin from the bath dominates over other interactions. We willassumethat thebathqubitisinitially inthemaximally mixedstate, which can be thought of as an equilibrium state at high temperature. From Eq. (249) one 114 can verify that if the system is initially in the state |0i, the state of the system plus the bath at any moment will have the form ρ(t) =(α(t)|0ih0|+(1−α(t))|1ih1|)⊗ I 2 −β(t)Y ⊗ X 2 . (256) In the tensor product, the first operator belongs to the Hilbert space of the system and the second to the Hilbert space of the bath. We have α(t) ∈ [0,1], and |β(t)| ≤ p α(t)(1−α(t)),β(t) ∈ R. The reduced density matrix of the system has the same formastheonefortheMarkovian case. Thetraceless termproportionaltoβ(t)can be thought of as a “hidden”part, which nevertheless plays an important role in theerror- creation process, since errors can be thought of as being transferred to the “visible” part from the “hidden” part (and vice versa). This can be seen from the fact that during an infinitesimal time stepdt, the Hamiltonian changes the parametersα andβ as follows: α→α−2βγdt, β→β+(2α−1)γdt. (257) The effect of an infinitesimal error-correcting operation is α→α+(1−α)κdt, β→β−βκdt. (258) 115 Note that the hidden part is also being acted upon. Putting it all together, we get the system of equations dα(t) dt =κ(1−α(t))−2γβ(t), dβ(t) dt =γ(2α−1)−κβ(t). (259) The solution for the fidelity α(t) is α(t) = 2γ 2 +κ 2 4γ 2 +κ 2 +e −κt κγ 4γ 2 +κ 2 sin2γt+ 2γ 2 4γ 2 +κ 2 cos2γt . (260) We see that as time increases, the fidelity stabilizes at the value α NM ∗ = 2+R 2 4+R 2 =1− 2 4+R 2 , (261) whereR =κ/γ istheratiobetweentheerror-correctionrateandthecouplingstrength. In Fig. 1 we have plotted the fidelity as a function of the dimensionless parameter γt for three different values of R. For error-correction rates comparable to the coupling strength (R = 1), the fidelity undergoes a few partial recurrences before it stabilizes close to α NM ∗ . For larger R = 2, however, the oscillations are already heavily damped and for R = 5 the fidelity seems confined above α NM ∗ . As R increases, the evolution becomes closer to a decay like the one in the Markovian case. A remarkable difference, however, is that the asymptotic weight outside the code space (1−α NM ∗ ) decreases with κ as 1/κ 2 , whereas in the Markovian case the same quantity decreases as 1/κ. The asymptotic value can be obtained as an equilibrium pointatwhichtheinfinitesimalweight flowingoutofthecodespaceduringatimestep dtisequaltotheweight flowingintoit. Thelatter correspondstovanishingright-hand sides in Eqs. (252) and (259). In Section 5.4, we will show that the difference in the 116 1 2 3 4 Γt 0.4 0.6 0.7 0.8 0.9 1 Α R=5 R=2 R=1 Figure 17: Fidelity of the single-qubit code with continuous bit-flip errors and cor- rection, as a function of dimensionless time γt, for three different values of the ratio R =κ/γ. equilibrium code-space fidelity for the two different types of decoherence arises from the difference in the corresponding evolutions during initial times. For multi-qubit codes, error correction cannot preserve a high fidelity with the initial codeword for all times, because there will be multi-qubit errors that can lead to errors within the code space itself. But it is natural to expect that the code-space fidelity can be kept above a certain value, since the effect of the error-correcting map (239) istoopposeits decrease. Ifsimilarly tothesingle-qubit codethereis aquadratic difference in the code-space fidelity for the cases of Markovian and non-Markovian decoherence, this could lead to a different performance of the error-correction scheme with respect to the rate of accumulation of uncorrectable errors inside the code space. This is because multi-qubit errors that can lead to transformations entirely within the code space during a time step dt are of order O(dt 2 ). This means that if the state is kept constantly inside the code space (as in the limit of an infinite error- correctionrate), uncorrectableerrorswillneverdevelop. Butifthereisafinitenonzero portion of correctable errors, by the error mechanism it will give rise to errors not 117 distinguishable or misinterpreted by the code. Therefore, the weight outside the code space can be thought of as responsible for the accumulation of uncorrectable errors, and consequently a difference in its magnitude may lead to a difference in the overall performance. In the following sections we will see that this is indeed the case. 5.3 The three-qubit bit-flip code 5.3.1 A Markovian error model Even though the three-qubit bit-flip code can correct only bit-flip errors, it captures most of the important characteristics of nontrivial stabilizer codes. Before we look at anon-Markovian model, wewillreview theMarkovian case whichwas studiedin[124]. Let the system decohere through identical independent bit-flip channels, i.e., L(ρ) is of the form (244) withλ 1 =λ 2 =λ 3 =λ. Then one can verify that the density matrix at any moment can be written as ρ(t)=a(t)ρ(0)+b(t)ρ 1 +c(t)ρ 2 +d(t)ρ 3 , (262) where ρ 1 = 1 3 (X 1 ρ(0)X 1 +X 2 ρ(0)X 2 +X 3 ρ(0)X 3 ), ρ 2 = 1 3 (X 1 X 2 ρ(0)X 1 X 2 +X 2 X 3 ρ(0)X 2 X 3 +X 1 X 3 ρ(0)X 1 X 3 ), (263) ρ 3 =X 1 X 2 X 3 ρ(0)X 1 X 2 X 3 , are equally-weighted mixtures of single-qubit, two-qubit and three-qubit errors on the original state. 118 The effect of decoherence for a single time step dt is equivalent to the following transformation of the coefficients in Eq. (262): a→a−3aλdt+bλdt, b→b+3aλdt−3bλdt+2cλdt, c→c+2bλdt−3cλdt+3dλdt, d→d+cλdt−3dλdt. (264) If the system is initially inside the code space, combining Eq. (264) with the effect of the weak error-correcting map ρ → (1−κdt)ρ+κdtΦ(ρ), where Φ(ρ) is given in Eq. (240), yields thefollowing system of first-orderlinear differential equations for the evolution of the system subject to decoherence plus error correction: da(t) dt =−3λa(t)+(λ+κ)b(t), db(t) dt =3λa(t)−(3λ+κ)b(t)+2λc(t), dc(t) dt =2λb(t)−(3λ+κ)c(t)+3λd(t), dd(t) dt =(λ+κ)c(t)−3λd(t). (265) The exact solution has been found in [124]. Here we just note that for the initial conditions a(0) = 1,b(0) =c(0) =d(0) = 0, the exact solution for the weight outside the code space is b(t)+c(t) = 3 4+r (1−e −(4+r)λt ), (266) where r =κ/λ. We see that similarly to what we obtained for the trivial code in the previous section, the weight outside the code space quickly decays to its asymptotic value 3 4+r which scales as 1/r. But note that here the asymptotic value is roughly three times greater than that for the single-qubit model. This corresponds to the fact that there are three single-qubit channels. More precisely, it can be verified that if 119 for a given κ the uncorrected weight by the single-qubit scheme is small, then the uncorrected weight by a multi-qubit code using the same κ and the same kind of decoherence for each qubit scales approximately linearly with the number of qubits. Similarly, the ratio r required to preserve a given overlap with the code space scales linearly with the number of qubits in the code. The most important difference from the single-qubit model is that in this model there are uncorrectable errors that cause a decay of the state’s fidelity inside the code space. Due to the finiteness of the resources employed by our scheme, there always remains a nonzero portion of the state outside the code space, which gives rise to uncorrectable three-qubit errors. To understand how the state decays inside the code space, we ignore the terms of the order of the weight outside the code space in the exact solution. We obtain: a(t)≈ 1+e − 6 r 2λt 2 ≈1−d(t), (267) b(t)≈c(t)≈0. (268) Comparing this solution to the expression for the fidelity of a single decaying qubit without error correction—which can be seen from Eq. (253) for κ = 0—we see that the encoded qubit decays roughly as if subject to bit-flip decoherence with rate 6λ/r. Therefore, for large r this error-correction scheme can reduce the rate of decoherence approximately r/6 times. In the limit r → ∞, it leads to perfect protection of the state for all times. 5.3.2 A non-Markovian error model We consider a model where each qubit independently undergoes the same kind of non-Markovian decoherence as the one we studied for the single-qubit code. Here the system we look at consists of six qubits - three for the codeword and three for the 120 environment. We assume that all system qubits are coupled to their corresponding environment qubits with the same coupling strength, i.e., the Hamiltonian is H =γ 3 X i=1 X S i ⊗X B i , (269) where the operators X S act on the system qubits and X B act on the corresponding bath qubits. The subscripts label the particular qubit on which they act. Obviously, the types of effective single-qubit errors on the density matrix of the system that can result from this Hamiltonian at any time, whether they are CP or not, will have operator elements whicharelinear combinations ofI andX S , i.e., theyarecorrectable by the procedure according to [139]. Considering the forms of the Hamiltonian (372) and the error-correcting map (240), one can see that the density matrix of the entire system at any moment is a linear combination of terms of the following type: ̺ lmn,pqr ≡X l 1 X m 2 X n 3 ρ(0)X p 1 X q 2 X r 3 ⊗ X l+p 1 2 ⊗ X m+q 2 2 ⊗ X n+r 3 2 . (270) Here the first term in the tensor product refers to the Hilbert space of the system, and the following three refer to the Hilbert spaces of the bath qubits that couple to the first, second and third qubits from the code, respectively. The powers l,m,n,p,q,r take values 0 and 1 in all possible combinations, and X 1 = X, X 0 = X 2 = I. Note that ̺ lmn,pqr should not be mistaken for the components of the density matrix in the computational basis. Collecting these together, we can write the density matrix in the form ρ(t) = X l,m,n,p,q,r (−i) l+m+n (i) p+q+r C lmn,pqr (t)×̺ lmn,pqr , (271) where the coefficients C lmn,pqr (t) are real. The coefficient C 000,000 is less than or equal to the codeword fidelity (with equality whenρ(0) =| ¯ 0ih ¯ 0| or ρ(0) =| ¯ 1ih ¯ 1|). Since the 121 schemeisintendedtoprotectanunknowncodeword,weareinterestedinitsworst-case performance; we will therefore useC 000,000 as a lower bound on the codeword fidelity. Usingthesymmetrywithrespecttopermutationsofthedifferentsystem-bathpairs of qubits and the Hermiticity of the density matrix, we can reduce the description of the evolution to a system of equations for only 13 of the 64 coefficients. (In fact, 12 coefficients aresufficient ifweinvoke thenormalization conditionTrρ=1, butwehave found it more convenient to work with 13.) The equations are linear, and we write them as a single 13-dimensional vector equation: d dt C 000,000 C 100,000 C 110,000 C 100,010 C 100,100 C 110,001 C 111,000 C 110,100 C 110,110 C 110,011 C 111,100 C 111,110 C 111,111 =γ 0 −6 0 0 3R 0 0 0 0 0 0 0 0 1 −R −2 −2 −1 0 0 0 0 0 0 0 0 0 2 −R 0 0 −1 −1 −2 0 0 0 0 0 0 2 0 −R 0 −2 0 −2 0 0 0 0 0 0 2 0 0 −R 0 0 −4 0 0 0 0 0 0 0 1 2 0 −R 0 0 0 −2 −1 0 0 0 0 3 0 0 −3R 0 0 0 0 −3 0 0 0 0 1 1 1 0 0 −R −1 −1 −1 0 0 0 0 0 0 0 0 0 4 −R 0 0 −2 0 0 0 0 0 0 2 0 2 0 −R 0 −2 0 0 0 0 0 0 1 1 2 0 0 −R −2 0 0 0 0 0 0 0 0 0 1 2 2 −R −1 0 0 0 0 0 0 0 0 3R 0 0 6 0 · C 000,000 C 100,000 C 110,000 C 100,010 C 100,100 C 110,001 C 111,000 C 110,100 C 110,110 C 110,011 C 111,100 C 111,110 C 111,111 (272) where R = κ/γ. Each nonzero component in this matrix represents an allowed tran- sition process for the quantum states; these transitions can be driven either by the decoherence process or the continuous error-correction process. We plot these allowed transitions in Fig. 2. We can use the symmetries of the process to recover the 64 coefficients of the full state. Eachofthe13coefficientsrepresentsasetofcoefficientshavingthesamenumber of 1s on the left and the same number of 1s on the right, as well as the same number of places which have 1 on both sides. All such coefficients are equal at all times. For example, the coefficient C 110,011 is equal to all coefficients with two 1s on the left, two 122 Figure 18: These are the allowed transitions between the different components of the system (272) and their rates, arising from both the decoherence (bit-flip) process (with rate γ) and the continuous error-correction process (with rate κ). Online, the transitions due to decoherence are black, and the transitions due to error correction are red. 123 1s on the right and exactly one place with 1 on both sides; there are exactly six such coefficients: C 110,011 =C 110,101 =C 101,011 =C 101,110 =C 011,110 =C 011,101 . In determining the transfer rate from one coefficient to another in Fig. 2, one has to take into account the number of different coefficients of the first type which can make a transition to a coefficient of the second type of orderdt according to Eq. (249). The sign of the flow is determined from the phases in front of the coefficients in Eq. (271). The eigenvalues of the matrix in Eq. (272) up to the first two lowest orders in 1/κ are presented in Table I. Table 1: Eigenvalues of the matrix Eigenvalues λ 0 =0 λ 1,2 =−κ λ 3,4 =−κ±i2γ λ 5,6 =−κ±i4γ λ 7,8 =−κ±i( √ 13+3)γ +O(1/κ) λ 9,10 =−κ±i( √ 13−3)γ +O(1/κ) λ 11,12 =±i(24/R 2 )γ−(144/R 3 )γ+O(1/κ 4 ) Obviously all eigenvalues except the first one and the last two describe fast decays with rates ∼ κ. They correspond to terms in the solution which will vanish quickly after the beginning of the evolution. The eigenvalue 0 corresponds to the asymptotic (t→∞) solution, since all other terms will eventually decay. The last two eigenvalues are those that play the main role in the evolution on a time scale t≫ 1 κ . We see that 124 on such a time scale, the solution will contain an oscillation with an angular frequency approximately equal to (24/R 2 )γ which is damped by a decay factor with a rate of approximately(144/R 3 )γ. InFig.3wehaveplottedthecodewordfidelityC 000,000 (t)as afunctionofthedimensionlessparameterγtforR =100. Thegraphindeedrepresents this type of behavior, except for very short times after thebeginning(γt∼0.1), where one can see a fast but small in magnitude decay (Fig. 4). The maximum magnitude of this quickly decaying term obviously decreases withR, since in the limit ofR→∞ the fidelity should remain constantly equal to 1. 500 1000 1500 2000 2500 3000 Γ t 0.2 0.4 0.6 0.8 1 C 000,000 Figure19: Long-timebehaviorofthree-qubitsystemwithbit-flipnoiseandcontinuous error correction. The ratio of correction rate to decoherence rate is R =κ/γ =100. From the form of the eigenvalues one can see that as R increases, the frequency of themainoscillation decreasesas1/R 2 whiletherateofdecaydecreasesfaster, as1/R 3 . Thus in the limit R → ∞, the evolution approaches an oscillation with an angular frequency(24/R 2 )γ. (We formulatethis statement morerigorously below.) Thisis the same type of evolution as that of a single qubit interacting with its environment, but the coupling constant is effectively reduced by a factor of R 2 /12. While the coupling constant serves to characterize the decoherence process in this particular case, this is not valid in general. To handle the more general situation, we 125 0.1 0.2 0.3 0.4 Γ t 0.9994 0.9995 0.9996 0.9997 0.9998 0.9999 1 C 000,000 Figure20: Short-timebehaviorofthree-qubitsystemwithbit-flipnoiseandcontinuous error correction. The ratio of correction rate to decoherence rate is R =κ/γ =100. propose to use the instantaneous rate of decrease of the codeword fidelity F cw as a measure of the effect of decoherence: Λ(F cw (t)) =− dF cw (t) dt . (273) (In the present case, F cw =C 000,000 .) This quantity does not coincide with the deco- herence rate in the Markovian case (which can be defined naturally from the Lindblad equation), but it is a good estimate of the rate of loss of fidelity and can be used for any decoherence model. From now on we will refer to it simply as an error rate, but we note that there are other possible definitions of instantaneous error rate suitable for non-Markovian decoherence, which in general may depend on the kind of errors they describe. Since the goal of error correction is to preserve the codeword fidelity, the quantity (273) is a useful indicator for the performance of a given scheme. Note that Λ(F cw ) is a function of the codeword fidelity and therefore it makes sense to use it for a comparison between different cases only for identical values of F cw . For our example, the fact that the coupling constant is effectively reduced approximately 126 R 2 /12 times implies that the error rate for a given value of F cw is also reducedR 2 /12 times. Similarly, the reduction of λ by the factor r/6 in the Markovian case implies a reduction of Λ by the same factor. We see that the effective reduction of the error rate increases quadratically with κ 2 in the non-Markovian case, whereas it increases only linearly with κ in the Markovian case. Nowletusrigorouslyderivetheapproximatesolutiontothismodelofnon-Markovian decoherence with continuous error correction. Assuming that γ ≪κ (or equivalently, R≫ 1), the superoperator driving the evolution of the system during a time step δt can be written as e Lδt = e Lκδt + δt Z 0 dt ′ e Lκ(δt−t ′ ) L γ e Lκt ′ + δt Z 0 dt ′ δt Z t ′ dt ′′ e Lκ(δt−t ′′ ) L γ e Lκ(t ′′ −t ′ ) L γ e Lκt ′ + + δt Z 0 dt ′ δt Z t ′ dt ′′ δt Z t ′′ dt ′′′ e Lκ(δt−t ′′′ ) L γ e Lκ(t ′′′ −t ′′ ) L γ e Lκ(t ′′ −t ′ ) L γ e Lκt ′ +... (274) We have denoted the Liouvillian by L = L γ +L κ , where L κ ρ = κΓ(ρ), and L γ ρ = −i[H,ρ]. Let γδt ≪ 1 ≪ κδt. We will derive an approximate differential equation for the evolution of ρ(t) by looking at the terms of order δt in the change of ρ according to Eq. (274). When κ = 0, we have dρ/dt =L γ ρ, so the effect of L γ on the state of the system can beseen fromEq. (272) withκtaken equal to 0. By the action of exp(L κ t), the different terms of the density matrix transform as follows: ̺ 000,000 ,̺ 111,000 ,̺ 111,111 remainunchanged,̺ 100,100 →e −κt ̺ 100,100 +(1−e −κt )̺ 000,000 ,̺ 110,110 →e −κt ̺ 110,110 + (1−e −κt )̺ 111,111 , ̺ 110,001 →e −κt ̺ 110,001 −(1−e −κt )̺ 111,000 , and all other terms are changed as ̺→e −κt ̺. Since κδt≫ 1, we will ignore terms of order e −κδt . But from Eq. (274) it can be seen that all terms except̺ 000,000 ,̺ 111,000 ,̺ 000,111 ,̺ 111,111 will get multiplied bythefactore −κδt bytheaction of exp(L κ δt)inEq. (274). Theintegrals in Eq. (274) also yield negligible factors, since every integral either gives rise to a factor 127 of order δt when the integration variable is trivially integrated, or a factor of 1/κ when the variable participates nontrivially in the exponent. Therefore, in the above approximation these terms of the density matrix can be neglected, which amounts to aneffectiveevolutionentirelywithinthecodespace. AccordingtoEq. (272),theterms ̺ 000,000 ,̺ 111,000 ,̺ 111,111 can couple to each other only by a triple or higher application ofL γ . This means that if we consider the expansion up to the lowest nontrivial order in γ, we only need to look at the triple integral in Eq. (274). Let us consider the effect of exp(Lδt) on C 000,000 . Any change can come di- rectly only from ̺ 111,000 and ̺ 000,111 . The first exponent e Lκt ′ acts on these terms as the identity. Under the action of the first operator L γ each of these two terms can transform to six terms that can eventually be transformed to ̺ 000,000 . They are ̺ 110,000 , ̺ 101,000 , ̺ 011,000 , ̺ 111,100 , ̺ 111,010 , ̺ 111,001 , and ̺ 000,110 , ̺ 000,101 , ̺ 000,011 , ̺ 100,111 , ̺ 010,111 , ̺ 001,111 , with appropriate factors. The action of the second exponent is to multiply each of these new terms bye −κ(t ′′ −t ′ ) . After the action of the secondL γ , the action of the third exponent on the relevant resultant terms will beagain to multi- ply them by a factor e −κ(t ′′′ −t ′′ ) . Thus the second and the third exponents yield a net factor ofe −κ(t ′′′ −t ′ ) . After the second and the thirdL γ , the relevant terms that we get are̺ 000,000 and̺ 100,100 ,̺ 010,010 ,̺ 001,001 ,eachwithacorrespondingfactor. Finally, the last exponent acts as the identity on̺ 000,000 and transforms each of the terms̺ 100,100 , ̺ 010,010 , ̺ 001,001 into (1−e −κ(δt−t ′′′ ) )̺ 000,000 . Counting the number of different terms that arise at each step, and taking into account the factors that accompany them, we obtain: C 000,000 → C 000,000 + δt Z 0 dt ′ δt Z t ′ dt ′′ δt Z t ′′ dt ′′′ (24e −κ(t ′′′ −t ′ ) −36e −κ(δt−t ′ ) )C 111,000 +··· ≈ C 000,000 +C 111,000 24 R 2 γδt+O(δt 2 ). (275) 128 Using that C 000,000 +C 111,111 ≈1, in a similar way one obtains C 111,000 →C 111,000 −(2C 000,000 −1) 12 R 2 γδt+O(δt 2 ). (276) For times much larger than δt, we can write the approximate differential equations dC 000,000 dt = 24 R 2 γC 111,000 , dC 111,000 dt =− 12 R 2 γ(2C 000,000 −1). (277) Comparing with Eq. (257), we see that the encoded qubit undergoes approximately the same type of evolution as that of a single qubit without error correction, but the couplingconstantiseffectively decreasedR 2 /12times. ThesolutionofEq. (277)yields for the codeword fidelity C 000,000 (t) = 1+cos( 24 R 2 γt) 2 . (278) This solution is valid only with precision O(1/R) for times γt≪R 3 . This is because we ignored terms whose magnitudes are always of order O(1/R) and ignored changes of order O(γδt/R 3 ) per time step δt in the other terms. The latter changes could accumulate with time and become of the order of unity for times γt ≈ R 3 , which is why the approximate solution is invalid for such times. In fact, if one carries out the expansion (274) to fourth order in γ, one obtains the approximate equations dC 000,000 dt = 24 R 2 γC 111,000 − 72 R 3 γ(2C 000,000 −1), dC 111,000 dt =− 12 R 2 γ(2C 000,000 −1)− 144 R 3 γC 111,000 , (279) 129 which yield for the fidelity C 000,000 (t) = 1+e −144γt/R 3 cos(24γt/R 2 ) 2 . (280) We see that in addition to the effective error process which is of the same type as that of a single qubit, there is an extra Markovian bit-flip process with rate 72γ/R 3 . This Markovian behavior is due to the Markovian character of our error-correcting procedurewhich, at this level of approximation, is responsiblefor thedirect transferof weightbetween̺ 000,000 and̺ 111,111 ,andbetween̺ 111,000 and̺ 000,111 . Theexponential factor explicitly reveals the range of applicability of the solution (278): with precision O(1/R), it is valid only for times γt of up to order R 2 . For times of the order of R 3 , the decay becomes significant and cannot be neglected. The exponential factor may also play an important role for short times of up to order R, where its contribution is bigger than that of the cosine. But in the latter regime the difference between the cosine and the exponent is of order O(1/R 2 ), which is negligible for the precision that we consider. In general, the effective evolution that one obtains in the limit of high error- correctionratedoesnothavetoapproachaformidenticaltothatofasingledecohering qubit. The reason we obtain such behavior here is that for this particular model the lowest order of uncorrectable errors that transform the state within the code space is 3, and three-qubit errors have the form of an encoded X operation. Furthermore, the symmetry of the problem ensured an identical evolution of the three qubits in the code. For general stabilizer codes, the errors that a single qubit can undergo are not limited to bit flips only. Therefore, different combinations of single-qubit errors may lead to different types of lowest-order uncorrectable errors inside the code space, none of which inprinciplehasto represent anencoded version of thesingle-qubit operations that composeit. Inaddition, ifthenoiseis differentforthedifferent qubits, thereisno 130 unique single-qubit error model to compare to. Nevertheless, we will show that with regard to the effective decrease in the error-correction rate, general stabilizer codes will exhibit the same qualitative performance. 5.4 Relation to the Zeno regime Theeffective continuous evolution (277) was derived undertheassumptionthatγδt≪ 1≪κδt. The first inequality implies that δt can be considered within the Zeno time scale of the system’s evolution without error correction. On the other hand, from the relation between κ and τ c in (242) we see that τ c ≪ δt. Therefore, the time for implementing a weak error-correcting operation has to be sufficiently small so that on the Zeno time scale the error-correction procedure can be described approximately as a continuous Markovian process. This suggests a way of understanding the quadratic enhancement in the non-Markovian case based on the properties of the Zeno regime. Let us consider again the single-qubit code from Section 5.2, but this time let the error model be any Hamiltonian-driven process. We assume that the qubit is initially in the state |0i, i.e., the state of the system including the bath has the form ρ(0) = |0ih0|⊗ρ B (0). For times smaller than the Zeno time δt Z , the evolution of the fidelity without error correction can be described by Eq. (247). Equation (247) naturally defines the Zeno regime in terms of α itself: α≥α Z ≡1−Cδt 2 Z . (281) For a single time step Δt≪δt Z , the change in the fidelity is α→α−2 √ C √ 1−αΔt+O(Δt 2 ). (282) 131 On the other hand, the effect of error correction during a time step Δt is α→α+κ(1−α)Δt+O(Δt 2 ), (283) i.e., it tends to oppose the effect of decoherence. If both processes happen simulta- neously, the effect of decoherence will still be of the form (282), but the coefficient C may vary with time. This is because the presence of error-correction opposes the decrease of the fidelity and consequently can lead to an increase in the time for which the fidelity remains within the Zeno range. If this time is sufficiently long, the state of the environment could change significantly under the action of the Hamiltonian, thus giving rise to a different value for C in Eq. (282) according to Eq. (248). Note that the strength of the Hamiltonian puts a limit on C, and therefore this constant can vary only within a certain range. The equilibrium fidelity α NM ∗ that we obtained for the error model in Section 5.2, can be thought of as the point at which the effects of error and error correction cancel out. For a general model, where the coefficient C may vary with time, this leads to a quasi-stationary equilibrium. From Eqs. (282) and (283), one obtains the equilibrium fidelity α NM ∗ ≈1− 4C κ 2 . (284) Inagreement withwhatweobtainedinSection5.2, theequilibriumfidelitydiffersfrom 1 by a quantity proportional to 1/κ 2 . This quantity is generally quasi-stationary and can vary within a limited range. If one assumes a Markovian error model, for short times the fidelity changes linearly with time which leads to 1−α M ∗ ∝ 1/κ. Thus the difference can be attributed to the existence of a Zeno regime in the non-Markovian case. 132 But what happens in the case of non-trivial codes? As we saw, there the state decays inside the code space and therefore can be highly correlated with the environ- ment. Can wetalk about aZenoregime then? It turnsout that theanswer is positive. Assuming that each qubit undergoes an independent error process, then up to first order in Δt the Hamiltonian cannot map terms in the code space to other terms with- out detectable errors. (This includes both terms in the code space and terms from the hidden part, like ̺ 111,000 in the example of the bit-flip code.) It can only transform terms from the code space into traceless terms from the hidden part which correspond to single-qubit errors (like̺ 100,000 in thesame example). Let| ¯ 0i,| ¯ 1i bethe two logical codewords and |ψ i i be an orthonormal basis that spans the space of all single-qubit errors. Then in the basis | ¯ 0i, | ¯ 1i, |ψ i i, all the terms that can be coupled directly to terms inside the code space are| ¯ 0ihψ i |,|ψ i ih ¯ 0|,| ¯ 1ihψ i |,|ψ i ih ¯ 1|. From the condition of positivity of the density matrix, one can show that the coefficients in front of these terms are at most p α(1−α) in magnitude, where α is the code-space fidelity. This implies that for small enough 1−α, the change in the code-space fidelity is of the type (282), which is Zeno-like behavior. Then using only the properties of the Zeno behavior as we did above, we can conclude that the weight outside the code space will be kept at a quasi-stationary value of order 1/κ 2 . Since uncorrectable errors enter the code space through the action of the error-correction procedure, which misinterprets some multi-qubit errors in the error space, the effective error rate will be limited by a factor proportional to the weight in the error space. That is, this will lead to an effective decrease of the error rate at least by a factor proportional to 1/κ 2 . The accumulation of uncorrectable errors in the Markovian case is similar, except that in this case there is a direct transfer of errors between the code space and the visible part of the error space. In both cases, the error rate is effectively reduced by a factor whichisroughlyproportionaltotheinverseoftheweight intheerrorspace, and therefore the difference in the performance comes from the difference in this weight. 133 The quasi-stationary equilibrium value of the code-space fidelity establishes a quasi- stationary flow between the code space and the error space. One can think that this flow effectively takes non-erroneous weight from the code space, transports it through the error space where it accumulates uncorrectable errors, and brings it back into the code space. Thus by minimizing the weight outside the code space, error correction createsa“bottleneck”whichreducestherateatwhichuncorrectableerrorsaccumulate. Finally, a brief remark about the resources needed for quadratic reduction of the error rate. As pointed out above, two conditions are involved: one concerns the rate of error correction; the other concerns the time resolution of the weak error-correcting operations. Both of these quantities must be sufficiently large. There is, however, an interplay between the two, which involves the strength of the interaction required to implement the weak error-correcting map (241). Let us imagine that the weak map is implemented by making the system interact weakly with an ancilla in a given state, after which the ancilla is discarded. The error-correction procedureconsists of a sequence of such interactions, and can be thought of as a cooling process which takes away the entropy accumulated in the system as a result of correctable errors. If the time for which a single ancilla interacts with the system is τ c , one can verify that the parameter ǫ in Eq. (241) would be proportional to g 2 τ 2 c , where g is the coupling strength between the system and the ancilla. From Eq. (242) we then obtain that κ∝g 2 τ c . (285) Thetwoparametersthat canbecontrolled aretheinteraction timeandtheinteraction strength, and they determine the error-correction rate. Thus if g is kept constant, a decrease in the interaction time τ c leads to a proportional decrease in κ, which may be undesirable. In order to achieve a good working regime, one may need to adjust both τ c and g. But it has to be pointed out that in some situations decreasing 134 τ c alone can prove advantageous, if it leads to a time resolution revealing the non- Markovian character of an error model which was previously described as Markovian. The quadratic enhancement of the performance as a function of κ may compensate the decrease in κ, thus leading to a seemingly paradoxical result: better performance with a lower error-correction rate. 5.5 Summary and outlook In this chapter we studied the performance of a particular continuous quantum error- correction scheme for non-Markovian errors. We analyzed the evolution of the single- qubit code and the three-qubit bit-flip code in the presence of continuous error correc- tionforasimplenon-Markovianbit-fliperrormodel. Thisenabledustounderstandthe workingsoftheerror-correction scheme, andthemechanismwherebyuncorrectableer- rorsaccumulate. Thefidelityofthestatewiththecodespaceinbothexamplesquickly reaches an equilibrium value, which can be made arbitrarily close to 1 by a sufficiently high rate of error correction. The weight of the density matrix outside the code space scales as 1/κ in the Markovian case, while it scales as 1/κ 2 in the non-Markovian case. Correspondingly, the rate at which uncorrectable errors accumulate in the three-qubit code is proportional to 1/κ in the Markovian case, and to 1/κ 2 in the non-Markovian case. These differences have the same cause, since the equilibrium weight in the error space is closely related to the rate of uncorrectable error accumulation. The quadratic difference in the error weight between the Markovian and non- Markovian cases can be attributed to the existence of a Zeno regime in the non- Markovian case. Regardless of the correlations between the density matrix inside the code space and the environment, if the lowest-order errors are correctable by thecode, there exists a Zeno regime in the evolution of the code-space fidelity. The effective reduction of the error rate with the rate of error correction for non-Markovian error 135 models depends crucially on the assumption that the time resolution of the continu- ous error correction is much shorter than the Zeno time scale of the evolution without error correction. This suggests that decreasing the time for a single (infinitesimal) error-correcting operation can lead to an increase in the performance of the scheme, even if the average error-correction rate goes down. While here we have only considered codes for the correction of single-qubit errors, our results can be extended to other types of codes and errors as well. As long as the errorprocessonlyproduceserrorscorrectablebythecodetolowestorder,anargument analogous to the one given here shows that a Zeno regime will exist, which leads to an enhancement in the error-correction performance. Unfortunately, it is very difficult to describe the evolution of a system with a continuous correction protocol, based on a general error-correction code and subject to general non-Markovian interactions with the environment. This is especially true if one must include the evolution of a complicated environment in the description, as would be necessary in general. A more practical step in this direction might be to find an effective description for the evolution ofthereduceddensitymatrixofthesystemsubjecttodecoherencepluserror correction, using projection techniques like the Nakajima-Zwanzig or the TCL master equations. Since one is usually interested in the evolution during initial times before the codeword fidelity decreases significantly, a perturbation approach could be useful. This is a subject for further research. 5.6 Appendix: Implementationofthequantum-jumperror- correcting process via weak measurements and weak uni- tary operations Hereweshow howtheweak CPTPmap (241) for thebit-flip codecanbeimplemented using weak measurements and weak unitary operations. We also present a similar 136 scheme for codes that correct arbitrary-single qubit errors, which yields a weak map different from (241) but one that also results in the strong error-correcting map Φ(ρ) when exponentiated. To introduce our construction, we start again with the single- qubit code with stabilizerhZi. 5.6.1 The single-qubit model Consider the completely positive map corresponding to the strong error-correcting operation for the single-qubit code: Φ(ρ) =X|1ih1|ρ|1ih1|X +|0ih0|ρ|0ih0| =|0ih1|ρ|1ih0|+|0ih0|ρ|0ih0|. (286) Observe that this transformation can also be written as Φ(ρ) =|0ih+|ρ|+ih0|+|0ih−|ρ|−ih0| =ZR|+ih+|ρ|+ih+|RZ+XR|−ih−|ρ|−ih−|RX, (287) where|±i=(|0i±|1i)/ √ 2 and R = 1 √ 2 1 1 1 −1 (288) is the Hadamard gate. Therefore the same error-correcting operation can be imple- mented as a measurement in the|±i basis (measurement of the operatorX), followed by a unitary conditioned on the outcome: if the outcome is ’+’, we apply ZR; if the outcome is ’-’, we apply XR. This choice of unitaries is not unique—for example, we could apply just R instead of ZR after outcome ’+’. But this particular choice has a convenient geometricinterpretation—theunitaryZRcorrespondstoarotationaround the Y-axis by an angle π/2: ZR = e i π 2 Y 2 , and XR corresponds to a rotation around the same axis by an angle−π/2: ZR =e −i π 2 Y 2 . 137 Aweakversionoftheaboveerror-correctingoperationcanbeconstructedbytaking the corresponding weak measurement of the operator X, followed by a weak rotation around the Y-axis, whose direction is conditioned on the outcome: ρ→ I +iǫ ′ Y p 1+ǫ ′ 2 r I +ǫX 2 ρ r I +ǫX 2 I−iǫ ′ Y p 1+ǫ ′ 2 + + I−iǫ ′ Y p 1+ǫ ′ 2 r I−ǫX 2 ρ r I−ǫX 2 I +iǫ ′ Y p 1+ǫ ′ 2 . (289) Hereǫ andǫ ′ are small parameters. From the symmetry of this map it can be seen that if the map is applied to a state which lies on the Z-axis, the resultant state will still lie on the Z-axis. Whether the state will move towards |0ih0| or towards |1ih1|, depends on the relation between ǫ and ǫ ′ . Since our goal is to protect the state from drifting away from|0ih0| due to bit-flip decoherence, we will assume that the state lies on the Z-axis in the northern hemisphere (although the transformation we will obtain works for any kind of decohernce where the state need nor remain on the Z-axis). We would like, if possible, to choose the relation between the parameters ǫ and ǫ ′ in such a way that the effect of this map on any state on the Z-axis to be to move the state towards|0ih0|. Inordertocalculate theeffect ofthismaponagiven state, itisconvenient towrite the state in the |±i basis. For a state on the Z-axis, ρ = α|0ih0| +(1−α)|1ih1|, we have ρ= 1 2 |+ih+|+ 1 2 |−ih−|+(2α−1) 1 2 |+ih−|+ 1 2 |−ih+| . (290) For the action of our map on the state (290) we obtain: ρ→ 1 2 |+ih+|+ 1 2 |−ih−|+ (1−ǫ ′ 2 ) √ 1−ǫ 2 (2α−1)+2ǫǫ ′ 1+ǫ ′ 2 1 2 |+ih−|+ 1 2 |−ih+| . (291) 138 Thus we can think that upon this transformation the parameter α transforms to α ′ , where 2α ′ −1= (1−ǫ ′ 2 ) √ 1−ǫ 2 (2α−1)+2ǫǫ ′ 1+ǫ ′ 2 . (292) If it is possible to choose the relation between ǫ and ǫ ′ in such a way that α ′ ≥α for every 0≤α≤ 1, then clearly the state must remain invariant when α = 1. Imposing this requirement, we obtain ǫ= 2ǫ ′ 1+ǫ ′ 2 , (293) or equivalently ǫ ′ = 1− √ 1−ǫ 2 ǫ . (294) Substituting back in (292), we can express α ′ −α= 4ǫ ′ 2 (1+ǫ ′ 2 ) 2 (1−α)≥0. (295) We see that the coefficient α (which is the fidelity of our state with |0ih0|) indeed increases after every application of our weak completely positive map (Fig.1). The amount by which it increases for fixed ǫ ′ depends on α and becomes smaller as α approaches 1. Since we will be taking the limit ǫ→0, we can write Eq. (294) as ǫ ′ = ǫ 2 +O(ǫ 3 ). (296) 139 If we define the relation between the time step τ c and ǫ as in Eq. (242), for the effect of the CPTP map (289) on an arbitrary state of the form ρ = α|0ih0| +β|0ih1| + β ∗ |1ih0|+(1−α)|1ih1|, α∈R, β∈C, we obtain α→α+(1−α)κτ c , (297) β→ √ 1−κτ c β =β− 1 2 κβτ c +O(τ c 2 ). (298) This is exactly the map (241) for Φ(ρ) given by Eq. (286). 5.6.2 The bit-flip code While in the toy model from the previous section we had to protect a given state from errors, here we have to protect the whole subspace spanned by|0i and|1i. This makes a geometric visualization of the problem significantly more difficult than in the previous case, which is why we will take a different approach. In the single-qubit model we saw how to protect a qubit in state |0i from bit-flip errors. Similarly we could protect a qubit in state |1i; the only difference is that the weak unitaries following the two outcomes of the weak measurement of X have to be exchanged. For the three-qubit bit-flip code, every block of the code lies in the subspace spanned by the codewords |000i and |111i, i.e., each qubit is in state |0i when the other two qubits are in state |00i, or in state |1i when the other qubits are in state|11i. This correlation is what makes it possible for the code to correct single- qubit bit-flip errors without ever acquiring information about the actual state of the system. We propose to utilize this correlation in a three-qubit scheme which protects each qubit by applying to it the corresponding single-qubit scheme for either |0i or |1i depending on the value of the other two qubits. This, of course, has to be done without acquiring information about the encoded state. 140 Just as in the single-qubit case, the scheme consist of weak measurements followed by weak unitaries conditioned on the outcomes of the measurements. For error correc- tion onthefirstqubit, weproposetheweak measurementwithmeasurement operators M 1 ± = r I±ǫX 2 ⊗(|00ih00|+|11ih11|)+ I √ 2 ⊗(|01ih01|+|10ih10|), (299) where q I±ǫX 2 are thesame weak measurement operators that we usedin (289), acting on the first qubit. This measurement can be thought of as a weak measurement of the operator X⊗(|00ih00| +|11ih11|). In order to understand its effect better, consider the expansion of the density matrix of our system in the computational basis of the three qubits in a given block of the code. Assuming that the state begins inside the code space and that the system decoheres through single-qubit bit-flip channels, the density matrix at any time can be written as a linear combination of the following terms: |000ih000|,|000ih111|, |111ih000|, |111ih111|, |100ih100|, |100ih011|, |011ih100|, |011ih011|, |010ih010|, |010ih101|, |101ih010|, |101ih101|, |001ih001|, |001ih110|, |110ih001|,|110ih110|. Forthosetermsintheexpansionforwhichthesecondandthird qubits are in the subspace spanned by |00i and |11i, the effect of this measurement will be the same as the effect of a weak single-qubit measurement of X on the first qubit. Those terms in which the second and third qubits are in the subspace spanned by |01i and |10i will not be affected by the measurement. This is because the three- qubit bit-flip code cannot distinguish multi-qubit errors from single-qubit errors; the subspaces corresponding to two- and three-qubit errors are the same as the subspaces corresponding to single-qubit or no errors. This is why, if the second and third qubits havedifferentvalues,theerror-correctionschemewillassumethatanerrorhasoccurred on one of these two qubits and will not apply any correction on the first qubit. 141 The unitary operation conditioned on the outcome of the measurement is U 1 ± = I±iǫ ′ Y p 1+ǫ ′ 2 ⊗|00ih00|+ I∓iǫ ′ Y p 1+ǫ ′ 2 ⊗|11ih11|+I⊗(|01ih01|+|10ih10|). (300) ThisisaweakunitarydrivenbytheHamiltonian±Y⊗(|00ih00|−|11ih11|). Again,itis designed in such a way that those components of the density matrix which correspond to an error on the second or third qubits will undergo no transformation, while the terms for which the second and third qubits have the same value (these are the same terms that have undergone non-trivial transformation during the measurement) will undergo a rotation of the first qubit analogous to that from the single-qubit model. One can verify that the only terms that undergo non-trivial transformation after the completely positive map ρ→U 1 + M 1 + ρM 1 + U 1 + † +U 1 − M 1 − ρM 1 − U 1 − † are: |100ih100|→(1−κτ c )|100ih100|+κτ c |000ih000|, |100ih011|→(1−κτ c )|100ih011|+κτ c |000ih111|, |011ih100|→(1−κτ c )|011ih100|+κτ c |111ih000|, |011ih011|→(1−κτ c )|011ih011|+κτ c |111ih111|. |100ihφ|→(1− 1 2 κτ c )|100ihφ|, |φih100|→(1− 1 2 κτ c )|φih100| |011ihφ|→(1− 1 2 κτ c )|011ihφ|, |φih011|→(1− 1 2 κτ c )|φih011| (301) where |φi is any state orthogonal to the subspace spanned by |100i and |011i. We see that the effect of this operation on the terms that correspond to bit flip on the 142 first qubit is to correct these terms by the same amount as in the single-qubit error- correction scheme. All other terms remain unchanged. If we write the state of the system at a given moment as ρ=aρ(0)+b 1 X 1 ρ(0)X 1 +b 2 X 2 ρ(0)X 2 +b 3 X 3 ρ(0)X 3 + (302) +c 1 X 2 X 3 ρ(0)X 2 X 3 +c 2 X 1 X 3 ρ(0)X 1 X 3 +c 3 X 1 X 2 ρ(0)X 1 X 2 +dX 1 X 2 X 3 ρ(0)X 1 X 2 X 3 , whereρ(0) is the initial state, then the effect of the above completely positive map is: a→a+b 1 4κτ c , b 1 →b 1 −b 1 4κτ c , b 2 →b 2 , b 3 →b 3 , c 1 →c 1 −c 1 4κτ c , c 2 →c 2 , c 3 →c 3 , d→d+c 1 4κτ c . (303) We apply the same correction (ρ→U i + M i + ρM i + U i + † +U i − M i − ρM i − U i − † ) to each of the other two qubits (i = 2,3) as well. One can easily see that the effect of all three corrections (up to first order in Δt) is equivalent to the map (241) with Φ(ρ) given in Eq. (240). 5.6.3 General single-error-correcting stabilizer codes We now proceed to generalizing this scheme to error-correcting codes that correct arbitrarysingle-qubiterrors. Astabilizercodewhichisabletocorrect arbitrarysingle- qubit errors, has the property that a single-qubitX,Y orZ error on a state inside the code space, sends that state to a subspace orthogonal to the code space [67]. One can verify that this implies that any two orthogonal codewords can be written as |0i= 1 √ 2 |0i|ψ 0 0 i+ 1 √ 2 |1i|ψ 0 1 i, |1i= 1 √ 2 |0i|ψ 1 0 i+ 1 √ 2 |1i|ψ 1 1 i, (304) 143 where|ψ i j i,i,j =0,1 form an orthonormal set. Here we have expanded the codewords in the computational basis (the eigenbasis of Z) of the first qubit, but the same can be done with respect to any qubit in the code. Note that an X, Y, or Z error on one of the other qubits sends each of the vectors|ψ i j i to a subspace orthogonal to the subspace spanned by|ψ i j i, i,j = 0,1. This can be shown to follow from the fact that different single-qubit errors send the code space to different orthogonal subspaces. An exception is the case of degenerate codes where the error in question has the same effect on a codeword as an error on the first qubit. In such a case, however, we can assumethattheerrorhasoccuredonthefirstqubit. Theweakoperationforcorrecting bit flips on a given qubit (say the first one) is therefore constructed similarly to that for the bit-flip code. We first apply the weak measurement M 1 ± = r I±ǫX 2 ⊗(|ψ 0 0 ihψ 0 0 |+|ψ 0 1 ihψ 0 1 |+|ψ 1 0 ihψ 1 0 |+|ψ 1 1 ihψ 1 1 |)+ + I √ 2 ⊗(I n−1 −|ψ 0 0 ihψ 0 0 |−|ψ 0 1 ihψ 0 1 |−|ψ 1 0 ihψ 1 0 |−|ψ 1 1 ihψ 1 1 |), (305) where I n−1 is the identity on the space of all qubits in the code except the first one. ThiscanbethoughtofasaweakmeasurementoftheoperatorX(|ψ 0 0 ihψ 0 0 |+|ψ 0 1 ihψ 0 1 |+ |ψ 1 0 ihψ 1 0 |+|ψ 1 1 ihψ 1 1 |). The measurement is followed by the unitary U 1 ± = I±iǫ ′ Y p 1+ǫ ′ 2 ⊗(|ψ 0 0 ihψ 0 0 |+|ψ 1 0 ihψ 1 0 |)+ I∓iǫ ′ Y p 1+ǫ ′ 2 ⊗(|ψ 0 1 ihψ 0 1 |+|ψ 1 1 ihψ 1 1 |)+ +I⊗(I n−1 −|ψ 0 0 ihψ 0 0 |−|ψ 0 1 ihψ 0 1 |−|ψ 1 0 ihψ 1 0 |−|ψ 1 1 ihψ 1 1 |) (306) conditioned on the outcome. The Hamiltonian driving this unitary is ±Y(|ψ 0 0 ihψ 0 0 |+ |ψ 1 0 ihψ 1 0 |−|ψ 0 1 ihψ 0 1 |−|ψ 1 1 ihψ 1 1 |). It is easy to verify that the effect of the corresponding 144 completely positive map is analogous to that for the bit-flip code. The action of each of the operators U 1 + M 1 + and U 1 − M 1 − can be summarized as follows: U 1 ± M 1 ± |ii|φi = 1 √ 2 |ii|φi, for |φi∈I n−1 − X j,k |ψ j k ihψ j k |, (307) U 1 ± M 1 ± |ji|ψ i k i= r 1−κτ c 2 |ji|ψ i k i± r κτ c 2 |ki|ψ i k i, for j6=k, (308) U 1 ± M 1 ± |ji|ψ i k i=|ji|ψ i k i, for j =k. (309) This implies that the effect of the map σ→U 1 + M 1 + σM 1 + U 1 + † +U 1 − M 1 − σM 1 − U 1 − † on a bit-flip error on the first qubit of a codeword ρ is: X 1 ρX 1 → (1−κτ c )X 1 ρX 1 +κτ c ρ, (310) X 1 ρ→(1− 1 2 κτ c )X 1 ρ, (311) ρX 1 →(1− 1 2 κτ c )ρX 1 . (312) Just like in the bit-flip code, the error-correcting procedure for the case where each qubit decoheres through an independent bit-flip channel consists of simultaneous cor- rections of all qubits (i = 1,2,...,n) by continuous application of the maps σ → U i + M i + σM i + U i + † +U i − M i − σM i − U i − † . From(304) itcanbeseenthatthecodewordshaveanalogousformswhenexpanded in the eigenbasis of another Pauli operator (X or Y) acting on a given qubit: |0i = 1 √ 2 |x + i|ψ 0 x + i+ 1 √ 2 |x − i|ψ 0 x − i= 1 √ 2 |y + i|ψ 0 y + i+ 1 √ 2 |y − i|ψ 0 y − i, |1i= 1 √ 2 |x + i|ψ 1 x + i+ 1 √ 2 |x − i|ψ 1 x − i= 1 √ 2 |y + i|ψ 1 y + i+ 1 √ 2 |y − i|ψ 1 y − i. (313) Here |x ± i=(−i) 1∓1 2 |0i±|1i √ 2 (314) 145 and |y ± i= |0i±i|1i √ 2 (315) are eigenbases of X and Y respectively, and |ψ i x ± i=i 1∓1 2 |ψ i 0 i±|ψ i 1 i √ 2 , i=0,1 (316) and |ψ i y ± i= |ψ i 0 i∓i|ψ i 1 i √ 2 , i=0,1 (317) are orthonormal sets. The reason why we have chosen these particular overall phases in the definition of the eigenvectors of X and Y, is that we want to have our expres- sions explicitly symmetric with respect to cyclic permutations of X, Y and Z. More precisely, the expansions of the operators X, Y, Z in the |0,1i basis are the same as the expansions ofY, Z,X in the|x ± i basis, and the same as the expansions of Z,X, Y inthe|y ± i basis. ThismeansthatY andZ errorsinthecomputational basiscan be treated asX errors in the bases|x ± i and|y ± i, and therefore can be corrected accord- ingly. The weak measurement and unitary for the correction of Y errors on the first qubit (let’s call themM 1 y± andU 1 y± ) are obtained from (305) and (306) by making the substitutions X → Y, Y → Z, |0,1i →|x ± i, |ψ i 0,1 i→|ψ i x ± i. The operations for the correction of Z errors (M 1 z± and U 1 z± ) are obtained from (305) and (306) by X →Z, Y →X,|0,1i→|y ± i,|ψ i 0,1 i→|ψ i y ± i. Theoperations for correction ofY andZ errors on any qubit (M i y± , U i y± , and M i z± , U i z± , i=1,2,...,n) are defined analogously. To prove that the weak error-correcting map resulting from the application of the described weak measurements and unitary operations is equal to Eq. (241), we are going to look at its effect on different components of the density matrix. Any density matrix can be written as a linear combination of terms of the type |φihχ|, where each of the vectors |φi and |χi belongs to one of the orthogonal subspaces on 146 which a state gets projected if we measure the stabilizer generators of the code. Let us denote the code space by C and the subspaces corresponding to different single- qubit errors by C X i , C Y i , and C Z i , where the subscript refers to the type of error (X, Y, or Z) and the number of the qubit on which it occurred. The code space and the subspaces corresponding to single-qubit errors in general do not cover the whole Hilbert space. Some of the outcomes of the measurement of the stabilizer generators may project the state onto subspaces corresponding to multi-qubit er- rors. We are going to denote the direct sum of these subspaces by C M . Our weak error-correcting operation consists of a simultaneous application of the weak maps ρ→U i + M i + ρM i + U i + † +U i − M i − ρM i − U i − † ,ρ→U i y+ M i y+ ρM i y+ U i y+ † +U i y− M i y− ρM i y− U i y− † , ρ→U i z+ M i z+ ρM i z+ U i z+ † +U i z− M i z− ρM i z− U i z− † , i=1,2,...,n. The order of application is irrelevant since we consider only contributions of up to first order in Δt. Using (307)-(309) and the symmetry under cyclic permutations of X, Y and Z, one can show that this map has the following effect: |φihχ|→|φihχ|, if |φi,|χi∈C⊕C M , (318) |φihχ|→(1−2κτ c )|φihχ|+κτ c X i |φihχ|X i +κτ c Z i |φihχ|Z i , if |φi,|χi∈C X i , (319) |φihχ|→(1−2κτ c )|φihχ|+κτ c Y i |φihχ|Y i +κτ c X i |φihχ|X i , if |φi,|χi∈C Y i , (320) |φihχ|→(1−2κτ c t)|φihχ|+κτ c Z i |φihχ|Z i +κτ c Y i |φihχ|Y i , if |φi,|χi∈C Z i , (321) |φihχ|→(1−κτ c )|φihχ|, if |φi∈C X i ⊕C Y i ⊕C Z i , |χi∈C⊕C M , (322) |φihχ|→(1−2κτ c )|φihχ|+κτ c X i |φihχ|X i , if |φi∈C X i ,|χi∈C Y i , (323) |φihχ|→(1−2κτ c )|φihχ|+κτ c Y i |φihχ|Y i , if |φi∈C Y i ,|χi∈C Z i , (324) |φihχ|→ (1−2κτ c )|φihχ|+κτ c Z i |φihχ|Z i , if |φi∈C Z i ,|χi∈C X i , (325) |φihχ|→(1−2κτ c )|φihχ|, if |φi∈C X i ,⊕C Y i ⊕C Z i ,|χi∈C X j ⊕C Y j ⊕C Z j , i6=j. (326) 147 This is sufficient to determine the effect of the error-correcting map on any density matrix. One can easily see that this map is not equal to the the map (241) because of the last terms on the right-hand sides of Eqs. (319)-(321) and (323)-(325). These terms appear because the operation we proposed for correctingX errors, for example, cannot distinguish between X and Y errors and corrects both. This gives rise to the last terms in Eqs. (320) and (323). The same holds for the operations we proposed for correctingY andZ errors. Nevertheless, this map is also a weak error-correcting map in the sense that in the limit of infinitely many applications, it corrects single-qubit errors fully, i.e., it results in the strong error-correcting map Φ(ρ). Toseethis, considerall possiblesingle-qubit errorson adensity matrixρ∈C. The most general form of a single-qubit error on the i th qubit is ρ i = 4 X j=1 M i,j ρM † i,j , (327) where the Kraus operators M i,j are complex linear combinations of I, X i , Y i and Z i that satisfy 4 P j=1 M † i,j M i,j = I. Observe that ρ i is a real superposition of the following terms: ρ,X i ρX i ,Y i ρY i ,Z i ρZ i ,i(X i ρ−ρX i ),i(Y i ρ−ρY i ),i(Z i ρ−ρZ i ),X i ρY i +Y i ρX i , Y i ρZ i +Z i ρY i , X i ρZ i +Z i ρX i . Each of the first four terms has trace 1 and the rest of the terms are traceless. From (318)-(326) one can see that the weak map does not couple the first four terms with the rest. Therefore, their evolution under continuous application of the map (without decoherence) can be treated separately. If we write the single-qubit error (327) as ρ i =aρ+bX i ρX i +cY i ρY i +dZ i ρZ i +traceless terms, (328) 148 a single application of the weak map causes the transformation a→a+(b+c+d)κτ c , (329) b→b−b2κτ c +cκτ c , (330) c→c−c2κτ c +dκτ c , (331) d→d−d2κτ c +bκτ c . (332) Using that at any moment a+b+c+d = 1 and taking the limit τ c → 0, from (329) we obtain that the evolution of a is described by da(t) dt =κ(1−a(t)). (333) The solution is a(t) =1−(1−a(0))e −κt , (334) i.e., in the limit of t → ∞ we obtain a(t) → 1 (and therefore b,c,d → 0). We don’t needtolook attheevolution ofthetraceless termsinρ i becauseourmapiscompletely positive and therefore the transformed ρ i is also a density matrix, which implies that if a = 1,b = 0,c = 0,d = 0, all traceless terms have to vanish. This completes the proof that in the limit of infinitely many applications, our weak error-correcting map is able to correct arbitrary single-qubit errors. It is interesting whether a similar implementation in terms of weak measurements and weak unitary operations can be found for the map (241) for general codes. One way to approach this problem might be to look at the error-correcting operations in the decoded basis. Another interesting question is whether the scheme we presented can be modified to include feedback which depends more generally on the history of measurement outcomes and not only on the outcome of the last measurement. It is 149 natural to expect that using fully the available information about the state could lead to a better performance. These questions are left open for future investigation. 150 Chapter 6: Correctable subsystems under continuous decoherence In the previous chapter, we were concerned with a situation in which the informa- tion stored in an error-correcting code was only approximately correctable. For the model we considered, there were non-correctable multi-qubit errors that accumulated with time, albeit with a slower rate. This is, in practice, the general situation—the probability for non-correctable errors is never truly zero and in order to deal with higher-order terms we need to use concatenation and fault-tolerant techniques (see Section 8.3). But as we saw in the previous chapter, the idea of perfect error cor- rection can be crucial for understanding the approximate process. In view of this, in this chapter we ask the question of the conditions under which a code is perfectly correctable during an entire time interval of continuous decoherence. We consider the most general form of quantum codes—operator, or subsystem codes. 6.1 Preliminaries Operator quantum error correction (OQEC) [93, 94, 24] is a unified approach to error correction which uses the most general encoding for the protection of information— encodinginsubsystems[87,161](seealso[26]). Thisapproachcontainsasspecialcases thestandardquantumerror-correctionmethod[142,148,22,88]aswellasthemethods of decoherence-free subspaces [55, 174, 104, 102] and subsystems [89, 51, 83, 172]. In 151 the OQEC formalism, noise is represented by a completely positive trace-preserving (CPTP) linear map or a noise channel, and correctability is defined with respect to such channels. In practice, however, noise is a continuous process and if it can be represented by a CPTP map, that map is generally a function of time. Correctability is therefore a time-dependent property. Furthermore, the evolution of an open system iscompletely positiveifthesystemandtheenvironmentareinitially uncorrelated, and necessary and sufficient conditions for CPTPdynamicsarenotknown. As pointed out inthepreviouschapter,formoregeneralcasesonemightneedanotionofcorrectability that can capture non-CP transformations [139]. Whether completely positive or not, the noise map is a result of the action of the generator driving the evolution and possibly of the initial state of the system and the environment. Therefore,ourgoalwillbetounderstandtheconditionsforcorrectability in terms of the generator that drives the evolution. We will consider conditions on the system-environment Hamiltonian, or in the case of Markovian evolution—on the Lindbladian. Conditions on the generator of evolution have been derived for decoherence-free subsystems (DFSs) [138], which are a special type of operator codes. DFSs are fixed subsystems of the system’s Hilbert space, inside which all states evolve unitarily. One generalization of this concept are the so called unitarily correctable subsystems [94]. Thesearesubsystems,allstatesinsideofwhichcanbecorrectedviaaunitaryoperation up to an arbitrary transformation inside the gauge subsystem. Unlike DFSs, the unitary evolution followed by states in a unitarily correctable code are not restricted to the initial subsystem. An even more general concept is that of unitarily recoverable subsystems [93, 94], for which states can be recovered by a unitary transformation up to an expansion of the gauge subsystem. It was shown that any correctable subsystem is in fact a unitarily recoverable subsystem [95]. This reflects the so called subsystem principle [87, 161], according to which protected information is always contained in a 152 subsystem of the system’s Hilbert space. The connection between DFSs and unitarily recoverable subsystems suggests that similar conditions on the generators of evolution to those for DFSs can be derived in the case of general correctable subsystems. This is the subject of the present study. The chapter is organized as follows. In Section 6.2 we review the definitions of correctable subsystems and unitarily recoverable subsystems. In Section 6.3, we dis- cuss the necessary and sufficient conditions for such subsystems to exist in the case of CPTP maps. In Section 6.4, we derive conditions for the case of Markovian decoher- ence. The conditions for general correctability in this case are essentially the same as those for unitary correctability except that the dimension of the gauge subsystem is allowed to suddenly increase. For the case when the evolution is non-correctable, we conjecture a procedurefor tracking the subsystem which contains the optimal amount of undissipatedinformationanddiscussitspossibleimplications fortheproblemof op- timal error correction. In Section 6.5, we derive conditions on the system-environment Hamiltonian. In this case, the conditions for unitary correctability concern only the effect oftheHamiltonian onthesystem, whereastheconditionsforgeneral correctabil- ity concern the entire system-environment Hamiltonian. In the latter case, the state of the the noisy subsystem plus environment belongs to a particular subspace which plays an important role in the conditions. We extend the conditions to the case where theenvironmentisinitialized insideaparticularsubspace. InSection 6.6, weconclude. 6.2 Correctable subsystems For simplicity, weconsider thecase whereinformation is stored inonly one subsystem. Then there is a corresponding decomposition of the Hilbert space of the system, H S =H A ⊗H B ⊕K, (335) 153 where the subsystem H A is used for encoding of the protected information. The subsystem H B is referred to as the gauge subsystem, and K denotes the rest of the Hilbert space. In the formulation of OQEC [93, 94], the noise process is a completely positivetrace-preserving(CPTP)linearmapE :B(H S )→B(H S ),whereB(H)denotes the set of linear operators on a finite-dimensional Hilbert space H. Let the operator- sum representation of the mapE be E(σ) = X i M i σM † i , for all σ∈B(H S ), (336) where the Kraus operators{M i }⊆B(H S ) satisfy X i M † i M i =I S . (337) ThesubsystemH A inEq.(335) iscalled noiseless withrespect tothenoiseprocess E, if Tr B {(P AB ◦E)(σ)} =Tr B {σ}, (338) for all σ∈B(H S ) such that σ =P AB (σ), where P AB (·) =P AB (·)P AB (339) with P AB being the projector ofH S ontoH A ⊗H B , P AB H S =H A ⊗H B . (340) 154 Similarly, a correctable subsystem is one for which there exists a correcting CPTP map R : B(H S ) → B(H S ), such that the subsystem is noiseless with respect to the mapR◦E: Tr B {(P AB ◦R◦E)(σ)} =Tr B {σ}, (341) for all σ∈B(H S ) such that σ =P AB (σ). When the correcting map R is unitary, R = U, the subsystem is called unitarily correctable: Tr B {(P AB ◦U◦E)(σ)} =Tr B {σ}, (342) for all σ∈B(H S ) such that σ =P AB (σ). A similar but more general notion is that of a unitarily recoverable subsystem, for which the unitaryU need not bring the erroneous state back to the original subspace H A ⊗H B but can bring it in a subspaceH A ⊗H B ′ such that Tr B ′{(P AB ′ ◦U◦E)(σ)} =Tr B {σ}, for all σ∈B(H S ) such that σ =P AB (σ). Obviously, ifH A isunitarilyrecoverable, it isalsocorrectable, sinceonecanalways apply a local CPTP mapE B ′ →B :B(H B ′ )→B(H B ) which brings all states fromH B ′ to H B . (In fact, if the dimension of H B ′ is smaller or equal to that of H B , this can always be done by a unitary map, i.e.,H A is unitarily correctable.) In Ref. [95] it was shown that the reverse is also true—if H A is correctable, it is unitarily recoverable. This equivalence will provide the basis for our derivation of correctability conditions for continuous decoherence. 155 Before we proceed with our discussion, we point out that condition () can be equivalently written as [93, 94] U◦E(ρ⊗τ) =ρ⊗τ ′ , τ ′ ∈B(H B ′ ), for all ρ∈B(H A ), τ ∈B(H B ). (343) 6.3 Completely positive linear maps Let H S and H E denote the Hilbert spaces of a system and its environment, and let H =H S ⊗H E bethetotalHilbertspace. Aswepointedoutearlier, acommonexample of a CP map is the transformation that the state of a system undergoes if the system is initially decoupled from its environment,ρ(0) =ρ S (0)⊗ρ E (0), and both the system and environment evolve according to the Schr¨ odinger equation: dρ(t) dt =−i[H(t),ρ(t)]. (344) Equation (344) gives rise to the unitary transformation ρ(t) =V(t)ρ(0)V † (t), (345) with V(t)=Texp(−i Z t 0 H(τ)dτ), (346) whereT denotes time ordering. Under the assumption of an initially-decoupled state of the system and the environment, the transformation of the state of the system is described by the time-dependent CPTP map ρ S (0)→ρ S (t)≡Tr E (ρ(t)) = X i M i (t)ρ S (0)M † i (t), 156 with Kraus operators M i (t) = p λ ν hμ|V(t)|νi, i=(μ,ν) (347) where {|μi} is a basis in which the initial environment density matrix is diagonal, ρ B (0) = P μ λ μ |μihμ|. We already saw one example of such a map in Chapter 4 where we studied the evolution of a qubit coupled to a spin bath (Eq. (164)). The Kraus representation (336) applies to any CP linear map which need not nec- essarily arise from evolution of the type (344). This is why in the following theorem we derive conditions for discrete CP maps. For correctability under continuous de- cohrence, the same conditions must apply at any moment of time, i.e., one can think that the quantitiesM i ,U,C i , as well as the subsystemH B ′ in the theorem are implic- itly time-depedent. Theorem 1: The subsystem H A in the decomposition (335) is correctable under a CP linear map in the form (336), if and only if there exists a unitary operator U ∈B(H S ) such that the Kraus operators satisfy M i P AB =U † I A ⊗C B→B ′ i , C B→B ′ i :H B →H B ′ , ∀i. (348) Proof: Thesufficiency of condition (348) is obvious—using thatρ⊗τ in Eq. (343) satisfies ρ⊗τ =P AB ρ⊗τP AB , it can be immediately verified that Eq. (348) implies Eq. (343) with U = U(·)U † . Now assume that H A is unitarily recoverable and the recovery map is U = U(·)U † . The map U ◦E in Eq. (343) can then be thought of as having Kraus operators UM i . In particular, condition (343) has to be satisfied for ρ =|ψihψ|, τ =|φihφ| where|ψi∈H A and|φi∈H B are pure states. Notice that the 157 image of|ψihψ|⊗|φihφ| under the mapU◦E would be of the form|ψihψ|⊗τ ′ , only if all terms in Eq. (336) are of the form UM i |ψihψ|⊗|φihφ|M † i U † =|g i (ψ)| 2 |ψihψ|⊗|φ ′ i (ψ)ihφ ′ i (ψ)|, g i (ψ)∈C, (349) where for now we assume that g i and|φ ′ i i may depend on|ψi. In other words, UM i |ψi|φi =g i (ψ)|ψi|φ ′ i (ψ)i, g i (ψ)∈C, ∀i. (350) But if we impose (350) on a linear superposition|ψi=a|ψ 1 i+b|ψ 2 i, (a,b6=0), we obtain g i (ψ 1 ) =g i (ψ 2 ) and|φ ′ i (ψ 1 )i=|φ ′ i (ψ 2 )i i.e., g i (ψ)≡g i , |φ ′ i (ψ)i≡|φ ′ i i, ∀|ψi∈H A , ∀i. (351) Since Eq. (350) has to be satisfied for all|ψi∈H A and all|φi∈H B , we obtain UM i P AB =I A ⊗C B→B ′ i , C B→B ′ i :H B →H B ′ , ∀i. (352) ApplyingU † from the left yields condition (348). Weremarkthatcondition(348)isequivalenttotheconditionsobtainedinRef.[94]. 6.4 Markovian dynamics The most general continuous completely positive time-local evolution of the state of a quantum system is described by a semi-group master equation in the form (243) but with time dependent coefficients, dρ(t) dt =−i[H(t),ρ(t)]− 1 2 X j (2L j (t)ρ(t)L † j (t) −L † j (t)L j (t)ρ(t)−ρ(t)L † j (t)L j (t))≡L(t)ρ(t). (353) 158 (For a discussion of the situations in which such time-dependent Markovian evolution can arise, see, e.g., Ref. [100].) HereH(t) is a system Hamiltonian,L j (t) are Lindblad operators, and L(t) is the Liouvillian superoperator corresponding to this dynamics. (Thedecoherenceratesλ j thatappearinEq.(243), herehavebeenabsorbedinL j (t).) The general evolution of a state is given by ρ(t 2 ) =Texp( Z t 2 t 1 L(τ)dτ)ρ(t 1 ), t 2 >t 1 . (354) Wewillfirstderivenecessary andsufficientconditionsforunitarilycorrectable sub- systems under the dynamics (353), and then will extend them to the case of unitarily recoverable subsystems. In the case of continuous dynamics, the error mapE and the error-correcting map U in Eq. (342) are generally time dependent. If we set t = 0 as the initial time at which the system is prepared, the error map resulting from the dynamics (353) is E(t)(·) =Texp Z t 0 L(τ)dτ (·). (355) Let the U(t) = U(t)(·)U † (t) be the unitary error-correcting map in Eq. (342). We can define the rotating frame corresponding to U † (t) as the transformation of each operator as O(t)→ e O(t) =U(t)O(t)U † (t). (356) In this frame, the Lindblad equation (353) can be written as de ρ(t) dt =−i[ e H(t)+H ′ (t),e ρ(t)]− 1 2 X j (2 e L j (t)e ρ(t) e L † j (t) − e L † j (t) e L j (t)e ρ(t)−e ρ(t) e L † j (t) e L j (t))≡ e L(t)e ρ(t), (357) 159 where H ′ (t) is defined through i dU(t) dt =H ′ (t)U(t), (358) i.e., U(t)=Texp −i Z t 0 H ′ (τ)dτ . (359) The CPTP map resulting from the dynamics (357) is e E(t)(·) =Texp Z t 0 e L(τ)dτ (·). (360) Theorem 2: Let e H(t) and e L j (t) be the Hamiltonian and the Lindblad operators in the rotating frame (356) with U(t) given by Eq. (358). Then the subsystem H A in the decomposition (335) is correctable by U(t) during the evolution (353), if and only if e L j (t)P AB =I A ⊗C B j (t), C B j (t)∈B(H B ), ∀j (361) and P AB ( e H(t)+H ′ (t)) =I A ⊗D B (t), D B (t)∈B(H B ) (362) and P AB ( e H(t)+H ′ (t)+ i 2 X j e L † j (t) e L j (t))P K =0 (363) for all t, where P K denotes the projector on K. Proof: Since by definition U(t) is an error-correcting map for subsystem H A , if P AB (ρ(0)) = ρ(0), we have Tr B {P AB ◦ e E(e ρ(0))} = Tr B {P AB (e ρ(t))} = Tr B {P AB ◦ 160 U(t)◦E(t)(ρ(0))} = Tr B {ρ(0)} = Tr B {˜ ρ(0)}, i.e, H A is a noiseless subsystem under the evolution in the rotating frame (357). Then the theorem follows from Eq. (357) and the conditions for noiseless subsystems under Markovian decoherence obtained in [138]. Comment: Conditions (362) and (363) can be used to obtain the operator H ′ (t) (and henceU(t)) if the initial decomposition (335) is known. Note that there is a free- dom in the definition ofH ′ (t). For example,D B (t) in Eq. (362) can be any Hermitian operator. In particular, we can chooseD B (t) =0. Also, the termP K H ′ (t)P K does not play a role and can be chosen arbitrary. Using that P K =I−P AB , we can choose H ′ (t) =− e H(t)− i 2 P AB X j e L † j (t) e L j (t) + i 2 X j e L † j (t) e L j (t) P AB , (364) which satisfies Eq. (362) and Eq. (363). Using Eq. (356), Eq. (358) and Eq. (364), we obtain the following first-order differential equation for U(t): i dU(t) dt =−U(t)H(t)− i 2 P AB U(t) X j L † j (t)L j (t) + i 2 U(t) X j L † j (t)L j (t) U † (t)P AB U(t). (365) This equation can be used to solve for U(t) starting from U(0) =I. NoticethatsinceH A isunitarilycorrectablebyU(t), attimettheinitiallyencoded information can be thought of as contained in the subsystemH A (t) defined through H A (t)⊗H B (t)≡U † (t)H A ⊗H B , (366) i.e., this subsystem is obtained from H A in Eq. (335) via the unitary transformation U † (t). One can easily verify that the fact that the right-hand side of Eq. (361) acts trivially onH A together with Eq. (362) are necessary and sufficient conditions for an 161 arbitrary state encoded in subsystem H A (t) to undergo trivial dynamics at time t. Therefore, these conditions can be thought of as the conditions for lack of noise in the instantaneous subsystem that contains the protected information. On the other hand, the fact that the right-hand side of Eq. (361) maps states fromH A ⊗H B toH A ⊗H B together with Eq. (363) are necessary and sufficient conditions for states inside the time-dependent subspace U † (t)H AB not to leave this subspace during the evolution. Thus the conditions of the theorem can be thought off as describing a time-varying noiseless subsystemH A (t). Wenowextendtheaboveconditionstothecaseofunitarilyrecoverablesubsystems. Aswepointedoutearlier, thedifferencebetweenaunitarilycorrectableandaunitarily recoverable subsystem is that in the latter the dimension of the gauge subsystem may increase. Since the dimension of the gauge subsystem is an integer, this increase can happen only in a jump-like fashion at particular moments. Between these moments, the evolution is unitarily correctable. Therefore, we can state the following Theorem 3: The subsystem H A in Eq. (335) is correctable during the evolution (353), if and only if there exist times t i , i = 0,1,2,..., t 0 = 0, t i <t i+1 , such that for each interval between t i−1 and t i there exists a decomposition H S =H A ⊗H B i ⊕K i , H B i ∋H B i−1 , (367) with respect to which the evolution during this interval is unitarily correctable. Remark: An increase of the gauge subsystem at time t i happens if the operator C j (t) in Eq. (361) obtains non-zero components that map states from H B i to H B i+1 . From that moment on, t i ≤t≤t i+1 , Eq. (361) must hold for the new decomposition H S =H A ⊗H B i+1 ⊕K i+1 . TheunitaryU(t) isdeterminedfromEq.(362) andEq.(363) as described earlier. 162 The conditions derived in this section provide insights into the mechanism of information preservation under Markovian dynamics, and thus could have implica- tions for the problem of error correction when perfect correctability is not possi- ble [132, 171, 58, 91]. For example, it is possible that the unitary operation con- structed according to Eq. (358) with the appropriate modification for the case of in- creasing gauge subsystem, may be useful for error-correction also when the conditions of the theorems are only approximately satisfied. Notice that the generator driving the effective evolution of the subspace U † (t)H A ⊗H B whose projector we denote by P AB (t)≡U † (t)P AB U(t), can be written as L(t)(·) =−i[H eff (t),·]+D(t)(·)+S(t)(·), (368) where H eff (t) =H(t)+ i 2 P AB (t) X j L † j (t)L j (t) − i 2 X j L † j (t)L j (t) P AB (t) (369) is an effective Hamiltonian, D(t)(·) = X j L j (t)(·)L † j (t) (370) is a dissipator, and S(t)(·) =− 1 2 P AB (t) X j L † j (t)L j (t) P AB (t)(·) − 1 2 (·)P AB (t) X j L † j (t)L j (t) P AB (t) (371) 163 is a superoperator acting on B(U † (t)H AB ). The dissipator most generally causes an irreversible loss of the information contained in the current subspace, which may in- volve loss of the information stored in subsystemH A (t) as well as an increase of the gauge subsystem. The superoperator S(t)(·) gives rise to a transformation solely in- side the current subspace. In the case when the evolution is correctable, this operator acts locally on the gauge subsystem, but in the general case it may act non-trivially on H A (t). The role of the effective Hamiltonian is to rotate the current subspace by an infinitesimal amount. If one could argue that the information lost under the ac- tion of D(t) and S(t) is in principle irretrievable, then heuristically one could expect that after a single time stepdt, the corresponding factor of the infinitesimally rotated (possibly expanded) subspace will contain the maximal amount of the remaining en- coded information. Note that to keep track of the increase of the gauge subsystem one would need to determine the operatorC j on the right-hand side of Eq. (361) that optimally approximates the left-hand side. Of course, since the dissipator generally causes leakage of states outside of the current subspace, the error-correcting map at the end would have to involve more than just a unitary recovery followed by a CPTP map on the gauge subsystem. In order to maximize the fidelity [116] of the encoded information with a perfectly encoded state, one would have to bring the state of the system fully inside the subspace H A ⊗H B . These heuristic arguments, however, re- quire a rigorous analysis. It is possible that the action of the superoperatorsD(t) and S(t) may be partially correctable and thus one may have to modify the unitary (358) in order to optimally track the retrievable information. We leave this as a problem for future investigation. 164 6.5 Conditions on the system-environment Hamiltonian We now derive conditions for correctability of a subsystem when the dynamics of the system and the environment is described by the Schr¨ odinger equation (344). While the CP-map conditions can account for such dynamics when the states of the system and the environemnt are initially disentangled, they depend on the initial state of the environment. Below, we will first derive conditions on the system-environment Hamiltonian that hold for any state of the environment, and then extend them to the case when the environment is initialized inside a particular subspace. We point out that the equivalence between unitary recoverable subsystems and correctablesubsystemshasbeenprovenforCPTPmaps. Here,wecouldhaveanon-CP evolution since the initial state of the system and the environment may be entangled. Nevertheless, since correctability must hold for the case when the initial state of the systemandtheenvironmentisseparable, theconditionsweobtainarenecessary. They are obviously also sufficient since unitary recoverability implies correctability. Let us write the system-environmnt Hamiltonian as H SE (t) =H S (t)⊗I E +I E ⊗H E (t)+H I (t), (372) whereH S (t)andH E (t)arethesystemandtheenvironmentHamiltoniansrespectively, and H I (t) = X i S j (t)⊗E j (t), (373) is the interaction Hamiltonian. From the point of view of the Hilbert space of the system plus environment, the decomposition (335) reads H =(H A ⊗H B ⊕K)⊗H E =H A ⊗H B ⊗H E ⊕K⊗H E . (374) 165 6.5.1 Conditions independent of the state of the environment We will consider again conditions for unitary correctability first, and then conditions for general correctability. In the rotating frame (356), the Schr¨ odinger equation (344) becomes de ρ(t) dt =−i[ e H SE (t)+H ′ (t),e ρ(t)]. (375) Sincein this picturea unitarily-correctable subsystemis noiseless, we can state the following Theorem 4: Consider the evolution (344) driven by the Hamiltonian (372). Let e H S (t) and e S j (t) be the system Hamiltonian and the interaction operators (373) in the rotating frame (356) with U(t) given by Eq. (358). Then the subsystem H A in the decomposition (335) is correctable by U(t) during this evolution, if and only if e S j (t)P AB =I A ⊗C B j (t), C B j (t)∈B(H B ), ∀j (376) and ( e H S (t)+H ′ (t))P AB =I A ⊗D B (t), D B (t)∈B(H B ). (377) Proof: With respect to the evolution in the rotating frame (356), the subsystem H A isnoiseless. Thetheoremfollowsfromtheconditionsfornoiselesssubsystemsunder Hamiltonian dynamics [138] applied to the Hamiltonian in the rotating frame. Note that the fact that the operator on the right-hand side of Eq. (377) sends states from H A ⊗H B toH A ⊗H B implies that theoff-diagonal terms of e H S (t)+H ′ (t) in theblock basiscorrespondingtothedecomposition(335)vanish,i.e.,P AB ( e H S (t)+H ′ (t))P K =0. 166 Comment: The Hamiltonian H ′ (t) can be obtained from conditions (376) and (377). We can choose D B (t) = 0 and define H ′ (t) = − e H S (t), which together with Eq. (358) yields i dU(t) dt =−U(t)H S (t), (378) i.e., U † (t) =Texp −i Z t 0 H S (τ)dτ . (379) This simply means that the evolution of the subspace that contains the encoded infor- mation is driven by the system Hamiltonian. Theconditionsagain canbeseparated intotwo parts. Thefact that theright-hand sides of Eq. (376) and Eq. (377) act trivially onH A is necessary and sufficient for the information stored in the instantaneous subsystemH A (t) to undergo trivial dynamics at time t. The fact that the right-hand-sides of these equations do not take states outside of H A ⊗H B is necessary and sufficient for states not to leave the subspace U † (t)H A ⊗H B as it evolves. The conditions for general correctability, however, are not obtained directly from Theorem 4 in analogy to the case of Markovian decoherence. Such conditions would certainly be sufficient, but it turns out that they are not necessary. This is because after applying the unitary recovery operation, the state of the gauge subsystem H B ′ (which is generally larger than the initial gauge subsystemH B ) plus the environment would generally belong to a proper subspace of H B ′ ⊗H E which cannot be factored into a subsystem belonging to H S and a subsystem belonging to H E . Thus it is not necessary that the Hamiltonian acts trivially on the factorH A inH A ⊗H B ′ ⊗H E , but only on the factor H A inH A ⊗ e H BE , where e H BE is the proper subspace in question. In the case of unitary correctability, tracing out the environment provides necessary conditions becauseH B ′ =H B , and henceH B ⊗H E is fully occupied. 167 Let H S =H A ⊗H B ′ ⊕K ′ (380) be a decomposition of the Hilbert space of the system such that the factorH B ′ ∋H B has the largest possible dimension. Since the evolution of the state of the system plus the environment is unitary, at time t the initial subspaceH A ⊗H B ⊗H E will be transformed to some other subspaceH A (t)⊗H B (t)⊗H E (t) which is unitarily related to the initial one. Applying the unitary recovery operationU(t) returns this subspace to the form H A ⊗ e H BE (t), where e H BE (t) is a subspace of H B ′ ⊗H E . Clearly, there exists a unitary operator W(t) :H B ′ ⊗H E →H B ′ ⊗H E that maps this subspace to the initial subspaceH B ⊗H E : W(t) e P BE (t)W † (t) =P BE . (381) (Here e P BE (t) denotes the projector on e H BE (t).) Note that as an operator on the entire Hilbert space, this unitary has the form W(t)≡I A ⊗W B ′ E (t)⊕I K ′⊗I E . Let us define the frame b O(t) =W(t)O(t)W † (t), (382) where i dW(t) dt =H ′′ (t)W(t). (383) Then the evolution driven by a Hamiltonian G(t), in this frame will be driven by b G(t)+H ′′ (t). Theorem 5: Let e O(t) denote the image of an operator O(t) ∈ B(H) under the transformation (356) with U(t) ∈ B(H S ) given by Eq. (358) (H ′ (t) ∈ B(H S )), and let b O(t) denote the image of O(t) under the transformation (382) with W(t) given by Eq. (383). Let P ABE be the projector on H A ⊗H B ⊗H E . The subsystem H A in the 168 decomposition (374) is recoverable by U(t) during the evolution driven by the system- environment Hamiltonian H SE (t), if and only if there exists H ′′ (t) ∈ B(H B ′ ⊗H E ), where H B ′ was defined in (380), such that ( b e H SE (t)+ b H ′ (t)+H ′′ (t))P ABE =I A ⊗D BE (t), (384) D BE (t)∈B(H B ⊗H E ), ∀t. Proof: Assume that the information encoded in H A is unitarily recoverable by U(t). Consider the evolution in the frame defined through the unitary operation W(t)U(t), whereW(t) is definedas in Eq. (381). In this frame, which can beobtained by consecutively applying the transformations (356) and (382), the Hamiltonian is b e H SE (t)+ b H ′ (t)+H ′′ (t). Under this Hamiltonian, the subsystemH A must benoiseless and no states should leave the subspaceH A ⊗H B ⊗H E . It is straightforward to see that the first requirement means that H A must be acted upon trivially by all terms of the Hamiltonian, hence the factor I A on the right-hand side of Eq. (384). At the same time, the subspace H B ⊗H E must be preserved by the action of the Hamilto- nian, which implies that the factor D BE (t) on the right-hand side of Eq. (384) must send states fromH B ⊗H E to H B ⊗H E . Note that this implies that the off-diagonal terms of the Hamiltonian in the block form corresponding to the decomposition (374) must vanish, i.e., P ABE ( b e H SE (t)+ b H ′ (t)+H ′′ (t))P ABE ⊥ =0, whereP ABE ⊥ denoted the projector onK⊗H E . Obviously, these conditions are also sufficient, since they ensure that in the frame defined by the unitary transformation W(t)U(t), the evolution of H A is trivial and states insidethesubspaceH B ⊗H E evolve unitarily underthe action of the HamiltonianD BE (t). SinceW(t) acts onH B ′ ⊗H E , subsystemH A is invariant also in the rotating frame (356). This means that H A is recoverable by the unitary U(t). 169 Comment: Similarly to the previous cases, the unitary operators U(t) and W(t) can be obtained iteratively from Eq. (384) if the decomposition (335) is given. Since H ′′ (t) acts on H B ′ ⊗H E , from Eq. (384) it follows that the operator b e H SE (t)+ b H ′ (t) must satisfy ( b e H SE (t)+ b H ′ (t))P ABE =I A ⊗F B ′ E (t), F B ′ E (t)∈B(H B ′ ⊗H E ). (385) At the same time, we can choose H ′′ (t) so that D BE (t)=0. This corresponds to W(t) e H BE (t)=H B ⊗H E , (386) where e H BE (t) was definedin the discussion before Theorem 5. To ensureD BE (t)=0, we can choose H ′′ (t) =− b e H SE (t)− b H ′ (t)+P ABE ⊥ b e H SE (t)+ b H ′ (t) , (387) where P ABE ⊥ (·) = P ABE ⊥ (·)P ABE ⊥ . For t = 0 (U(0) = I, W(0) = I), we can find a solution for b H ′ (0) =H ′ (0) from Eq. (385), given the Hamiltonian b e H SE (0) =H SE (0). Plugging the solution in Eq. (387), we can obtain H ′′ (0). For the unitaries after a single time step dt we then have U(dt) =I−iH ′ (0)dt+O(dt 2 ), (388) W(dt) =I−iH ′′ (0)dt+O(dt 2 ). (389) UsingU(dt)andW(dt)wecancalculate b e H SE (dt)accordingtoEq.(356)andEq.(382). Then we can solve Eq. (385) for b H ′ (dt) = W(dt)H ′ (dt)W † (dt), which we can use in Eq. (387) to find H ′′ (dt), and so on. Note that here we cannot specify a simple 170 expression for b H ′ (t) in terms of b e H SE (t), since we do not have the freedom to choose fully F B ′ E (t) in Eq. (385) due to the restriction that H ′ (t) acts locally onH S . We point out that condition (384) again can be understood as consisting of two parts—thefact that theright-hand sideacts trivially onH A is necessary andsufficient for the instantaneous dynamics undergone by the subsystem U † (t)W † (t)H A at time t to be trivial, while the fact that it preservesH A ⊗H B ⊗H E is necessary and sufficient for states not to leave U † (t)W † (t)H A ⊗H B ⊗H E as it evolves. It is tempting to perform an argument similar to the one we presented for the Markovian case about the possible relation of the specified recovery unitary operation U(t) and theoptimal error-correcting map in the case of approximate error correction. If theencoded information is not perfectly preserved, wecan construct theunitary op- erationU(t)asexplainedinthecomment afterTheorem5byoptimally approximating Eq. (385) and Eq. (387). However, in this case the evolution is not irreversible and the information that leaks out of the system may return to it. Thus we cannot argue that theunitarymapspecifiedinthismannerwouldoptimally track theremainingencoded information. 6.5.2 Conditions depending on the initial state of the environment We can easily extend Theorem 5 to the case when the initial state of the environment belongs to a particular subspace H E 0 ∈ H E . The only modification is that instead of P ABE in Eq. (384), we must have P ABE 0 , where P ABE 0 is the projector on H A ⊗ H B ⊗H E 0 , and on the right-hand side must have D BE 0 (t)∈B(H B ⊗H E 0 ). The following two theorems follow by arguments analogous to those for Theorem 5. We assume the same definitions as in Theorem 5 (Eq. (356), Eq. (358), Eq. (382), Eq. (383) ), except that in the second theorem we restrict the definition of H ′′ (t). Theorem6: LetP ABE 0 betheprojector onH A ⊗H B ⊗H E 0 , whereH E 0 ∈H E . The subsystem H A in the decomposition (374) is recoverable by U(t) ∈ B(H S ) during the 171 evolution driven by the system-environment Hamiltonian H SE (t) when the state of the environment is initialized insideH E 0 , if and only if there exists H ′′ (t)∈B(H B ′ ⊗H E ) such that ( b e H SE (t)+ b H ′ (t)+H ′′ (t))P ABE 0 =I A ⊗D BE 0 (t), (390) D BE 0 (t)∈B(H B ⊗H E 0 ), ∀t. The conditions for unitary correctability in this case require the additional restric- tion thatW(t) acts onH B ⊗H E and not onH B ′ ⊗H E , since in this case U(t) brings the state inside H A ⊗H B ⊗H E . Notice that when the state of the environment is initialized in a particular subspace, we cannot use conditions for unitary correctability similar to those in Theorem 4. This is because after the correction U(t), the state of the gauge subsystem plus environment may belong to a proper subspace ofH B ⊗H E and tracing out the environment would not yield necessary conditions. Theorem7: LetP ABE 0 betheprojector onH A ⊗H B ⊗H E 0 , whereH E 0 ∈H E . The subsystem H A in the decomposition (374) is correctable by U(t) ∈ B(H S ) during the evolution driven by the system-environment Hamiltonian H SE (t) when the state of the environment is initialized insideH E 0 , if and only if there exists H ′′ (t)∈B(H B ⊗H E ) such that ( b e H SE (t)+ b H ′ (t)+H ′′ (t))P ABE 0 =I A ⊗D BE 0 (t), (391) D BE 0 (t)∈B(H B ⊗H E 0 ), ∀t. Notice that the conditions of Theorem 6 and Theorem 7 do not depend on the particularinitialstateoftheenvironmentbutonlyonthesubspacetowhichitbelongs. 172 This can be understood by noticing that different environment states inside the same subspacegiverisetoKrausoperators(347)whicharelinearcombinationsofeachother. The discretization of errors in operator quantum error correction [93, 94] implies that all such maps will be correctable. Theconditionsforcorrectable dynamicsdependentonthestateoftheenvironment could be useful if we are able to prepare the state of the environment in the necessary subspace. Theenvironment,however,isgenerallyoutsideoftheexperimenter’scontrol. Nevertheless, it is conceivable that the experimenter may have some control over the environment(forexample, byvaryingitstemperature), whichforcertainHamiltonians could bring the environment state close to a subspace for which the evolution of the system is correctable. It is important to point out that according to the result we derive in the next chapter, the error dueto imperfect initialization of thebath will not increase under the evolution. 6.6 Summary and outlook We have derived conditions for correctability of subsystems undercontinuous decoher- ence. We first presented conditions for the case when the evolution can be described by a CPTP linear map. These conditions are equivalent to those known for operator codes [93, 94] except that we consider them for time-dependent noise processes. We thenderived condition forthecaseof Markovian decoherenceandgeneral Hamiltonian evolution of the system and the environment. We derived conditions for both unitary correctability andgeneral correctability, usingthefactthat correctable subsystemsare unitarily recoverable [95]. Theconditionsforcorrectability inbothMarkovian andHamiltonian evolution can be understood as consisting of two parts—the first is necessary and sufficient for lack of noise inside the instantaneous subsystem that contains the information, and the 173 second is necessary and sufficient for states not to leave the subsystem as it evolves with time. In this sense, the new conditions can be thought of a generalizations of the conditionsfornoiselesssubsystemstothecasewherethesubsystemistime-dependent. In the Hamiltonian case, the conditions for unitary correctability concern only the action of the Hamiltonian on the system, whereas the conditions for general cor- rectability concern theentire system-bath Hamiltonian. Thereason for thisis that the state of the gauge subsystem plus the environment generally belongs to a particular subspace, which does not factor into sectors belonging separately to the system and the environment. We also derived conditions in the Hamiltonian case that depend on the initial state of the environment. These conditions could be useful, in principle, since errors due to imperfect initialization of the environment do not increase under the evolution. Furthermore, these conditions could provide a better understanding of correctability under CPTP maps, since a CPTP map that results from Hamiltonian evolution depends on both the Hamiltonian and the initial state of the environment. An interesting generalization of this work would be to derive similar condition for the case of the Nakajima-Zwanzig or the TCL master equations. We discussed possible implications of the conditions we derived for the problem of optimal recovery in the case of imperfectly preserved information. We hope that the results obtained in this study will provide insight into the mechanisms of information flowunderdecoherencethatcouldbeusefulintheareaofapproximateerrorcorrection as well. 174 Chapter 7: Robustness of operator quantum error correction against initialization errors Theconditions we derived in the previouschapter, as well as the standardOQECcon- ditionsfordiscreteerrors,dependontheassumptionthatstatesareperfectlyinitialized inside the subspace factored by the correctable subsystem. In practice, however, per- fect initialization of the state may not be easy to achieve. Hence, it is important to understandtowhatextentthepreparationrequirementcanberelaxed. Inthischapter, we examine the performance of OQEC in the case of imperfect encoding. 7.1 Preliminaries Ascanbeseenfromthedefinitions(338) and(341), theconcept ofnoiseless subsystem is a cornerstone in the theory of OQEC; it serves as a basis for the definition of cor- rectable subsystem and error correction in general. As shown in Ref. [138], in order to ensure perfect noiselessness of a subsystem in the case of imperfect initialization, the noise process has to satisfy more restrictive conditions than those required in the case of perfect initialization. It was believed that these conditions are necessary if a noise- less (or more generally decoherence-free) subsystem is to be robust against arbitrarily large initialization errors. The fundamental relation between a noiseless subsystem and a correctable subsystem implies that in the case of imperfect initialization, more restrictive conditions would be needed for OQEC codes as well. 175 In this chapter we show that with respect to the ability of a code to protect from errors, more restrictive conditions are not necessary. For this purpose, we define a measure of the fidelity between the encoded information in two states for the case of subsystem encoding. We first give an intuitive motivation for the definition, and then study the properties of the measure. We then show that the effective noise that can arise inside the code due to imperfect initialization under the standard conditions, is such that it can only increase the fidelity of the encoded information with the infor- mation encoded in a perfectly prepared state. This robustness against initialization errors is shown to hold also when the state is subject to encoded operations. 7.2 Review of the noiseless-subsystem conditions on the Kraus operators For simplicity, we consider again the case where information is stored in only one subsystem, i.e., we consider the decomposition (335). The definition of noiseless sub- system (338) implies that the information encoded in B(H A ) remains invariant after theprocessE, iftheinitial densityoperatorofthesystemρ(0) belongstoB(H A ⊗H B ). If, however, one allows imperfect initialization,ρ(0)6=P AB (ρ(0)), this need not bethe case. Consider the “initialization-free” analogue of the definition (338): Tr B {(P AB ◦E)(σ)} =Tr B {P AB (σ)}, for all σ∈B(H S ). Obviously Eq. () implies Eq. (338), but the reverse is not true. As shown in [138], the definition () imposes more restrictive conditions on the channelE than those imposed by (338). To see this, consider the form of the Kraus operators M i ofE ((336)) in the block basis corresponding to the decomposition (335). From a result derived in [138] 176 it follows that the subsystem H A is noiseless in the sense of Eq. (338), if and only if the Kraus operators have the form M i = I A ⊗C B i D i 0 G i , (392) where the upper left block corresponds to the subspace H A ⊗H B , and the lower right block corresponds to K. The completeness relation (337) implies the following conditions on the operators C B i , D i , and G i : X i C †B i C B i =I B , (393) X i I A ⊗C †B i D i =0, (394) X i (D † i D i +G † i G i )=I K . (395) In the same block basis, a perfectly initialized stateρ and its image underthe map (392) have the form ρ= ρ 1 0 0 0 , E(ρ) = ρ ′ 1 0 0 0 , (396) where ρ ′ 1 = P i I A ⊗C B i ρ 1 I A ⊗C †B i . Using the linearity and cyclic invariance of the trace together with Eq. (393), we obtain Tr B {(P AB ◦E)(ρ)} =Tr B { X i I A ⊗C B i ρ 1 I A ⊗C †B i } =Tr B {ρ 1 X i I A ⊗C †B i C B i | {z } I A ⊗I B }=Tr B {P AB (ρ)}, (397) i.e., the reduced operator onH A remains invariant. 177 On the other hand, an imperfectly initialized state ˜ ρ and its image have the form ˜ ρ= ˜ ρ 1 ˜ ρ 2 ˜ ρ † 2 ˜ ρ 3 , E(˜ ρ)= ˜ ρ ′ 1 ˜ ρ ′ 2 ˜ ρ ′† 2 ˜ ρ ′ 3 . (398) Here ˜ ρ 2 and/or ˜ ρ 3 are non-vanishing, and ˜ ρ ′ 1 = X i (I A ⊗C B i ˜ ρ 1 I A ⊗C †B i +D i ˜ ρ † 2 I A ⊗C †B i +I A ⊗C B i ˜ ρ 2 D † i +D i ˜ ρ 3 D † i ), (399) ˜ ρ ′ 2 = X i (I A ⊗C B i ˜ ρ 2 G † i +D i ˜ ρ 3 G † i ), (400) ˜ ρ ′ 3 = X i G i ˜ ρ 3 G † i . (401) Inthiscase,usingthelinearityandcyclicinvarianceofthetracetogetherwithEq.(393) and Eq. (394), we obtain Tr B {(P AB ◦E)(˜ ρ)} = Tr B { X i (I A ⊗C B i ˜ ρ 1 I A ⊗C †B i +D i ˜ ρ † 2 I A ⊗C †B i +I A ⊗C B i ˜ ρ 2 D † i +D i ˜ ρ 3 D † i )} (402) = Tr B {˜ ρ 1 X i I A ⊗C †B i C B i | {z } I A ⊗I B }+Tr B {( X i I A ⊗C †B i D i | {z } 0 )˜ ρ † 2 } +Tr B {˜ ρ 2 ( X i I A ⊗C †B i D i | {z } 0 ) † }+Tr B { X i D i ˜ ρ 3 D † i } = Tr B ˜ ρ 1 +Tr B { X i D i ˜ ρ 3 D † i }6=Tr B ˜ ρ 1 ≡Tr B {P AB (˜ ρ)}, i.e., the reduced operator on H A is not preserved. It is easy to see that the reduced operator would be preserved for every imperfectly initialized state if and only if we impose the additional condition D i =0, for all i. (403) 178 Thisfurtherrestriction to theform of the Krausoperators is equivalent to the require- mentthattherearenotransitionsfromthesubspaceKtothesubspaceH A ⊗H B under the process E. This is in addition to the requirement that no states leave H A ⊗H B , which is ensured by the vanishing lower left blocks of the Kraus operators (392). Con- dition (403) automatically imposes an additional restriction on the error-correction conditions, since ifR is an error-correcting map in this “initialization-free” sense, the mapR◦E would have to satisfy Eq. (403). But is this constraint necessary from the point of view of the ability of the code to correct further errors? Notice that since ˜ ρis apositiveoperator, ˜ ρ 3 ispositive, andhenceTr B { P i D i ˜ ρ 3 D † i } is positive. The reduced operator on subsystem H A , although unnormalized, can be regarded as a (partial) probability mixture of states on H A . The noise process modifies the original mixture (Tr B ˜ ρ 1 ) by adding to it another partial mixture (the positive operator Tr B { P i D i ˜ ρ 3 D † i }). Since the weight of any state already present in the mixture can only increase by this process, this should not worsen the faithfulness with which information is encoded in ˜ ρ. In order to make this argument rigorous, however, we need a measure that quantifies the faithfulness of the encoding. 7.3 Fidelitybetweentheencodedinformationintwostates 7.3.1 Motivating the definition If we consider two states with density operators τ and υ, a good measure of the faithfulness with which one state represents the other is given by the fidelity between the states: F(τ,υ) =Tr q √ τυ √ τ. (404) This quantity can be thought of as a square root of a generalized “transition proba- bility” between the two states τ and υ as defined by Uhlmann [155]. Another inter- pretation dueto Fuchs [59] gives an operational meaning of the fidelity as the minimal 179 overlap between the probability distributions generated by all possible generalized measurements on the states: F(τ,υ) = min {M i } X i p Tr{M i τ} p Tr{M i υ}. (405) Here, minimumistaken over all positiveoperators{M i }thatformapositiveoperator- valued measure (POVM) [92], P i M i =I S . In our case, we need a quantity that compares the encoded information in two states. Clearly, the standard fidelity between the states will not do since it measures thesimilaritybetweenthestatesontheentireHilbertspace. Theencodedinformation, however, concerns only the reduced operators on subsystem H A . In view of this, we propose the following Definition 1: Let τ and υ be two density operators on a Hilbert space H S with decomposition (335). The fidelity between the information encoded in subsystemH A in the two states is given by: F A (τ,υ) =max τ ′ ,υ ′ F(τ ′ ,υ ′ ), (406) where maximum is taken over all density operators τ ′ and υ ′ that have the same re- duced operators onH A asτ andυ: Tr B {P AB (τ ′ )}=Tr B {P AB (τ)}, Tr B {P AB (υ ′ )}= Tr B {P AB (υ)}. The intuition behindthis definition is that by maximizing over all states that have the same reduced operators on H A as the states being compared, we ensure that the measuredoesnotpenalizefordifferencesbetweenthestatesthatarenotduespecifically to differences between the reduced operators. 180 7.3.2 Properties of the measure Property 1 (Symmetry): Since the fidelity is symmetric with respect to its inputs, it is obvious from Eq. (406) that F A is also symmetric: F A (τ,υ) =F A (υ,τ). (407) Although intuitive, the definition (406) does not allow for a simple calculation of F A . We now derive an equivalent form for F A , which is simple and easy to compute. LetP K (·) =P K (·)P K denote the superoperator projector onB(K), and let ρ A ≡Tr B {P AB (ρ)}/Tr{P AB (ρ)} (408) denote the normalized reduced operator of ρ onH A . Theorem 1: The definition (406) is equivalent to F A (τ,υ) =f A (τ,υ)+ p Tr{P K (τ)}Tr{P K (υ)}, (409) where f A (τ,υ) = q Tr{P AB (τ)}Tr{P AB (υ)}F(τ A ,υ A ). (410) Proof: Letτ ∗ andυ ∗ betwo states for which themaximum on the right-hand side of Eq. (406) is attained. From the monotonicity of the standard fidelity under CPTP maps [16] it follows that F A (τ,υ) =F(τ ∗ ,υ ∗ )≤F(Π(τ ∗ ),Π(υ ∗ )), (411) 181 where Π(·) =P AB (·)+P K (·). But the states Π(τ ∗ ) and Π(υ ∗ ) satisfy Tr B {P AB (Π(τ ∗ ))}=Tr B {P AB (τ)}, (412) Tr B {P AB (Π(τ ∗ ))}=Tr B {P AB (υ)}, (413) i.e., they are among those states over which the maximum in Eq. (406) is taken. Therefore, F A (τ,υ) =F(Π(τ ∗ ),Π(υ ∗ )). (414) UsingEq.(404)andthefactthatintheblockbasiscorrespondingtothedecomposition (335) the states Π(τ ∗ ) and Π(υ ∗ ) have block-diagonal forms, it is easy to see that F(Π(τ ∗ ),Π(υ ∗ )) = ˇ F(P AB (τ ∗ ),P AB (υ ∗ ))+ ˇ F(P K (τ ∗ ),P K (υ ∗ )), (415) where ˇ F is a function that has the same expression as the fidelity (404), but is defined over all positive operators. From Eq. (412) and Eq. (413) it can be seen that Tr{P AB (τ ∗ )} = Tr{P AB (τ)}, Tr{P AB (υ ∗ )} = Tr{P AB (υ)}, which also implies that Tr{P K (τ ∗ )} = Tr{P K (τ)} = 1− Tr{P AB (τ)}, Tr{P K (υ ∗ )} = Tr{P K (υ)} = 1− Tr{P AB (υ)}. The two terms on the right-hand side of Eq. (415) can therefore be written as ˇ F(P AB (τ ∗ ),P AB (υ ∗ )) = q Tr{P AB (τ)}Tr{P AB (υ)} ×F P AB (τ ∗ ) Tr{P AB (τ)} , P AB (υ ∗ ) Tr{P AB (υ)} , (416) ˇ F(P K (τ ∗ ),P K (υ ∗ )) = p Tr{P K (τ)}Tr{P K (υ)} ×F P K (τ ∗ ) Tr{P K (τ)} , P K (υ ∗ ) Tr{P K (υ)} . (417) 182 Sinceτ ∗ andσ ∗ shouldmaximizetheright-handsideofEq.(415), andtheonlyrestric- tion on P K (τ ∗ ) and P K (υ ∗ ) is Tr{P K (τ ∗ )} = Tr{P K (τ)}, Tr{P K (υ ∗ )} = Tr{P K (υ)}, we must have F P K (τ ∗ ) Tr{P K (τ)} , P K (υ ∗ ) Tr{P K (υ)} =1, (418) i.e., P K (τ ∗ ) Tr{P K (τ)} = P K (υ ∗ ) Tr{P K (υ)} . (419) Thus we obtain ˇ F(P K (τ ∗ ),P K (υ ∗ ))= p Tr{P K (τ)}Tr{P K (υ)}. (420) The term (416) also must be maximized. Applying again the monotonicity of the fidelity under CPTP maps for the map Γ(ρ AB ) = Tr B {ρ AB }⊗|0 B ih0 B | defined on operators over H A ⊗H B , where|0 B i is some state inH B , we see that the term (416) must be equal to ˇ F(P AB (τ ∗ ),P AB (υ ∗ )) = q Tr{P AB (τ)}Tr{P AB (υ)}F(τ A ,υ A )≡f A (τ,υ). (421) This completes the proof. Wenextprovideanoperational interpretation ofthemeasureF A . For thisweneed the following Lemma: The function f A (τ,υ) defined in Eq. (410) equals the minimum overlap betweenthestatistical distributionsgenerated byall local measurementsonsubsystem H A : f A (τ,υ) = min {M i } X i p Tr{M i τ} p Tr{M i υ}, (422) 183 where M i =M A i ⊗I B , P i M i =I A ⊗I B , M A i >0, for all i. Note that since the operators M i do not form a complete POVM on the entire Hilbert space, the probability distributions p τ (i) = Tr{M i τ} and p υ (i) = Tr{M i υ} generated by such measurements generally do not sum up to 1. This reflects the fact that a measurement on subsystem H A requires a projection onto the subspace H A ⊗H B , i.e., it is realized through post-selection. Proof: Using that Tr{M i τ}=Tr{M A i ⊗I B P AB (τ)} =Tr{P AB (τ)}Tr{M A i ⊗I B P AB (τ) Tr{P AB (τ)} } =Tr{P AB (τ)}Tr{M A i τ A }, (423) we can write Eq. (422) in the form f A (τ,υ) = q Tr{P AB (τ)}Tr{P AB (υ)}min {M A i } X i q Tr{M A i τ A } q Tr{M A i υ A }. (424) From Eq. (405), we see that (424) is equivalent to (410). Theorem 2: F A (τ,υ) equals the minimum overlap F A (τ,υ) = min {M i } X i≥0 p Tr{M i τ} p Tr{M i υ} (425) between the statistical distributions generated by all possible measurements of the form M 0 =P K , M i =M A i ⊗I B for i≥1, P i≥0 M i =I S . Proof: The proof follows from Eq. (409) and Eq. (422). NotethatthemeasureF A comparestheinformationstoredinsubsystemH A ,which is the information extractable through local measurements onH A . The last result re- flects the intuition that extracting information encoded inH A involves a measurement that projects on the subspacesH A ⊗H B orK. 184 Property 2 (Normalization): From the definition (406) it is obvious that F A (τ,υ)≤F A (τ,τ) =1, τ 6=υ. (426) From Eq. (409) we can now see that F A (τ,υ) =1, iff Tr B {P AB (τ)} =Tr B {P AB (υ)}, (427) as one would expect from a measure that compares only the encoded information in H A . Proposition: Using that the maximum in Eq. (406) is attained for states of the form Π(τ ∗ ) and Π(υ ∗ ) (Eq. (414)) where τ ∗ and υ ∗ satisfy Eq. (419) and Eq. (421), without loss of generality we can assume that for all τ and υ, F A (τ,υ) =F(τ ∗ ,υ ∗ ), (428) where τ ∗ =Tr B {P AB (τ)}⊗|0 B ih0 B |+Tr{P K (τ)}|0 K ih0 K |, (429) υ ∗ =Tr B {P AB (υ)}⊗|0 B ih0 B |+Tr{P K (υ)}|0 K ih0 K |, (430) with|0 B i and|0 K i being some fixed states inH B andK, respectively. Property 3 (Strong concavity and concavity of the square of F A ): The formofF A givenbyEqs.(428)–(430) canbeusedforderivingvarioususefulproperties 185 of F A from the properties of the standard fidelity. For example, it implies that for all mixtures P i p i τ i and P i q i υ i we have F A ( X i p i τ i , X i q i υ i )=F( X i p i τ ∗ i , X i q i υ ∗ i ). (431) This means that the property of strong concavity of the fidelity [114] (and all weaker concavity properties that follow from it) as well as the concavity of the square of the fidelity [155], are automatically satisfied by the measureF A . Definition 2: Similarly to the concept of angle between two states [114] which can be defined from the standard fidelity, we can define an angle between the encoded information in two states: Λ A (τ,υ)≡arccosF A (τ,υ). (432) Property 4 (Triangle inequality): From Eqs. (428)–(430) it follows that just as the angle between states satisfies the triangle inequality, so does the angle between the encoded information: Λ A (τ,υ)≤Λ A (τ,φ)+Λ A (φ,υ). (433) Property 5 (Monotonicity of F A under local CPTP maps): We point out that the monotonicity under CPTP maps of the standard fidelity does not translate directly to the measure F A . Rather, as can be seen from Eq. (409), F A satisfies monotonicity under local CPTP maps onH A : F A (E(τ),E(υ))≥F A (τ,υ) (434) 186 for E =E A ⊗E B ⊕E K , (435) whereE A ,E B andE K are CPTP maps on operators overH A ,H B andK, respectively. Remark: There exist other maps under which F A is also non-decreasing. Such are the maps which take states from H A ⊗H B to K without transfer in the opposite direction. But in general, maps which couple states inH A ⊗H B with states inK, or states in H A with states in H B , do not obey this property. For example, a unitary map which swaps the states in H A and H B (assuming both subsystems are of the same dimension) could both increase or decrease the measure depending on the states in H B . Similarly, a unitary map exchanging states between H A ⊗H B and K could give rise to both increase or decrease of the measure depending on the states inK. Finally, the monotonicity of F A under local CPTP maps implies Property 6 (Contractivity of the angle under local CPTP maps): For CPTP maps of the form (435), Λ A satisfies Λ A (E(τ),E(υ))≤Λ A (τ,υ). (436) 7.4 Robustness of OQEC with respect to initialization errors Letusnowconsiderthefidelitybetweentheencodedinformationinanideallyprepared state (396) and in a state which is not perfectly initialized (398): F A (ρ,˜ ρ) = p Trρ 1 p Tr˜ ρ 1 F(ρ A ,˜ ρ A )+0 (437) =Tr q p Tr B ρ 1 Tr B ˜ ρ 1 p Tr B ρ 1 ≡ ˇ F(Tr B ρ 1 ,Tr B ˜ ρ 1 ). 187 After the noise process E with Kraus operators (392), the imperfectly encoded state transforms toE(˜ ρ). Its fidelity with the perfectly encoded state becomes F A (ρ,E(˜ ρ)) = ˇ F(Tr B ρ 1 ,Tr B ˜ ρ ′ 1 ) = ˇ F(Tr B ρ 1 ,Tr B ˜ ρ 1 +Tr B { X i D i ˜ ρ 3 D † i }), (438) where we have used the expressions for Tr B ρ ′ 1 and Tr B ˜ ρ ′ 1 obtained in Eq. (397) and Eq. (402). As we pointed out earlier, the operator Tr B { P i D i ˜ ρ 3 D † i } is positive. Then from the concavity of the square of the fidelity [155], it follows that ˇ F 2 Tr B ρ 1 ,Tr B ˜ ρ 1 +Tr B { X i D i ˜ ρ 3 D † i } ! =Trρ 1 Tr{˜ ρ 1 + X i D i ˜ ρ 3 D † i } (439) ×F 2 ρ A , Tr˜ ρ 1 Tr{˜ ρ 1 + P i D i ˜ ρ 3 D † i } ˜ ρ A + Tr{ P i D i ˜ ρ 3 D † i } Tr{˜ ρ 1 + P i D i ˜ ρ 3 D † i } Tr B { P i D i ˜ ρ 3 D † i } Tr{ P i D i ˜ ρ 3 D † i } ≥Trρ 1 Tr˜ ρ 1 F 2 (ρ A ,˜ ρ A )+Trρ 1 Tr{ X i D i ˜ ρ 3 D † i }F 2 ρ A , Tr B { P i D i ˜ ρ 3 D † i } Tr{ P i D i ˜ ρ 3 D † i } = ˇ F 2 (Tr B ρ 1 ,Tr B ˜ ρ 1 )+ ˇ F 2 (Tr B ρ 1 ,Tr B { X i D i ˜ ρ 3 D † i })≥ ˇ F 2 (Tr B ρ 1 ,Tr B ˜ ρ 1 ). (Here, the transition from the first to the second line is obtained by pulling out the normalization factors of the operators in ˇ F so that the latter can be expressed in terms of the fidelity F. The transition form the second to the third line is by using the concavity of the square of the fidelity. The last line is obtained by expressing the quantities again in terms of ˇ F). Therefore, we can state the following Theorem3: Thefidelitybetweentheencodedinformationinaperfectlyinitialized state (396) and an imperfectly initialized state (398) does not decrease under CPTP mapsE with Kraus operators of the form (392): F A (ρ,E(˜ ρ))≥F A (ρ,˜ ρ). (440) 188 We see that even if the “initialization-free” constraint (403) is not satisfied, no further decrease in the fidelity occurs as a result of the process. The effective noise (the term Tr B { P i D i ˜ ρ 3 D † i }) that arises due to violation of that constraint, can only decrease the initialization error. The above result can be generalized to include the possibility for information pro- cessing on the subsystem. Imagine that we want to perform a computational task which ideally corresponds to applying the CPTP map C A on the encoded state. In general,thesubsystemH A mayconsistofmanysubsystemsencodingseparateinforma- tion units(e.g., qubits), andthecomputational processmay involve many applications of error correction. The noise process itself generally acts continuously during the computation. Let us assume that all operations following the initialization are per- formed fault-tolerantly [143, 2, 85, 90, 67] so that the overall transformation C on a perfectly initialized state succeeds with an arbitrarily high probability (for a model of fault-tolerant quantum computation on subsystems, see, e.g., Ref. [11]). This means that the effect ofC on the reduced operator of a perfectly initialized state is tr B ρ 1 →C A (Tr B ρ 1 ) (441) up to an arbitrarily small error. Theorem 4: Let C be a CPTP map whose effect on reduced operator of every perfectly initialized state (396) is given by Eq. (441) with C A being a CPTP map on B(H A ). Then the fidelity between the encoded information in a perfectly initialized state (396) and an imperfectly initialized state (398) does not decrease underC: F A (C(ρ),C(˜ ρ))≥F A (ρ,˜ ρ). (442) Proof: From Eq. (441) it follows that the map C has Kraus operators with van- ishing lower left blocks, similarly to (392). If the state is not perfectly initialized, an 189 argument similar to the one performed earlier shows that the reduced operator on the subsystem transforms as Tr B ˜ ρ 1 →C A (Tr B ˜ ρ 1 )+ ˜ ρ A err , where ˜ ρ A err is a positive operator whichappearsasaresultofthepossiblynon-vanishingupperright blocksoftheKraus operators. Using an argument analogous to (439) and the monotonicity of the fidelity under CPTP maps [16], we obtain F A (C(ρ),C(˜ ρ)) = ˇ F(C A (Tr B ρ 1 ),C A (Tr B ˜ ρ 1 )+ ˜ ρ A err )+0 (443) ≥ ˇ F(C A (Tr B ρ 1 ),C A (Tr B ˜ ρ 1 )) = p Trρ 1 p Tr˜ ρ 1 F(C A (ρ A ),C A (˜ ρ A ) ≥ p Trρ 1 p Tr˜ ρ 1 F(ρ A ,˜ ρ A ) = ˇ F(Tr B ρ 1 ,Tr B ˜ ρ 1 ) =F A (ρ,˜ ρ). Again, the preparation error is not amplified by the process. The problem of how to deal with preparation errors has been discussed in the context of fault-tolerant computation on standard error-correction codes, e.g., in Ref. [130]. The situation for general OQEC is similar—if the initial state is known, the error can be eliminated by repeating the encoding. If the state to be encoded is unknown, the preparation error generally cannot be corrected. Nevertheless, encoding would still be worthwhile as long as the initialization error is smaller than the error which would result from leaving the state unprotected. 7.5 Summary and outlook In summary, we have shown that a noiseless subsystem is robust against initialization errors without the need for modification of the noiseless subsystem conditions. Simi- larly, we have argued that general OQEC codes are robust with respect to imperfect preparation in their standard form. This property is compatible with fault-tolerant methods of computation, which is essential for reliable quantum information process- ing. In order to rigorously prove our result, we introduced a measure of the fidelity F A (τ,υ) between the encoded information in two states. The measure is defined as 190 the maximum of the fidelity between all possible states which have the same reduced operators on the subsystem code as the states being compared. We derived a simple form of themeasureand discussedmany of its properties. We also gave an operational interpretation of the quantity. Sincetheconceptofencodedinformationiscentraltoquantuminformationscience, thefidelity measureintroducedinthisstudymay findvariousapplications. Itprovides a natural means for extending key concepts such as the fidelity of a quantum channel [88] or the entanglement fidelity [136] to the case of subsystem codes. 191 Chapter 8: A fault-tolerant scheme for holonomic quantum computation 8.1 Preliminaries There are two main sources of errors in quantum computers—environment-induced decoherence and imperfect control. According to the theory of fault tolerance [143, 53, 2, 85, 90, 68, 67, 130], if the errors of each type are sufficiently uncorrelated and their rates are bellow a certain threshold, it is possible to implement reliably an arbi- trarily longcomputational task withanefficient overhead of resources. Quantumerror correction thus provides a universal software strategy to combat noise in quantum computers. In addition to the software approach, there have also been proposals to deal with theeffectsofnoisebyhardwaremethodsthatproviderobustnessthroughtheirinherent properties. Onesuchmethodisholonomicquantumcomputation(HQC)[175,122]—an adiabatic,all-geometricmethodofcomputationwhichusesnon-Abeliangeneralizations [166] of the Berry phase [25]. It has been shown that due to its geometric nature, this approach is robust against various types of errors in the control parameters driving the evolution [43, 49, 146, 60, 177], and thus provides a degree of built-in resilience at the hardware level. 192 In Ref. [170] HQC was combined with the method of decoherence-free subspaces (DFSs) [55, 174, 104, 102], which was the first step towards systematic error protec- tion in conjunction with the holonomic approach. DFSs provide passive protection against certain types of correlated noise; however, they cannot protect against inde- pendent errors. The standard tool to deal with the latter is active error correction [142, 148]. Active error correction is also the basis of quantum fault tolerance, which is necessary for the scalability of any method of computation. Even if the system is perfectly isolated from its environment, when the size of the circuit increases, errors dueto imperfectoperations wouldaccumulate detrimentally unlessthey are corrected. Therefore, scalability of HQC requires combining the holonomic approach with active error correction. In this chapter, we presented a scheme which combines HQC with the techniques for fault-tolerant computation on stabilizer codes. This demonstrates that HQC is a scalable method of computation. The scheme uses Hamiltonians which are elements of the stabilizer, or in the case of subsystem codes—elements of the gauge group. Gates are implemented by slowly varying the Hamiltonians along suitable paths in parameter space, such that the resulting geometric transformation in each eigenspace of the Hamiltonian is transversal. On certain codes such as the 9-qubit Shor code [142] or its subsystem generalizations [13, 14], universal computation according to our scheme can be implemented with Hamiltonians of weight 2 and 3. This chapter is an expanded version of the work [120], where the scheme was first presented. Here we provide details on various proofs sketched in Ref. [120] and analyze properties of the scheme that were not discussed there. We examine in detail the construction for the Bacon-Shor code, discuss the adiabatic approximation for different parameterizations of the Hamiltonians, and provide explicit calculations of the holonomy for theZ gate for two different types of interpolation. 193 8.2 Holonomic Quantum Computation Let {H λ } be a family of Hamiltonians on an N-dimensional Hilbert space, which is continuously parameterized by a pointλ in a control-parameter manifoldM. Assume that the family has the same degeneracy structure, i.e., there are no level crossings. The Hamiltonians can then be written as H λ = P R n=1 ε n (λ)Π n (λ), where{ε n (λ)} R n=1 are the R different d n -fold degenerate eigenvalues of H λ , ( P R n=1 d n =N), and Π n (λ) are the projectors on the corresponding eigenspaces. If the parameter λ is changed adiabatically, a state which initially belongs to an eigenspace of the Hamiltonian will remain in the corresponding eigenspace as the Hamiltonian evolves. The unitary evo- lution that results from the action of the Hamiltonian H(t) :=H λ(t) is U(t) =Texp(−i Z t 0 dτH(τ)) =⊕ R n=1 e iωn(t) U λ An (t), (444) where ω n (t) = − R t 0 dτε n (λ(τ)) is a dynamical phase, and U λ An (t) is given by the fol- lowing path-ordered exponent: U λ An (t) =Pexp( Z λ(t) λ(0) A n ). (445) HereA n istheWilczek-Zeeconnection[166],A n = P μ A n,μ dλ μ ,whereA n,μ hasmatrix elements [166] (A n,μ ) αβ =hnα;λ| ∂ ∂λ μ |nβ;λi. (446) The parameters λ μ are local coordinates on M (1≤μ≤ dimM) and{|nα;λi} dn α=1 is an orthonormal basis of then th eigenspace of the Hamiltonian at the point λ. When the path λ(t) forms a loop γ(t), γ(0) =γ(T)=λ 0 , the unitary matrix U γ n ≡U λ An (T) =Pexp( I γ A n ) (447) 194 is called the holonomy associated with the loop. In the case when then th energy level is non-degenerate (d n = 1), the corresponding holonomy reduces to the Berry phase [25]. The set Hol(A) ={U γ /γ ∈L λ 0 (M)}, where L λ 0 (M) ={γ : [0,T]→M/γ(0) = γ(T) = λ 0 } is the space of all loops based on λ 0 , is a subgroup of U(d n ) called the holonomy group. In Refs. [175, 122] it was shown that if the dimension of the control manifold is sufficiently large, quantum holonomies can be used as a means of universal quantum computation. In the proposed approach, logical states are encoded in the degenerate eigenspace of a Hamiltonian and gates are implemented by adiabatically varying the Hamiltonian along suitable loops in the parameter manifold (for a construction of a universal set of gates, see also Ref. [115]). 8.3 Stabilizer codes and fault-tolerant computation A large class of quantum error-correcting codes can be described by the so called stabilizer formalism [66, 37, 38]. A stabilizer S is an Abelian subgroup of the Pauli groupG n on n qubits, which does not contain the element −I [114]. The Pauli group consists of all possible n-fold tensor products of the Pauli matrices X, Y, Z together with the multiplicative factors ±1, ±i. The stabilizer code corresponding to S is the subspace of all states |ψi which are left invariant under the action of every operator in S (G|ψi = |ψi, ∀G ∈ S). It is easy to see that the stabilizer of a code encoding k qubits into n has n−k generators. For the case of operator codes, the stabilizer leaves the subspace H A ⊗H B in the decomposition (335) invariant but the encoded information is invariant also under operations that act on the gauge subsystem. An operator stabilizer code encoding k qubits into n with r gauge qubits, has n−r−k stabilizer generators, while the gauge group has 2r generators [129]. According to the error-correction condition for stabilizer codes [114, 129], a set of errors {E i } in G n 195 (which without loss of generality are assumed to be Hermitian) is correctable by the code if and only if, for all i andj, E i E j anticommutes with at least one element of S, or otherwise belongs toS or to the gauge group. In this chapter we will be concerned with stabilizer codes for the correction of single-qubit errors and the techniques for fault-tolerant computation [143, 53, 2, 85, 90, 68, 67, 130] on such codes. A quantum information processing scheme is called fault-tolerant if a single er- ror occurring during the implementation of any given operation introduces at most one error per block of the code [68]. This property has to apply for unitary gates as well as measurements, including those that constitute the error-correcting opera- tions themselves. Fault-tolerant schemes for computation on stabilizer codes generally depend on the code being used—some codes, like the Bacon-Shor subsystem codes [13, 14] for example, are better suited for fault-tolerant computation than others [11]. Inspiteofthesedifferences, however, ithasbeenshownthatfault-tolerant information processing is possible on any stabilizer code [68, 67]. The general procedure can be described briefly as follows. DiVincenzo and Shor [53] demonstrated how to perform fault-tolerant measurements of the stabilizer for any stabilizer code. Their method makes use of an approach introduced by Shor [143], which involves the “cat” state (|0...0i +|1...1i)/ √ 2 which can be prepared and verified fault-tolerantly. As pointed out by Gottesman [68], by the same method one can measure any operator in the Pauli group. Since the encoded X, Y and Z operators belong to the Pauli group for any stabilizer code [68], one can prepare fault-tolerantly various superpositions of the logical basis states|0i and|1i, like|+i=(|0i+|1i)/ √ 2for example. Thelatter can be used to implement fault-tolerantly the encoded Phase and Hadamard gates, as long as afault-tolerant C-NOT gate is available [68]. Gottesman showed how theC-NOTgate can be implemented fault-tolerantly by first applying a transversal operation on four encoded qubits and then measuring the encoded X operator on two of them. Finally, 196 for universal computation one needs a gate outside of the Clifford group, e.g., the Tof- foli gate. TheToffoli construction wasdemonstratedfirstbyShorin[143]foraspecific type of codes—those obtained from doubly-even self-dual classical codes by the CSS construction [39, 149]. Gottesman showed [67] that a transversal implementation of the same procedure exists for any stabilizer code. Note that the described method for universal fault-tolerant computation on sta- bilizer codes uses almost exclusively transversal operations—these are operations for which each qubit in a block interacts only with the corresponding qubit from another block or from a special ancillary state such as the Shor’s “cat” state (see also Steane’s [150] and Knill’s [86] methods). Since single-qubit unitaries together with the C-NOT gateformauniversalsetofgates, fault-tolerant computationcanberealizedentirelyin terms of single-qubit operations and C-NOT operations between qubits from different blocks,assumingthattheancillarystatescanbepreparedreliably. Hence,ourgoalwill betoconstruct holonomic realizations of these operations as well as of thepreparation of the ancillary states. It is not evident that by doingsowe will obtain afault-tolerant construction, because the geometric approach requires that we use degenerate Hamil- tonians which generally couple qubits within the same block. Nevertheless, we will see that it is possible to design the scheme so that single-qubit errors do not propagate. 8.4 The scheme Consider an [[n,1,r,3]] stabilizer code. This is a code that encodes 1 qubit into n, has r gauge qubits, and can correct arbitrary single-qubit errors. In order to perform a holonomic operation on this code, we need a nontrivial starting Hamiltonian which leaves the code space invariant. It is easy to verify that the only Hamiltonians that satisfy this property are linear combinations of the elements of the stabilizer, or in the case of subsystem codes—elements of the gauge group. Note that the stabilizer and 197 the gauge group transform during the course of the computation under the operations being applied. At any stage when we complete an encoded operation, they return to their initial forms. During the implementation of a standard encoded gate, the Pauli groupG n on a given codeword may spread over other codewords, but it can beverified that this spreading can be limited to at most 4 other codewords counting the “cat” state. This is because the encoded C-NOT gate can be implemented fault-tolerantly on any stabilizer code by a transversal operation on 4 encoded qubits [67], and any encoded Clifford gate can berealized usingonly theencoded C-NOTprovided that we are able to do fault-tolerant measurements (the encoded Clifford group is generated by the encoded Hadamard, Phase and C-NOT gates). Encoded gates outside of the Clifford group, such as the encoded π/8 or Toffoli gates, can be implemented fault- tolerantly using encoded C-NOT gates conditioned on the qubits in a “cat” state, so theymayrequiretransversaloperationsonatotalof5blocks. Moreprecisely,thefault- tolerant implementation of the Toffoli gate requires the preparation of a special state of three encoded qubits [143], which involves a sequence of conditional encoded Phase operationsandconditionalencodedC-NOToperationswithconditioningonthequbits ina“cat”state[67]. ButtheencodedPhasegatehasauniversalimplementation using an encoded C-NOT between the qubit and an ancilla, so the conditional Phase gate may require applying a conditional encoded C-NOT. The procedure for implementing an encodedπ/8 gate involves applying an encodedSX gate conditioned on the qubits in a “cat” state [27], where S = 1 0 0 i , (448) is the Phase gate, but the encoded S gate generally also involves an encoded C-NOT on the qubit and an ancilla, so it may also require the interaction of 4 blocks. For 198 CSS codes, however, the spreading of the Pauli group of one block during the imple- mentation of a basic encoded operation can be limited to a total of 3 blocks, since the encoded C-NOT gate has a transversal implementation [67]. It also has to bepointed out that fault-tolerant encoded Clifford operations can be implemented using only Clifford gates on the physical qubits [67]. These operations transform the stabilizer and the gauge group into subgroups of the Pauli group, and their elements remain in the form of tensor products of Pauli matrices. The fault- tolerant implementation of encoded gates outside of the Clifford group, however, in- volves operations thattake thesegroupsoutsideofthePauli group. Wewill, therefore, consider separately two cases—encoded operations in the Clifford group, and encoded operations outside of the Clifford group. 8.4.1 Encoded operations in the Clifford group In Ref. [67] it was shown that every encoded operation in the Clifford group can be implemented fault-tolerantly using Clifford gates on physical qubits. The Clifford group is generated by the Hadamard, Phase and C-NOT gates, but in addition to these gates, we will also demonstrate the holonomic implementation of the X and Z gates which are standard for quantum computation. We will restrict our attention to implementing single-qubit unitaries on the first qubit in a block, as well as C-NOT operations between the first qubits in two blocks. The operations on the rest of the qubits can be obtained analogously. 8.4.1.1 Single-qubit unitary operations In order to implement a single-qubit holonomic operation on the first qubit in a block, we will choose as a starting Hamiltonian an element of the stabilizer (with a minus sign) or an element of the gauge group that acts non-trivially on that qubit. Since we areconsideringcodesthatcancorrectarbitrarysingle-qubiterrors,onecanalwaysfind 199 an element of the initial stabilizer or the initial gauge group that has a factor σ 0 =I, σ 1 =X, σ 2 =Y or σ 3 =Z acting on the first qubit, i.e., b G=σ i ⊗ e G, i =0,1,2,3 (449) where e G is atensor productof Pauli matrices andtheidentity ontherestn−1qubits. ItcanbeverifiedthatunderCliffordgatesthestabilizerandthegaugegrouptransform insuchawaythatthisisalwaysthecaseexceptthatthefactor e Gmayspreadonqubits inotherblocks. Fromnowon, wewilluse“hat”todenoteoperatorsonallthesequbits and “tilde” to denote operators on all the qubits excluding the first one. Without loss of generality wewill assumethat thechosen stabilizer or gauge-group element for that qubit has the form b G=Z⊗ e G. (450) As initial Hamiltonian, we will take the operator b H(0) =− b G=−Z⊗ e G. (451) Thus,if b Gisanelementofthestabilizer,thecodespacewillbelongtothegroundspace of b H(0). Ourgoal is to find paths in parameter space such that whenthe Hamiltonian is varied adiabatically along these paths, it gives rise to single-qubit transformations on the first qubit in each of its eigenspaces. Proposition: If the initial Hamiltonian (451) is varied adiabatically so that only the factor acting on the first qubit changes, b H(t)=−H(t)⊗ e G, (452) 200 where Tr{H(t)} =0, (453) the geometric transformation resulting in each of the eigenspaces of this Hamiltonian will be equivalent to a local unitary on the first qubit, b U(t)≈U(t)⊗ e I. Proof. Observe that (452) can be written as b H(t)=H(t)⊗ e P 0 −H(t)⊗ e P 1 , (454) where e P 0,1 = e I± e G 2 (455) are orthogonal complementary projectors. The evolution driven by b H(t) is therefore b U(t) =U 0 (t)⊗ e P 0 +U 1 (t)⊗ e P 1 , (456) where U 0,1 (t)=Texp(−i t Z 0 ±H(τ)dτ). (457) Let|φ 0 (t)i and|φ 1 (t)i betheinstantaneous groundandexcited states ofH(t) with eigenvalues E 0,1 (t) =∓E(t) (E(t)>0). Using Eq. (444) for the expressions (457), we obtain that in the adiabatic limit U 0,1 (t)=e iω(t) U A 0,1 (t)⊕e −iω(t) U A 1,0 (t), (458) where ω(t) = R t 0 dτE(τ) and U A 0,1 (t) =e R t 0 dτhφ 0,1 (τ)| d dτ |φ 0,1 (τ)i |φ 0,1 (t)ihφ 0,1 (0)|. (459) 201 The projectors on the instantaneous ground and excited eigenspaces of b H(t) are b P 0 =|φ 0 (t)ihφ 0 (t)|⊗ e P 0 +|φ 1 (t)ihφ 1 (t)|⊗ e P 1 (460) and b P 1 =|φ 1 (t)ihφ 1 (t)|⊗ e P 0 +|φ 0 (t)ihφ 0 (t)|⊗ e P 1 , (461) respectively. Using Eq. (458) and Eq. (459), one can see that the effect of the unitary (456) on each of these projectors is b U(t) b P 0 =e iω(t) (U A 0 (t)⊕U A 1 (t))⊗ e I b P 0 , (462) b U(t) b P 1 =e −iω(t) (U A 0 (t)⊕U A 1 (t))⊗ e I b P 1 , (463) i.e, up to an overall dynamical phase its effect on each of the eigenspaces is the same as that of the unitary b U(t) =U(t)⊗ e I, (464) where U(t) =U A 0 (t)⊕U A 1 (t). (465) We next show how by suitably choosing H(t) we can implement all necessary single-qubit gates. We will identify a set of points in parameter space, such that by interpolating between these points we can draw various paths resulting in the desired transformations. Weremarkthatifapathdoesnotformaloop,theresultinggeometric transformation (465) is an open-path holonomy [97]. Consider the single-qubit unitary operator V θ± = 1 √ 2 1 ∓e −iθ ±e iθ 1 , (466) 202 where θ is a real parameter (note that V θ− = V † θ+ ). Define the following sngle-qubit Hamiltonian: H θ± ≡V θ± ZV † θ± . (467) Let H(t) in Eq. (452) be a Hamiltonian which interpolates between H(0) = Z and H(T) =H θ± (up to a factor) as follows: H(t)=f(t)Z +g(t)H θ± ≡H θ±;f,g (t), (468) where f(0),g(T) > 0, f(T) = g(0) = 0. To simplify our notations, we will drop the indices f and g of the Hamiltonian, since the exact form of these functions is not important for our analysis as long as they are sufficiently smooth (see discussion below). This Hamiltonian has eigenvalues± p f(t) 2 +g(t) 2 and its energy gap is non- zero unless the entire Hamiltonian vanishes. We will show that in the adiabatic limit, the Hamiltonian (452) with H(t)=H θ± (t) gives rise to the geometric transformation b U θ± (T) =V θ± ⊗ e I. (469) To prove this, observe that −H θ± (t)=W θ H θ± (t)W θ , (470) where W θ is the Hermitian unitary W θ = 0 ie −iθ −ie iθ 0 . (471) The unitaries U θ±0,1 , given by Eq. (457) for H(t)=H θ± (t), are then related by U θ±0 =W θ U θ±1 W θ . (472) 203 Using that W θ |0i =−ie iθ |1i, W θ |1i =ie −iθ |0i, from Eq. (458) and Eq. (459) one can see that Eq. (472) implies U θ±A 0 =W θ U θ±A 1 W θ . (473) LetusdefinetheeigenstatesofH θ± (t)attimeT as|φ θ±0 (T)i=V θ± |0iand|φ θ±1 (T)i= V θ± |1i. Expression (459) can then be written as U θ±A 0 (T) =e iα θ±0 V θ± |0ih0|, U θ±A 1 (T) =e iα θ±1 V θ± |1ih1|, (474) where α θ±0 and α θ±1 are geometric phases. Without explicitly calculating the geo- metric phases, from Eq. (474) and Eq. (473) we obtain e iα θ±0 =e iα θ±1 . (475) Therefore, up to a global phase, Eq. (465) yields U θ± (T)∼V θ± . (476) We will use this result, to construct a set of standard gates by sequences of opera- tions of the form V θ± , which can be generated by interpolations of the type (468) run forward or backward. For single-qubit gates in the Clifford group, we will only need three values of the parameter θ: 0, π/2 and π/4. For completeness, however, we will also demonstrate how to implement the π/8 gate, which together with the Hadamard gate is sufficient to generate any single-qubit unitary transformation [27]. For this we will need θ =π/8. Note that H θ± =±(cosθX +sinθY), (477) 204 so for these values of θ we have H 0± =±X, H π/2± =±Y, H π/4± =±( 1 √ 2 X + 1 √ 2 Y), H π/8± =±(cos π 8 X +sin π 8 Y). Consider the adiabatic interpolations between the following Hamiltonians: −Z⊗ e G→−Y ⊗ e G→Z⊗ e G. (478) According to the above result, the first interpolation yields the transformation V π/2+ . The second interpolation can be regarded as the inverse of Z⊗ e G→−Y ⊗ e G which is equivalent to −Z⊗ e G → Y ⊗ e G since b H(t) and − b H(t) yield the same geometric transformations. Thus the second interpolation results in V † π/2− = V π/2+ . The net result is therefore V π/2+ V π/2+ = iX. We see that up to a global phase the above sequence results in a geometric implementation of the X gate. Similarly, one can verify that theZ gate can be realized via the loop −Z⊗ e G→−X⊗ e G→Z⊗ e G→Y ⊗ e G→−Z⊗ e G. (479) The Phase gate can be realized by applying −Z⊗ e G→−( 1 √ 2 X + 1 √ 2 Y)⊗ e G→Z⊗ e G, (480) followed by the X gate. The Hadamard gate can be realized by first applyingZ, followed by −Z⊗ e G→−X⊗ e G. (481) Finally, the π/8 gate can be implemented by first applyingXZ, followed by Z⊗ e G→−(cos π 8 X +sin π 8 Y)⊗ e G→−Z⊗ e G. (482) 205 8.4.1.2 A note on the adiabatic condition Before we show how to implement the C-NOT gate, let us comment on the conditions under which the adiabatic approximation assumed in the above operations is satisfied. Because of the form (456) of the overall unitary, the adiabatic approximation depends on the extent to which each of the unitaries (457) approximate the expressions (458). The latter depends only on the properties of the single-qubit Hamiltonian H(t), for which the adiabatic condition [109] reads ε Δ 2 ≪1, (483) where ε= max 0≤t≤T |hφ 1 (t)| dH(t) dt |φ 0 (t)i|, (484) and Δ= min 0≤t≤T (E 1 (t)−E 0 (t)) = min 0≤t≤T 2E(t) (485) is the minimum energy gap of H(t). Along the segments of the parameter paths we described, theHamiltonian is of the form (468) and its derivative is dH θ± (t) dt = df(t) dt Z + dg(t) dt H θ± , 0<t<T. (486) This derivative is well defined as long as df(t) dt and dg(t) dt are well defined. The curves we described, however, may not be differentiable at the points connecting two seg- ments. In order for the Hamiltonians (468) that interpolate between these points to be differentiable, the functions f(t) and g(t) have to satisfy df(T) dt = 0 and dg(0) dt = 0. This means that the change of the Hamiltonian slows down to zero at the end of each segment (except for a possible change in its strength), and increases again from zero 206 along the next segment. We point out that when the Hamiltonian stops changing, we can turn it off completely by decreasing its strength. This can be done arbitrarily fast and it would not affect a state which belongs to an eigenspace of the Hamiltonian. Similarly, we can turn on another Hamiltonian for the implementation of a different operation. The above condition guarantees that the adiabatic approximation is satisfied with precision O(( ε Δ 2 ) 2 ). It is known, however, that under certain conditions on the Hamil- tonian, we can obtain better results [70]. Let us write the Schr¨ odinger equation as i d dt |ψ(t)i =H(t)|ψ(t)i≡ 1 ǫ ¯ H(t)|ψ(t)i, (487) whereǫ>0 is small. If ¯ H(t) is smooth and all its derivatives vanish at the end points t =0 and t =T, the error would scale super-polynomially with ǫ, i.e., it will decrease with ǫ faster than O(ǫ N ) for any N. (Notice that ε Δ 2 ∝ǫ.) In our case, the smoothness condition translates directly to the functionsf(t) and g(t). Foranychoiceofthesefunctionswhichsatisfies thestandardadiabaticcondition, wecan ensurethatthestrongercondition issatisfiedbythereparameterizationf(t)→ f(y(t)), g(t) → g(y(t)) where y(t) is a smooth function of t which satisfies y(0) = 0, y(T) = T, and has vanishing derivatives at t = 0 and t = T. Then by slowing down the change of the Hamiltonian by a constant factor ǫ, which amounts to an increase of the total time T by a factor 1/ǫ, we can decrease the error super-polynomially in ǫ. We will use this result to obtain a low-error interpolation in Section 8.5 where we estimate the time needed to implement a holonomic gate with certain precision. 8.4.1.3 The C-NOT gate The stabilizer or the gauge group on multiple blocks of the code is a direct product of the stabilizers or the gauge groups of the individual blocks. Therefore, from Eq. (449) 207 it follows that one can always find an element of the initial stabilizer or gauge group on multiple blocks which has any desired combination of factors σ i , i = 0,1,2,3 on the first qubits in these blocks. It can be verified that applying transversal Clifford operations ontheblocks doesnotchange thisproperty. Therefore, wecanassumethat for implementing a C-NOT gate we can find an element of the stabilizer or the gauge group which has the form (450) where the factor Z acts on the target qubit and e G acts trivially on the control qubit. Then it is straightforward to verify that the C-NOT gate can be implemented by first applying the inverse of the Phase gate (S † ) on the control qubit, as well as the transformation V π/2+ on the target qubit, followed by the transformation −I c ⊗Y ⊗ e G→−Z c ⊗Z⊗ e G, (488) where the superscript c denotes the control qubit. The interpolation (488) is under- stood as in Eq. (468). To see that this yields the desired transformation, observe that the Hamiltonian corresponding to Eq. (488) can be written in the form b b H(t) =|0ih0| c ⊗H π/2+ (T −t)⊗ e G+|1ih1| c ⊗H π/2− (T −t)⊗ e G. (489) The application of this Hamiltonian from time t = 0 to time t = T results in the unitary b b U(T) =|0ih0| c ⊗ b U † π/2+ (T)+|1ih1| c ⊗ b U † π/2− (T), (490) where b U π/2± (T) =Texp(−i T Z 0 dτH π/2± (τ)⊗ e G). (491) 208 But the Hamiltonians H π/2+ (T−t)⊗ e G andH π/2− (T−t)⊗ e G have the same instan- taneous spectrum, and Eq. (444) implies that up to a dynamical phase, each of the eigenspaces of b b H will undergo the geometric transformation b b U g (T) =|0ih0| c ⊗V † π/2+ ⊗ e I +|1ih1| c ⊗V † π/2− ⊗ e I, (492) whereV † π/2± ⊗ e I are the geometric transformations generated byH π/2± (T−t)⊗ e G as shown earlier. This transformation was preceded by the operation S †c ⊗V π/2+ ⊗ e I, which means that the net result is b b U g (T)S †c ⊗V π/2+ ⊗ e I =|0ih0| c ⊗V † π/2+ V π/2+ ⊗ e I−i|1ih1| c ⊗V † π/2− V π/2+ ⊗ e I =|0ih0| c ⊗I⊗ e I +|1ih1| c ⊗X⊗ e I. (493) ThisisexactlytheC-NOTtransformation. Notethatbecauseoftheform(489)of b b H(t), theextent towhichtheadiabaticapproximation issatisfiedduringthis transformation dependsonlyontheadiabaticpropertiesofthesingle-qubitHamiltoniansH π/2± (T−t) which we discussed in the previous subsection. Ourconstruction allowed ustoprove theresultinggeometric transformationswith- out explicitly calculating the holonomies (447). It may be instructive, however, to demonstrate this calculation for at least one of the gates we described. In the ap- pendix at the end of this chapter we present an explicit calculation of the geometric transformation for the Z gate for the following two cases: f(t) = 1− t T , g(t) = t T (linear interpolation); f(t)=cos πt 2T , g(t) =sin πt 2T (unitary interpolation). 8.4.2 Encoded operations outside of the Clifford group Foruniversalfault-tolerant computationwealsoneedatleastoneencodedgateoutside of the Clifford group. The fault-tolerant implementation of such gates is based on the preparation of a special encoded state [143, 90, 67, 27, 176] which involves a 209 measurement of an encoded operator in the Clifford group. For example, theπ/8 gate requiresthepreparationofthestate |0i+exp(iπ/4)|1i √ 2 ,whichcanberealizedbymeasuring the operator e −iπ/4 SX [27]. Equivalently, the state can be obtained by applying the operation RS † , where R denotes the Hadamard gate, on the state cos(π/8)|0i+sin(π/8)|1i √ 2 whichcanbepreparedbymeasuringtheHadamardgate[90]. TheToffoli gaterequires the preparation of the three-qubit encoded state |000i+|010i+|100i+|111i 2 and involves a similarprocedure[176]. Inalltheseinstances,themeasurementoftheCliffordoperator is realized by applying transversally the operator conditioned on the qubits in a “cat” state. We now show a general method that can be used to implement holonomically any conditional transversal Clifford operation with conditioning on the “cat” state. Let O beaCliffordgate acting onthefirstqubitsfromsomeset ofblocks. Aswediscussedin the previous section, under this unitary the stabilizer and the gauge group transform in such a way that we can always find an element with an arbitrary combination of Pauli matrices on the first qubits. If we write this element in the form b G=G 1 ⊗G 2,...,n , (494) whereG 1 isatensorproductofPaulimatricesactingonthefirstqubitsfromtheblocks, and G 2,...,n is an operator on the rest of the qubits, then applying O conditioned on the first qubit in a “cat” state transforms this stabilizer or gauge-group element as follows: I c ⊗G 1 ⊗G 2,...,n =|0ih0| c ⊗G 1 ⊗G 2,...,n +|1ih1| c ⊗G 1 ⊗G 2,...,n →|0ih0| c ⊗G 1 ⊗G 2,...,n +|1ih1| c ⊗OG 1 O † ⊗G 2,...,n , (495) where the superscriptc denotes the control qubit from the “cat” state. We can imple- ment this operation by choosing the factor G 1 the same as the one we would use if we 210 wantedtoimplementtheoperationO accordingtothepreviouslydescribedprocedure. Then we can apply the following Hamiltonian: b b H C(O) (t) =−|0ih0| c ⊗G 1 ⊗G 2,...,n −α(t)|1ih1| c ⊗H O (t)⊗G 2,...,n , (496) where H O (t)⊗G 2,...,n is the Hamiltonian that we would use for the implementation of the operation O and α(t) is a real parameter chosen such that at every moment the operator α(t)|1ih1| c ⊗H O (t)⊗G 2,...,n has the same instantaneous spectrum as the operator |0ih0| c ⊗G 1 ⊗G 2,...,n . This guarantees that the overall Hamiltonian is degenerate and the geometric transformation in each of its eigenspaces is b b U g (t) =|0ih0| c ⊗I 1 ⊗I 2,...,n +|1ih1| c ⊗U O (t)⊗I 2,...,n , (497) whereU O (t) is the geometric transformation on the first qubits generated byH O (t)⊗ G 2,...,n . Since we presented the constructions of our basic Clifford operations up to an overall phase, the operation U O (t) may differ from the desired operation by a phase. This phase can be corrected by applying a suitable gate on the control qubit from the “cat” state (we explain how this can be done in the next section). We remark that a Hamiltonian of the type (496) requires finetuning of the parameterα(t) and generally can be complicated. Our goal in this section is to prove that universal fault-tolerant holonomic computation is possibleinprinciple. InSection 8.6 weshow that depending on the code one can find more natural implementations of these operations. If we want to apply a second conditional Clifford operation Q on the first qubits in the Block, we can do this via the Hamiltonian b b H C(Q) (t) =−|0ih0| c ⊗G 1 ⊗G 2,...,n −β(t)|1ih1| c ⊗H Q (t)⊗G 2,...,n , (498) 211 whereH Q (t)⊗G 2,...,n isnowtheHamiltonianwewouldusetoimplementtheoperation Q, had we implemented the operation O before that. Here again, the factor β(t) guaranteesthatthereisnosplittingoftheenergylevelsoftheHamiltonian. Subsequent operations are applied analogously. Using this general method, we can implement holonomically any transversal Clifford operation conditioned on the “cat” state. 8.4.3 Using the “cat” state In addition to transversal operations, a complete fault-tolerant scheme requires the ability to prepare, verify and use a special ancillary state such as the “cat” state (|00...0i +|11...1i)/ √ 2 proposed by Shor [143]. This can also be done in the spirit of our holonomic scheme. Since the “cat” state is known and its construction is non- fault-tolerant, we can prepare it by simply treating each initially prepared qubit as a simplecode(with e G inEq. (450) beingtrivial), andupdatingthestabilizer of thecode viatheapplied geometric transformation as theoperation progresses. Thestabilizer of the prepared “cat” state is generated by Z i Z j , i <j. Transversal unitary operations between the “cat” state and other codewords are applied as described in the previous section. Wealsohavetobeabletomeasuretheparityofthestate,whichrequirestheability to apply successively C-NOT operations from two different qubits in the “cat” state to one and the same ancillary qubit initially prepared in the state|0i. We can regard the qubit in state |0i as a simple code with stabilizer hZi, and we can apply the first C-NOT as described before. Even though after this operation the state of the target qubit is unknown, the second C-NOT gate can be applied via the same interaction, since the transformation in each eigenspace of the Hamiltonian is the same and at the end when we measure the qubit we project on one of the eigenspaces. 212 8.4.4 Fault tolerance of the scheme We showedhowwecan generate anytransversal operation onthecodespaceholonom- ically, assuming that the state is non-erroneous. But what if an error occurs on one of the qubits? At any moment, we can distinguish two types of errors—those that result in tran- sitions between the ground and the excited spaces of the current Hamiltonian, and those that result in transformations inside the eigenspaces. Due to the discretization of errors in QEC, it suffices to prove correctability for each type separately. The key property of our construction is that in each of the eigenspaces, the geometric transfor- mation is the same and it is transversal. Because of this, if we are applying a unitary on the first qubit, an error on that qubit will remain localized regardless of whether it causes an excitation or not. If the error occurs on one of the other qubits, at the end of the transformation the result would be the desired single-qubit unitary gate plus the error on the other qubit, which is correctable. It is remarkable that even though the Hamiltonian couples qubits within the same block, single-qubit errors do not propagate. This is because the coupling between the qubits amounts to a change in the relative phase between the ground and excited spaces,butthelatterisirrelevantsinceitiseitherequivalenttoagaugetransformation, or when we apply a correcting operation we project on one of the eigenspaces. In the case of the C-NOT gate, an error can propagate between the control and the target qubits, but it never results in two errors within the same codeword. 213 8.5 Effects on the accuracy threshold for environment noise Since the method we presented conforms completely to a given fault-tolerant scheme, it would not affect the error threshold per operation for that scheme. Some of its features, however, would affect the threshold for environment noise. First, observe that when applying the Hamiltonian (452), we cannot at the same time apply operations on the other qubits on which the factor e G acts non-trivially. Thus, some operations at the lowest level of concatenation that would otherwise be implemented simultaneously might have to be implemented serially. The effect of this isequivalent toslowingdownthecircuit byaconstant factor. (Note that wecouldalso vary the factor e G simultaneously withH(t), but in order to obtain the same precision as that we would achieve by a serial implementation, we would have to slow down the change of the Hamiltonian by the same factor.) The slowdown factor resulting from thislossofparallelismisusuallysmallsincethisproblemoccursonlyatthelowestlevel of concatenation. For example, for the Bacon-Shor code, we can implement operation on up to 6 out of the 9 qubits in a block simultaneously. As we show in Section 8.6, we can address any two qubits in a row or column using our method by taking e G in Eq. (452) to be a single-qubit operator Z or X on the third qubit in the same row or column. The Hamiltonians used for applying operations on the two qubits commute with each other at all times and do not interfere. The same hold for the operations involving the “cat” state. Thus for the Bacon-Shor code we have a slowdown due to parallelism by a factor of 1.5. A more significant slowdown results from the fact that the evolution is adiabatic. In order to obtain a rough estimate of the slowdown due specifically to the adiabatic requirement,wewillcomparethetimeT h neededfortheimplementationofaholonomic gate with precision 1−δ to the timeT d needed for a dynamical realization of the same 214 gate with the same strength of the Hamiltonian. We will consider a realization of the X gate via the interpolation b H(t) =−V X (τ(t))ZV † X (τ(t))⊗ e G, V X (τ(t)) =exp iτ(t) π 2T h X , (499) where τ(0) = 0, τ(T h ) = T h . Thus the energy gap of the Hamiltonian is always at maximum. The optimal dynamical implementation of the same gate is via the Hamiltonian −X for time T d = π 2 . As we argued in Section 8.4, the accuracy with which the adiabatic approximation holds for the Hamiltonian (499) is the same as that for the Hamiltonian H(t)=V X (τ(t))ZV † X (τ(t)). (500) We now present estimates for two different choices of the function τ(t). The first one is τ(t)=t. (501) InthiscasetheSchr¨ odingerequationcanbeeasilysolvedintheinstantaneouseigenba- sis of the Hamiltonian (500). For the probability that the initial ground state remains a ground state at the end of the evolution, we obtain p= 1 1+ε 2 + ε 2 1+ε 2 cos 2 ( π 4ε p 1+ε 2 ), (502) where ε= T d T h . (503) 215 Expandingin powers ofε and averaging the square of the cosine whose period is much smaller than T h , we obtain the condition ε 2 ≤2δ. (504) Assuming, for example, that δ ≈ 10 −4 (approximately the threshold for the 9-qubit Bacon-Shor [11]), we obtain that the time of evolution for the holonomic case must be about 70 times longer than that in the dynamical case. It is known, however, that ifH(t) is smooth and its derivatives vanish att=0 and t=T h , the adiabatic error decreases super-polynomially withT h [70]. To achieve this, we will choose τ(t) = 1 a Z t 0 dt ′ e −1/sin(πt ′ /T h ) , a= Z T h 0 dt ′ e −1/sin(πt ′ /T h ) . (505) For this interpolation, by a numerical solution we obtain that when T h /T d ≈ 17 the error is already of the order of 10 −6 , which is well below the threshold values obtained for the Bacon-Shor codes [11]. Thisis a remarkable improvement in comparison to the previous interpolation which shows that the smoothness of the Hamiltonian plays an important role in the performance of the scheme. An additional slowdown in comparison to a perfect dynamical scheme may result fromthefactthattheconstructionsforsomeofthestandardgateswepresentedinvolve long sequences of loops. With more efficient parameter paths, however, it should be possible reduce this slowdown to minimum. An approach for finding loops presented in Ref. [115] may be useful in this respect. In comparison to a dynamical implementation, the allowed rate of environment noisefortheholonomic case woulddecrease byafactor similar totheslowdown factor. Inpractice, however, dynamicalgates arenotperfectandtheholonomicapproachmay be advantageous if it allows for a better precision. 216 WefinallypointoutthatanerrorinthefactorH(t)intheHamiltonian(452)would result in an error on the first qubit according to Eq. (465). Such an error clearly has tobebelow theaccuracy threshold. Moredangerouserrors, however, arealsopossible. For example, if the degeneracy of the Hamiltonian is broken, this can result in an unwanted dynamical transformation affecting all qubits on which the Hamiltoniain acts non-trivially. Such multi-qubit errors have to be of higher order in the threshold, which imposes more severe restrictions on the Hamiltonian. 8.6 Fault-tolerantholonomiccomputationwithlow-weight Hamiltonians The weight of the Hamiltonians needed for the scheme we described depend on the weight of the stabilizer or gauge-group elements. Remarkably, certain codes possess stabilizer or gauge-group elements of low weight covering all qubits in the code, which allows us to perform holonomic computation using low-weight Hamiltonians. Here we will consider as an example a subsystem generalization of the 9-qubit Shor code [142]—the Bacon-Shor code [13, 14]—which has particularly favorable properties for fault-tolerant computation[10,11]. Inthe9-qubitBacon-Shorcode, thegaugegroupis generatedbytheweight-two operatorsZ k,j Z k,j+1 andX j,k X j+1,k ,wherethesubscripts label the qubits by row and column when they are arranged in a 3×3 square lattice. Since the Bacon-Shor code is a CSS code, the C-NOT gate has a direct transversal implementation. We now show that the C-NOT gate ca be realized using at most weight-three Hamiltonians. If we want to apply a C-NOT gate between two qubits each of which is, say, in the first row and column of its block, we can use as a starting Hamiltonian −Z t 1,1 ⊗Z t 1,2 , where the superscript t signifies that these are operators in the target block. We can then apply the C-NOT gate as described in the previous section. After the operation, 217 however, thisgauge-groupelementwilltransformto−Z t 1,1 ⊗Z c 1,1 ⊗Z t 1,2 . Ifwenowwant to implement a C-NOT gate between the qubits with index{1,2} using as a starting Hamiltonian the operator−Z t 1,1 ⊗Z c 1,1 ⊗Z t 1,2 according to the same procedure, we will have to use a four-qubit Hamiltonian. Of course, at this point we can use the starting Hamiltonian −Z t 1,2 ⊗Z t 1,3 , but if we had also applied a C-NOT between the qubits labelled {1,3}, this operator would not be available—it would have transformed to −Z t 1,2 ⊗Z t 1,3 ⊗Z c 1,3 . What we can do instead, is to use as a starting Hamiltonian the operator−Z t 1,1 ⊗ Z t 1,2 ⊗Z c 1,2 which is obtained from the gauge-group element Z t 1,1 ⊗Z c 1,1 ⊗Z t 1,2 ⊗Z c 1,2 after the application of the C-NOT between the qubits with index {1,1}. Since the C-NOT gate is its own inverse, we can regard thefactorZ t 1,1 as e G in Eq. (488) anduse this starting Hamiltonian to apply our procedure backwards. Thus we can implement any transversal C-NOT gate using at most weight-three Hamiltonians. Since the encoded X, Y and Z operations have a bitwise implementation, we can always applythemaccordingtoourprocedureusingHamiltonians of weight 2. For the Bacon-Shor code, the encoded Hadamard gate can be applied via bitwise Hadamard transformations followed by a rotation of the grid to a 90 degree angle [11]. Thephase gate can be implemented by using the C-NOT and an ancilla. We point out that the preparation and measurement of the “cat” state can also be done using Hamiltonians of weight 2. To prepare the “cat” state, we prepare first all qubitsinthestate|f 0 + i =(|0i+|1i)/ √ 2,whichcanbedonebymeasuringeachofthem in the|0i,|1i basis (this ability is assumed for any type of computation) and applying the transformation −Z →−X or Z →−X depending on the outcome. To complete the preparation of the “cat” state, apply a two-qubit transformation between the first qubit and each of the other qubits (j >1) via the transformation −I 1 ⊗X j →−Z 1 ⊗Z j . (506) 218 Single-qubittransformationsonqubitsfromthe“cat”statecanbeappliedaccordingto the method described in the previous section using at most weight-two Hamiltonians. Tomeasuretheparityofthestate, weneedtoapplysuccessivelyC-NOToperations fromtwodifferentqubitsinthe“cat”statetothesameancillaryqubitinitiallyprepared in the state|0i. As described in the previous section, this can also be done according to our method and requires Hamiltonians of weight 2. For universal computation with the Bacon-Shor code, we also need to be able to apply one encoded transformation outside of the Clifford group. As we mentioned earlier, in order to implement the Toffoli gate or theπ/8 gate, it is sufficient to beable to implement a C-NOT gate conditioned on a “cat” state. For the Bacon-Shor code, thetheC-NOTgatehasatransversal implementation, sotheconditionedC-NOTgate canberealized byaseries oftransversalToffoli operationsbetweenthe“cat” state and the two encoded states. We now show that the latter can be implemented using at most three-qubit Hamiltonians. Ref. [114] provides a circuit for implementing the Toffoli gate as a sequence of one- and two-qubit gates. We will use the same circuit, except that we flip the control and target qubits in every C-NOT gate using the identity R 1 ⊗R 2 C 1,2 R 1 ⊗R 2 =C 2,1 , (507) where R i denotes a Hadamard gate on the qubit labelled by i and C i,j denotes a C- NOT gate between qubits i and j with i being the control and j being the target. Let Toffoli i,j,k denote the Toffoli gate on qubits i, j and k with i and j being the two control qubits and k being the target qubit, and let S i and T i denote the Phase and 219 π/8 gates on qubiti, respectively. Then the Toffoli gate on three qubits (the first one of which we will assume to belong to the “cat” state), can be written as: Toffoli 1,2,3 =R 2 C 3,2 R 3 T † 3 R 3 R 1 C 3,1 R 3 T 3 R 3 C 3,2 R 3 T † 3 R 3 C 3,1 × R 3 T 3 R 3 R 2 T † 2 R 2 C 2,1 R 2 T † 2 R 2 C 2,1 R 2 S 2 R 1 T 1 . (508) To show that each of the above gates can be implemented holonomically using Hamil- tonians of weight at most 3, we will need an implementation of the C-NOT gate which is suitable for the case when we have a stabilizer or gauge-group element of the form b G=X⊗ e G, (509) where the factor X acts on the target qubit and e G acts trivially on the control qubit. By a similar argument to the one in Section 8.4, one can verify that in this case the C-NOT gate can be implemented as follows: apply the operation S † on the control qubit (we describe how to do this for our particular case below) together with the transformation −X⊗ e G→−Z⊗ e G→X⊗ e G (510) on the target qubit, followed by the transformation I c ⊗X⊗ e G→−(|0ih0| c ⊗Z +|1ih1| c ⊗Y)⊗ e G→−I c ⊗X⊗ e G. (511) Since the second and the third qubits belong to blocks encoded with the Bacon- Shor code, there are weight-two elements of the initial gauge group of the formZ⊗Z covering all qubits. The stabilizer generators on the “cat” state are also of this type. Followingthetransformationoftheseoperatorsaccordingtothesequenceofoperations (508), one can see that before every C-NOT gate in this sequence, there is an element 220 of the form (509) with e G = Z which can be used to implement the C-NOT gate as described provided that we can implement the gate S † on the control qubit. We also point out that all single-qubit operations on qubit 1 in this sequence can be implemented according to the procedure describes in Section 8.4, since at every step we have a weight-two stabilizer element on that qubit with a suitable form. Therefore, all we need to show is how to implement the necessary single-qubit operations on qubits 2 and 3. Due to the complicated transformation of the gauge-group elements during the sequence of operations (508), we will introduce a method of applying a single-qubit operation with a starting Hamiltonian that acts trivially on the qubit. For implementing single-qubit operations on qubits 2 and 3 we will use as a starting Hamiltonian the operator b b H(0) =−I i ⊗X 1 ⊗ e Z, i=2,3 (512) wherethefirstfactor (I i )acts onthequbitonwhichwewanttoapplytheoperation(2 or 3), andX 1 ⊗ e Z is the transformed (after the Hadamard gate R 1 ) stabilizer element of the “cat” state that acts non-trivially on qubit 1 (the factor e Z acts on some other qubit in the “cat” state). To implement a single-qubit gate on qubit 3 for example, we first apply the inter- polation −I 3 ⊗X 1 ⊗ e Z→−Z 3 ⊗Z 1 ⊗ e Z. (513) This results in a two-qubit geometric transformation U 1,3 on qubits 1 and 3. We do not have to calculate this transformation exactly since we will undo it later, but the fact that each eigenspace undergoes the same two-qubit geometric transformation can be verified similarly to the C-NOT gate we described in Section 8.4. At this point, the Hamiltonian is of the form (451) with respect to qubit 3, and we can apply any single-qubit unitary gate V 3 according to the method described in 221 Section 8.4. This transforms the Hamiltonian to −V 3 Z 3 V † 3 ⊗X 1 ⊗ e Z. We can now “undo” the transformation U 1,3 by the interpolation −V 3 Z 3 V † 3 ⊗Z 1 ⊗ e Z→−I 3 ⊗X 1 ⊗ e Z. (514) The latter transformation is the inverse of Eq. (513) up to the single-qubit unitary transformation V 3 , i.e., it results in the transformation V 3 U 1,3† V 3† . Thus the net result is V 3 U † 1,3 V † 3 V 3 U 1,3 =V 3 , (515) which is thedesired single-qubit unitary transformation on qubit 3. We point out that during this transformation, a single-qubit error can propagate between qubits 1 and 3, but this is not a problem since we are implementing a transversal Toffoli operation and such an error would not result in more that one error per block of the code. We showed that for the BS code our scheme can be implemented with at most 3-local Hamiltonians. This is optimal for the construction we presented, since there are nonon-trivial codes with stabilizer or gauge-group elements of weight smaller than 2 covering all qubits. One could argue that since the only Hamiltonians that leave the code space invariant are superpositions of elements of the stabilizer or the gauge group, one cannot do bettwer than this. However, it may be possible to approximate the necessary Hamiltonians with sufficient precision using 2-local interactions. A pos- sible direction to consider in this respect is the technique introduced in Ref. [84] for approximating three-local Hamiltonians by two-local ones. This is left as a problem for future investigation. 222 8.7 Conclusion We described a scheme for fault-tolerant holonomic computation on stabilizer codes, whichdemonstratesthatHQCisascalablemethodofcomputation. Theschemeopens thepossibilityofcombiningthesoftwareprotectionoferrorcorrectionwiththeinherent robustness of HQC against control imperfections. Our construction uses Hamiltonians that are elements of the stabilizer or the gauge group for the code. The Hamiltonians needed for implementing two-qubit gates are at least 3-local. We have shown that computation with at most 3-local Hamiltonians is possible with the Bacon-Shor code. It is interesting to point out that the adiabatic regime in which our scheme oper- ates is consistent with the model of Markovian decoherence. In Ref. [9] it was argued that the standard dynamical paradigm of fault tolerance is based on assumptions that are in conflict with the rigorous derivation of the Markovian limit. Although the thresholdtheoremhasbeenextendedtonon-Markovianmodels[154,12,3],theMarko- vian assumption is an accurate approximation for a wide range of physical scenarios [42]. It also allows for a much simpler description of the evolution in comparison to non-Markovian models, as we saw in Chapter 5. In Ref. [9] it was shown that the weak-coupling-limit derivation of the Markovian approximation is consistent with computational methods that employ slow transformations, such as adiabatic quantum computation [57] or HQC. A theory of fault tolerance for the adiabatic model of com- putation at present is not known, although significant steps in this direction have been undertaken [79, 101]. Our hybrid HQC-QEC scheme provides a solution for the case of HQC. We point out, however, that it is an open problem whether the Markovian approximation makes sense for a fixed value of the adiabatic slowness parameter when the circuit increases in size. Applying the present strategy to actual physical systems might require modifying our abstract construction in accordance with the available interactions, possibly using 223 superpositions of stabilizer or gauge-group elements rather than single elements as the basic Hamiltonians. Given that simple QEC codes and two-qubit geometric transfor- mations have been realized using NMR [47, 78] and ion-trap [50, 99] techniques, these systems seem particularly suitable for hybrid HQC-QEC implementations. We hope that the techniques presented in this study might prove useful in other areas aswell. It ispossiblethat somecombination oftransversal adiabatic transforma- tions and active correction could provide a solution to the problem of fault tolerance in the adiabatic model of computation. 8.8 Appendix: Calculating the holonomy for the Z gate 8.8.1 Linear interpolation We first demonstrate how to calculate the ground-space holonomy for the Z gate for the case of linear interpolation between the vertices of the path, i.e., when f(t) and g(t) in Eq. (468) are f(t)=1− t T , g(t) = t T . (516) In order to calculate the holonomy (447) corresponding to our construction of the Z gate, we need to define a single-valued orthonormal basis of the ground space of the Hamiltonian along the loop described by Eq. (479). Since the Hamiltonian has the form (454) at all times, it is convenient to choose the basis of the form |jk;λi =|χ j (λ)i| e ψ jk i, (517) j =0,1; k =1,...,2 n−2 , where|χ 0 (λ(t))i and |χ 1 (λ(t))i are ground and excited states of H(t), and | e ψ 0k i and | e ψ 1k i are fixed orthonormal bases of the subspaces that support the projectors e P 0 and e P 1 defined in Eq. (455), respectively. The eigenstates |χ 0 (λ(t))i and |χ 1 (λ(t))i are 224 defined upto an overall phase, but we have to chose the phasesuch that the states are single-valued along the loop. Observe that because of this choice of basis, the matrix elements (446) become (A μ ) jk,j ′ k ′ =hjk;λ| ∂ ∂λ μ |j ′ k ′ ;λi=hχ j (λ)| ∂ ∂λ μ |χ j ′(λ)i ×h e ψ jk | e ψ j ′ k ′i=hχ j (λ)| ∂ ∂λ μ |χ j ′(λ)iδ jj ′δ kk ′, (518) i.e., the matrixA μ is diagonal. (Since we are looking only at the ground space, we are not writing the index of the energy level). We can therefore drop the path-ordering operator. The resulting unitary matrix U γ jk,j ′ k ′ acting on the subspace spanned by {|jk;λ(0)i} is also diagonal and its diagonal elements are U γ jk,jk =exp I γ hχ j (λ)| ∂ ∂λ μ |χ j (λ)idλ μ . (519) These are precisely the Berry phases for the loops described by the states |χ j (λi). Since the loop in parameter space consist of four line segments, we can write the last expression as U γ jk,jk =exp 4 X i=1 Z γ i hχ j (λ)| ∂ ∂λ μ |χ j (λ)idλ μ ! , (520) where γ i , i = 1,2,3,4 denotes the segments in order in Eq. (479). If we parameterize each line segment by the dimensionless time 0≤s≤1, we get U γ jk,jk =exp 4 X i=1 Z 1 0 hχ i j (s)| d ds |χ i j (s)ids ! , (521) where the superscript i in |χ i j (s)i indicates the segment. In the |0i, |1i basis, we will write these states as |χ i j (s)i= a i j (s) b i j (s) , j =1,2 , i=1,2,3,4 , (522) 225 where|a i j (s)| 2 +|b i j (s)| 2 =1. Along the segment γ 1 , the states |χ 1 0 (s)i and |χ 1 1 (s)i are the ground and excited states of the Hamiltonian H 1 (t) =(1−s)Z +sX. (523) For these states we obtain a 1 0 (s) = (1−s+ √ 1−2s+2s 2 )e iω 1 0 (s) q 2−4s+4s 2 +(2−2s) √ 1−2s+2s 2 , (524) b 1 0 (s)= se iω 1 0 (s) q 2−4s+4s 2 +(2−2s) √ 1−2s+2s 2 , (525) a 1 1 (s) = (1−s− √ 1−2s+2s 2 )e iω 1 1 (s) q 2−4s+4s 2 −(2−2s) √ 1−2s+2s 2 , (526) b 1 1 (s)= se iω 1 1 (s) q 2−4s+4s 2 −(2−2s) √ 1−2s+2s 2 , (527) whereω 1 j (s)arearbitraryphaseswhichhavetobechosensothatwhenwecompletethe loop, the phases of the corresponding states will return to their initial values modulo 2π. We will definethe loops as interpolating between thefollowing intermediate states defined with their overall phases: |ψ 0 (λ)i: |0i→|f 0 + i→|1i→|f π/2 − i→|0i, (528) |ψ 1 (λ)i: |1i→|f 0 − i→|0i→|f π/2 + i→|1i, (529) where |f θ ± i= |0i±e iθ |1i √ 2 . (530) Inotherwords,weimposetheconditions|χ 1 0,1 (0)i =|0,1i,|χ 1 0,1 (1)i=|f 0 ± i=|χ 2 0,1 (0)i, |χ 2 0,1 (1)i =|1,0i =|χ 3 0,1 (0)i,|χ 3 0,1 (1)i =|f π/2 ∓ i=|χ 4 0,1 (0)i,|χ 4 0,1 (1)i =|0,1i. 226 From Eq. (524) and Eq. (525) we see that a 1 0 (0) = e iω 1 0 (0) , b 1 0 (0) = 0 and a 1 0 (1) = 1 √ 2 e iω 1 0 (1) , b 1 0 (1) = 1 √ 2 e iω 1 0 (1) , so we can choose ω 1 0 (s) =0, ∀s∈[0,1]. (531) Similarly, from Eq. (526) and Eq. (527) it can be seen that a 1 1 (0) = 0, b 1 1 (0) =e iω 1 1 (0) and a 1 1 (1) = − 1 √ 2 e iω 1 1 (1) , b 1 1 (1) = 1 √ 2 e iω 1 1 (1) . This means that ω 1 1 (s) has to satisfy e iω 1 1 (0) =1, e iω 1 1 (1) =−1. We can choose any differentiable ω 1 1 (s) that satisfies ω 1 1 (0) =0, ω 1 1 (1) =π. (532) In order to calculate R 1 0 hχ 1 j (s)| d ds |χ 1 j (s)ids, we also need d ds |χ 1 j (s)i= d ds a 1 j (s) d ds b 1 j (s) . (533) Differentiating Eqs. (524)-(527) yields d ds a 1 0 (s)=− s(1−s+ √ 1−2s+2s 2 ) 2 √ 2−4s+4s 2 [1−2s+2s 2 +(1−s) √ 1−2s+2s 2 ] 3 2 , (534) d ds b 1 0 (s) = 2−4s+3s 2 +(2−2s) √ 1−2s+2s 2 2 √ 2−4s+4s 2 [1−2s+2s 2 +(1−s) √ 1−2s+2s 2 ] 3 2 , (535) d ds a 1 1 (s) =− s(1−s− √ 1−2s+2s 2 )e iω 1 1 (s) 2 √ 2−4s+4s 2 [1−2s+2s 2 −(1−s) √ 1−2s+2s 2 ] 3 2 +a 1 1 (s)i d ds ω 1 1 (s), (536) d ds b 1 0 (s) =− (2−4s+3s 2 −(2−2s) √ 1−2s+2s 2 )e iω 1 1 (s) 2 √ 2−4s+4s 2 [1−2s+2s 2 −(1−s) √ 1−2s+2s 2 ] 3 2 +b 1 1 (s)i d ds ω 1 1 (s). (537) 227 By a straightforward substitution, we obtain hχ 1 0 (s)| d ds |χ 1 0 (s)i = a 1∗ 0 (s) d ds a 1 0 (s)+b 1∗ 0 (s) d ds b 1 0 (s) =0, (538) hχ 1 1 (s)| d ds |χ 1 1 (s)i = a 1∗ 1 (s) d ds a 1 1 (s)+b 1∗ 1 (s) d ds b 1 1 (s) =i d ds ω 1 1 (s). (539) (540) Thus the integrals are Z 1 0 hχ 1 0 (s)| d ds |χ 1 0 (s)ids = 0, (541) Z 1 0 hχ 1 1 (s)| d ds |χ 1 1 (s)ids = iω 1 1 (s)| 1 0 =iπ. (542) In the same manner, we calculate the contributions of the other three line segments. The results are: Z 1 0 hχ 2 0 (s)| d ds |χ 2 0 (s)ids = 0, (543) Z 1 0 hχ 2 1 (s)| d ds |χ 2 1 (s)ids = 0, (544) Z 1 0 hχ 3 0 (s)| d ds |χ 3 0 (s)ids = i π 2 , (545) Z 1 0 hχ 3 1 (s)| d ds |χ 3 1 (s)ids = 0, (546) Z 1 0 hχ 4 0 (s)| d ds |χ 4 0 (s)ids = 0, (547) Z 1 0 hχ 4 1 (s)| d ds |χ 4 1 (s)ids = i π 2 . (548) 228 Putting everything together, for the diagonal elements of the holonomy we obtain U γ 0k,0k =e i π 2 , U γ 1k,1k =e i 3π 2 . (549) The holonomy transforms any state in the ground space of the initial Hamiltonian as U γ X jk α jk |ji| e ψ jk i=e i π 2 X jk (−1) j α jk |ji| e ψ jk i, j =0,1. (550) From the point of view of the full Hilbert space, this is effectively aZ gate on the first qubit up to an overall phase. We point out that other single-qubit transformations like the Hadamard or the X gates, which do not form a complete loop in parameter space, can be obtained in a similar fashion by calculating the open-path expression (445). In principle, the result of that calculation depends on the choice of basis {|α;λi} which is defined up to a unitary gauge transformation. However, this ambiguity is removed by the notion of parallel transport between the initial and the final subspaces [97]. One can verify that this yields the correct result for our transformations. 8.8.2 Unitary interpolation The calculation is simpler if we choose a unitary interpolation, f(t)=cos πt 2T , g(t) =sin πt 2T . (551) 229 Such interpolation corresponds to a rotation of the Bloch sphere around a particular axis for each of the segments of the loop. The first two segments of the loop (479) are realized via the Hamiltonian b H 1,2 (t)=−V † Y (t)ZV Y (t)⊗ e G, V Y (t) =exp it π 2T Y , (552) appliedfor timeT, andthethirdandfourthsegments arerealized viatheHamiltonian b H 3,4 (t) =−V X (t)ZV † X (t)⊗ e G, V X (t) =exp it π 2T X , (553) again applied for time T. Let us define the eigenstates of the Hamiltonian along the first two segments as |χ 1,2 0 (t)i =V X (t)|0i, |χ 1,2 1 (t)i=V X (t)|1i, 0≤t≤T (554) and along the third and forth segments as |χ 3,4 0 (t)i =−iV † Y (t)Y|0i, |χ 3,4 1 (t)i=−iV † Y (t)Y|1i, 0≤t≤T. (555) Notice that |χ 1,2 0 (T)i =−iY|0i =|χ 3,4 0 (0)i, |χ 1,2 1 (T)i=−iY|1i=|χ 3,4 1 (0)i, (556) but |χ 1,2 0 (0)i =|0i6=|χ 3,4 0 (T)i=−i|0i, |χ 1,2 1 (0)i =|1i6=|χ 3,4 1 (T)i=i|1i, (557) 230 i.e., this basis is not single-valued. To make it single valued, we can modify it along the third and fourth segments as |χ 3,4 0 (t)i→|e χ 3,4 0 (t)i =e iω 0 (t) |χ 3,4 0 (t)i, |χ 3,4 1 (t)i→|e χ 3,4 1 (t)i=e iω 1 (t) |χ 3,4 0 (t)i, (558) where ω 0 (0) =0, ω 0 (T) = π 2 , (559) ω 1 (0) =0, ω 1 (T) =− π 2 . (560) The expression (521) then becomes U γ jk,jk =exp Z T 0 hχ 1,2 j (t)| d dt |χ 1,2 j (t)idt+ Z T 0 hχ 3,4 j (t)| d dt |χ 3,4 j (t)idt+(−1) j π 2 , j =0,1. (561) But hχ 1,2 j (t)| d dt |χ 1,2 j (t)i=−i π 2T hj|Y|ji =0, (562) and hχ 3,4 j (t)| d dt |χ 3,4 j (t)i=i π 2T hj|YXY|ji =0. (563) Therefore, we obtain (549). 231 Chapter 9: Conclusion In this thesis we obtained various results in the theory of open quantum systems and quantum information. These results have opened interesting questions and suggested promising directions for future research. The decomposition into weak measurements presents a practical prescription for the implementation of any generalized measurement using weak measurements and feedback control. It also presents a powerful mathematical tool for the study of mea- surement processes. In practice, there may exist limitations on the type of weak measurements an experimenter can implement, and hence it would be interesting to look at the inverse problem—given a set of weak measurements, what are the gen- eralized measurements that one can generate with them. It might be convenient to recastthisproblemintermsofthesystem-ancillainteractionsthatareavailableforthe implementation of such measurements. The decomposition may prove useful in other problems involving feedback control as well. One of its interesting features is that the evolution to which it corresponds is confined to a specific manifold (the simplex). In that sense, the procedureavoids dissipation into areas from which the state could drift away from the desired outcomes. This property could be helpful in designing optimal feedback-control protocols. The decomposition into weak measurements furthermore suggests that it may be possible to find a unified description of measurement protocols. The operations ap- plied at a given time during the measurement procedure for generating generalized 232 measurements, drive the evolution of a stochastic process on the simplex, i.e., they can be represented by a stochastic matrix on the coordinate space. This suggests that theremayexistageneralcoordinatespace, whichincludesallsuchsimplexes, onwhich the most general notion of a measurement protocol can be represented by a stochastic process. The basic object in such a description would not be a quantum state but a classical probability distribution on a space whose coordinates correspond to quantum states. Since stochastic processes are well understood, such a unifieddescription could be useful for studies of measurement-driven schemes. We also used the decomposition into weak measurements for deriving necessary and sufficient conditions for entanglement monotones. These conditions may be useful for proving monotonicity of conjectured monotones, finding new classes of entangle- mentmeasures,orfindingmeasureswithparticularlynicepropertiessuchasadditivity. Another interesting possibility suggested by the existence of necessary and sufficient differential conditions for monotonicity under all types of CPTP transformations, is that it may be possible to think of all quantum operations as generated by infinites- imal operations. It is known that CPTP maps cannot be generated by weak CPTP maps, however, the differential form of the convexity condition can be thought of as a condition for monotonicity under infinitesimal loss of information. Therefore, if we adopt the approach in which the basic objects are ensembles of states and loss of clas- sical information is a basic operation on these objects, it may be possible to arrive at a unified description of the most general form of quantum operations where every operation can be continuously connected to the identity. Our investigation of the deterministic evolution of open quantum systems and the differencebetweenMarkovian andnon-Markovian decoherencehasalsoopenedvarious interesting questions. While we compared the performance of different perturbative master equations, we have not compared their solutions to the perturbative expansion of the exact solution. Studying these equations is important in its own right as it 233 provides understanding of the actual dynamics driving the effective evolution. But for the purpose of obtaining an approximation of the exact solution starting from first principles, it may be more useful to expand the solution directly. Expanding the exact solution is justified in the same parameter regime—small αt—and requires computation ofthesamebathcorrelation functions,butit issignificantly simplersince it does not require deriving an equation and solving it. As we mentioned in Chapter 5, the TCL or NZ projection techniques might be useful also for the effective description of the reduced dynamics of a system subject to non-Markovian decoherence and continuous error correction. Here too, it would be interesting to consider expanding the solution directly. We presented a generalized notionofaZenoregimeapplicablefortheproblemoferrorcorrectionandidentifiedthe bottle-neck mechanism through which the performance of the error-correction scheme depends on this regime. As the Zeno regime plays a central role in the workings of another error-correction approach—dynamical decoupling(DD) [162, 56]—it might be useful to apply the insights developed here in the design of hybrid EC-DD schemes. Another direction for future research is expanding our scheme for continuous error correction based on weak measurement and weak unitary operations to include more sophisticated feedback. Since making full use of the available information about the state can only help, we expect that this approach would lead to schemes with better performance. Oneoftheproblemssuggestedbyourstudyoftheconditionsforexactcorrectability under continuous decoherence, was whether a similar approach to the one we used could be useful in studies concerning approximate error correction. A question we raised is whether the Markovian decoherence process during an infinitesimal time step can be separated into completely correctable and non-correctable parts. If this is the case, it could allow us to formulate conditions for optimal correctability by tracking the evolution of the maximal information that remains during the process. As we 234 argued, for non-Markovian decoherence such an approach cannot be optimal since the information may flow out to the environment and later return to the system, but it could nevertheless be helpful for finding locally optimal solutions. Even if the answer to this question is negative, the differential approach in studying information loss certainly seems promising. One interesting extension of this work would be to derive conditions for correctability in the context of the TCL or NZ master equations. Another promising tool introduced in this thesis is the measure of fidelity for en- coded information that we used to prove the robustness of operator error correction against imperfect encoding. As we pointed out, this measureprovides anatural means of extending concepts such as the fidelity of a quantum channel and the entanglement fidelity to the case of subsystem codes. As subsystem encoding provides the most general method of encoding, this measure could also be useful in studies concerning optimal quantum error correction. Its simple form makes it suitable for computation which is important in this respect. Finally, our scheme for fault-tolerant holonomic computation has also opened a number of interesting questions. We have shown that for universal computation this scheme requires three-local Hamiltonians. It may be possible, however, to use per- turbative techniques to approximate three-local Hamiltonians using two-local ones in a manner similar to the one introduced in Ref. [84]. Another direction for future re- search is suggested by the fact that the gap of the adiabatic Hamiltonian provides a natural protection against those types of errors that lead to excitations. It is in- teresting whether it is possible to design more efficient error correction schemes that make use of this property. Another question is whether the holonomic approach could provide a solution to the problem of the inconsistency between the standard fault- tolerance assumptions and the rigorous derivation of the Markovian limit. 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Abstract (if available)
Abstract
This thesis examines seven topics in quantum information and the theory of open quantum systems. The first one concerns weak measurements and their universality as a means of generating quantum measurements. It is shown that every generalized measurement can be decomposed into a sequence of weak measurements which allows us to think of measurements as resulting form continuous stochastic processes. The second topic concerns an application of the decomposition into weak measurements to the theory of entanglement. Necessary and sufficient differential conditions for entanglement monotones are derived, and are used tofind a new entanglement monotone for three-qubit states. The third topic examines the performance of different master equations for the description of non-Markovian dynamics. The system studied is a qubit coupled to a spin bath via the Ising interaction. The fourthtopic studies continuous quantum error-correction in the case of non-Markovian decoherence. It is shown that due to the existence of a Zeno regime in non-Markovian dynamics, the performance of continuous quantum error correction may exhibit a quadratic improvement if the time resolution of the error-correcting operations is sufficiently high.
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University of Southern California Dissertations and Theses
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Creator
Oreshkov, Ognyan
(author)
Core Title
Topics in quantum information and the theory of open quantum systems
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Physics
Publication Date
04/28/2008
Defense Date
03/28/2008
Publisher
University of Southern California
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University of Southern California. Libraries
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Tag
continuous quantum error correction,entanglement,holonomic quantum computation,non-Markovian dynamics,OAI-PMH Harvest,operator quantum error correction,weak measurements
Language
English
Advisor
Brun, Todd A. (
committee chair
), Dappen, Werner (
committee member
), Haas, Stephan (
committee member
), Lidar, Daniel (
committee member
), Zanardi, Paolo (
committee member
)
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oreshkov@usc.edu
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https://doi.org/10.25549/usctheses-m1201
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Oreshkov, Ognyan
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texts
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Tags
continuous quantum error correction
entanglement
holonomic quantum computation
non-Markovian dynamics
operator quantum error correction
weak measurements