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Quantum information flow in steganography and the post Markovian master equation

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Content Quantum Information Flow in Steganography and the Post Markovian Master Equation by Christopher Sutherland A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements of the Degree DOCTOR OF PHILOSOPHY (Physics) August, 2018 Copyright 2018 Christopher Sutherland Dedication To my parents. ii Acknowledgements First and most importantly I would like to thank my advisor Prof. Todd Brun. Without his guidance, patience, and support this document would not exist. From the moment he agreed to take me on as his student he has been an all around class act, for which I am very grateful. Secondly, thank you very much to Prof. Lidar, Prof. Haas, Prof. Pilch, Prof. Reichardt, and Prof. Zanardi for their support and mentorship at various stages throughout my time here at USC. Without them, I would not have made it through. Thank you also to Prof. Jonckheere for serving on my committee. Thank you to all my friends in the GTA, Waterloo, Los Angeles, and elsewhere for keeping me sane throughout my academic journey. Special thanks to my long time friends back home in Mississauga. Finally, a deep thank you to my family, whose consistent and continued support over the years has been incomparable. iii Table of Contents Dedication ii Acknowledgements iii List of Figures vi Abstract viii 1 Introduction 1 1.1 Coherent Information . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Distinguishability of Quantum States . . . . . . . . . . . . . . . . . 5 2 Quantum Steganography over Noiseless Channels: Achievability and Bounds 8 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Quantum Steganography: Achievability . . . . . . . . . . . . . . . . 11 2.2.1 The Bit Flip Channel . . . . . . . . . . . . . . . . . . . . . . 13 2.2.2 Depolarizing Channel . . . . . . . . . . . . . . . . . . . . . . 16 2.2.3 General Channels . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.4 Secret key consumption . . . . . . . . . . . . . . . . . . . . 22 2.3 Secrecy, Reliability, and Bounds . . . . . . . . . . . . . . . . . . . . 25 2.3.1 The information processing task . . . . . . . . . . . . . . . . 25 2.3.2 Upper bound on steganographic rate . . . . . . . . . . . . . 28 2.3.3 Upper bounds for specic channels . . . . . . . . . . . . . . 30 2.4 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . 33 3 Quantum Steganography over Noisy Channels: Achievability and Bounds 34 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 Achievability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2.1 Bit Flip Channel . . . . . . . . . . . . . . . . . . . . . . . . 37 iv 3.2.2 Depolarizing Channel . . . . . . . . . . . . . . . . . . . . . . 41 3.2.3 Secret key consumption . . . . . . . . . . . . . . . . . . . . 44 3.3 Secrecy, Reliability, and Bounds . . . . . . . . . . . . . . . . . . . . 45 3.3.1 The information processing task . . . . . . . . . . . . . . . . 45 3.3.2 Upper bound on steganographic rate . . . . . . . . . . . . . 48 3.3.3 Upper bounds for specic channels . . . . . . . . . . . . . . 50 3.4 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . 55 4 Non-Markovianity of the Post Markovian Master Equation 57 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Quantum non-Markovianity and the Post Markovian master equation 59 4.2.1 Quantum non-Markovianity . . . . . . . . . . . . . . . . . . 59 4.2.2 The post-Markovian master equation . . . . . . . . . . . . . 63 4.3 Previous Examples of the non-Markovianity of the PMME . . . . . 65 4.4 Non-Markovianity of the PMME via qubit dephasing . . . . . . . . 68 4.5 Which kernels give rise to CP-divisible dynamics? . . . . . . . . . . 73 4.6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . 75 5 Conclusions 76 A An example with no information back ow but with a non-divisible map 78 References 81 v List of Figures 2.1 The information processing task we consider for Alice sending M stego qubits to Bob over a quantum channel (which is identity for the noiseless case). Alice encodes her message M and an innocent covertext c into a suitable quantum error-correcting code which has had typical errors applied to it, where the encoding depends on the secret key k. She sends this to Bob, who then decodes the message and covertext using his copy of the shared secret key k. Alice's message is entangled with a reference systemR. The ability to transmit entanglement implies the ability to do general quantum communication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1 The information processing task we consider, of Alice sending M stego qubits to Bob over a quantum channelN p (which is either the quantum bit ip channel or depolarizing channel), which Eve believes to be noisier. Eve's ignorance of the channel is characer- ized by the parameter p. Dependent on the secret key k, Alice encodes her message subsystem M and an innocent covertext c into a suitable quantum error-correcting code in such a way that once passed through the physical channel, it looks as though typical errors of the channelN p+p have been applied. Bob then decodes the message and covertext using his copy of the shared secret key k. Alice's message is entangled with a reference system R. The ability to transmit entanglement implies the ability to do general quantum communication. . . . . . . . . . . . . . . . . . . . . . . . 46 vi 4.1 The non-Markovianity measure given by Eq. (4.27) corresponding to the two dierent choices of kernelsk 1 (t) (top) andk 2 (t) (bottom). Onlyf 1;2 (t) d dt f 1;2 (t) is plotted since it characterizes the behaviour of 1;2 and we can ignore discontinuities induced by the denominator. Note the non-Markovianity regions shown by the shaded regions in both plots. Both solutions exhibit damped oscillations, and hence an innite number of non-Markovianity regions, however the plot parameters are chosen so that only one such region is displayed. For 1 , the zeroes are given byr n = 2 ! (n arctan[!=(a + )]) and s n = n ! where ! = p 4aA (a + ) 2 . For 2 the zeroes are given by t n = n= where = p 2 +Aa. The parameters A;a, and are chosen so that! is real or else no non-Markovian eects would be present. The values used for these plots are A = 6;a = 1; = 1:1; and =. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 vii Abstract In this thesis, by applying ideas and results from quantum information theory, I study the elds of quantum steganography and the post-Markovian master equation (PMME), which is used in the eld of open quantum systems to model system-bath dynamics. Quantum steganography is the study of hiding secret quantum information by encoding it into what an eavesdropper would perceive as an innocent-looking mes- sage. In the rst part of this thesis, I study an explicit steganographic encoding for Alice to hide her secret message in the syndromes of an error-correcting code, so that the encoding simulates a given noisy quantum channel. I calculate achievable rates of steganographic communication over noiseless quantum channels using this encoding. I give denitions of secrecy and reliability for the communication process, and with these assumptions derive upper bounds on the amount of steganographic communication possible, and show that these bounds match the communication rates achieved with our encoding. This gives a steganographic capacity for a noise- less channel emulating a given noisy channel. I then go on to study the case where the actual physical channel shared between Alice and Bob is noisy. I study in par- ticular the bit- ip and depolarizing channels, and derive results that generalize from the noiseless scenario. Finally, I discuss the possibility of steganographic commu- nication over more general quantum channels, and conjecture a general formula for the steganographic rate. In the second part of this thesis, I analyze an easily solvable quantum master equation known as the PMME that takes into account memory eects induced on the system by the bath, i.e., non-Markovian eects. I brie y review the PMME, and analyze a simple example where solutions obtained exhibit non-Markovianity. viii I apply the distinguishability measure introduced by Breuer et al. which encodes a notion of information ow between the system and bath, and I also explictly analyse the divisibility of the associated quantum dynamical maps. I give a math- ematical condition on the memory kernel used in the PMME that guarantees non- CP-divisible dynamics. ix Chapter 1 Introduction The theory of information developed by Shannon in 1948 [46] is more or less the basis for all technological communication we take for granted today. With increasingly powerful quantum computers and devices on the horizon [40], we can imagine a world where characterizing the ow of quantum information between them will be important. Fortunately, we already have a powerful theory of quantum information [59]. It is then our duty, as theorists, to imagine ourselves living in such a future, and to ask questions such as: Which equations should we use to model such devices? What tasks might we have these devices carry out? The theory of open quantum systems is a eld which concerns itself mainly with this former question [12]. Being able to model a quantum system interacting with its environment with easily solvable equations is clearly an important task, espe- cially given how fragile and delicate quantum systems tend to be [32]. As for future quantum protocols, ever since Shor's remarkable discovery that a quantum com- puter would be able to factor large numbers eciently [50] and thus break modern day internet encryption schemes such as RSA [43], interest in quantum cryptog- raphy has been intense. Just as classical steganography has been an interesting 1 and fruitful appendum to classical cryptography to ensure stronger encryption [28], one can envision a world where a theory of quantum steganography will become important. In the subsequent sections, we will review some elements of quantum information theory that will prove to be useful in the chapters to come. A more detailed exposition of the ideas to be presented in the next section is given in Mark Wilde's quantum Shannon theory book [59]. 1.1 Coherent Information Before dening what the coherent information is, we start with some elementary denitions to familiarize the reader with the subject and notation. It is often important in quantum Shannon theory to characterize the way in which quantum states evolve, which is dened in the following way: Denition 1 (Quantum Channel). A quantum channel is a linear, completely pos- itive, trace preserving map, corresponding to a quantum physical evolution. A quantum channel can be written in a particularly pleasant form, known as the Kraus decomposition: Theorem 1 (Kraus Decomposition). A mapN :L(H A )!L(H B ) (denoted also byN A!B ) is linear, completely positive, and trace-preserving if and only if it has a Kraus decomposition as follows: N A!B () = d1 X i=0 A i A y i ; (1.1) 2 where d1 X i=0 A y i A i =I A ; (1.2) and d need not be any larger than dim(H A ) dim(H B ). We will make frequent use of quantum channels and their Kraus decompositions in later chapters. Another important quantity in the study of quantum Shannon theory that we will make extensive use of is the von Neumann entropy: Denition 2 (von Neumann Entropy). Suppose that Alice prepares some quantum system A in a state A 2D(H A ). Then the entropy H(A) of the state is dened as follows: H(A) Trf A log A g: (1.3) The joint quantum entropyH(AB) of the density operator AB 2D(H A H B ) for a bipartite systemAB follows naturally from the denition of the von Neumann entropy: H(AB) Trf AB log AB g: (1.4) An interesting property of the von Neumann entropy is that it is zero for pure states, that is, when =j ih j for some quantum statej i2H. Now we can dene the coherent information: Denition 3 (Coherent Information). The coherent information I(AiB) of a bi- partite state AB 2D(H A H B ) is given by: I(AiB) H(B) H(AB) : (1.5) 3 The coherent information is a good measure of quantum correlations. A rst indicator of this is that for a maximally entangled stateji AB = 1 p d P d i=1 jii A jii B shared between Alice and Bob, we have that H(B) = 1 and H(AB) = 0 (because it is a pure state). Another indicator that the coherent information is an important information quantity is that it satises a data processing inequality, that is: Theorem 2 (Data Processing for Coherent Information). Let AB 2D(H A H B ) and letN :L(H B )!L(H B 0) be a quantum channel. Set AB 0 N B!B 0 ( AB ). Then the following quantum data-processing inequality holds I(AiB) I(AiB 0 ) : (1.6) Of course, without a specic information processing task that makes use of the coherent information, there is no telling whether it is a good quantum information theoretic quantity. It turns out that the coherent information is intimately related to one's ability to send quantum information over a quantum channelN . To give this precice relation, rst we must dene the following: Denition 4 (Coherent Information of a Quantum Channel). The coherent infor- mation Q(N ) of a quantum channel is the maximum of the coherent information with respect to all input pure states: Q(N ) max AA 0 I(AiB) : (1.7) We can now state one of the most important theorems in the eld of quantum information theory: 4 Theorem 3 (Quantum Capacity Theorem). The quantum capacity C Q (N ) of a quantum channelN A!B is equal to C Q (N ) = lim k!1 1 k Q(N k ): (1.8) It should be noted that the quantum capacity is not known for many channels because the RHS of Eq. (1.8) is not necessarily easy to calculate. In our study of quantum steganography, we are interested in not only secretly sending classical information, but quantum information as well. It is for this reason that we will make extensive use of the coherent information to prove capacity theorems for our steganographic protocol, to be outlined in Chapters 2 and 3. 1.2 Distinguishability of Quantum States The trace norm of an operator A is given byjjAjj 1 = TrjAj, wherejAj = p A y A. Using this, we can dene a metric on the state spaceD(H) of the underlying Hilbert spaceH. For quantum states 1 and 2 , consider the trace distance between them: D( 1 ; 2 ) = 1 2 jj 1 2 jj 1 : (1.9) It can be easily checked that D( 1 ; 2 ) = 0 i 1 = 2 and D( 1 ; 2 ) = 1 i 1 and 2 are orthogonal. As we will see in Ch. 4, the trace distance can be seen as a measure of distinguishability between quantum states. As discussed in Breuer et al.'s review on quantum non-Markovianity [11], we can consider a notion of information ow between a system and its environment using the trace distance given in Eq. (1.9). Consider a bipartite system-environment 5 quantum state SE (t). It is reasonable to dene the information inside the open system as I int (t) =D( S 1 (t); S 2 (t)); (1.10) and the information which lies outside the open system as I ext (t) =D( SE 1 (t); SE 2 (t))D( S 1 (t); S 2 (t)): (1.11) We can think ofI ext (t) as the information which is not accessible when only measure- ments on the open system can be performed. Because we are assuming initially fac- torized system and bath states, we have that D( SE 1 (0); SE 2 (0)) =D( S 1 (0); S 2 (0)). Furthermore, the dynamics of SE are unitary and so I int (t) +I ext (t) =I int (0) = constant: (1.12) From this we see that an increase of dinstinguishability between the system states, i.e. an increase of D( S 1 (t); S 2 (t)); over time corresponds to a ow of information from the environment to the system, and vice-versa when the system states become less distinguishable. We will make use of this notion of information ow between the system and environment extensively in Ch. 4 when we study the non-Markovianity of the post-Markovian master equation. In Ch. 2 we describe our protocol for quantum steganography over noiseless channels where Alice and Bob simulate a given noisy channel, prove bounds on the amount of secret communication possible, and give explicit encodings for achieving these bounds. In Ch. 3 we then extend these results to the case where the actual physical channel shared between Alice and Bob is noisy. Finally, in Ch. 4, we show 6 through a simple example that the easily solvable post-Markovian master equation can describe non-Markovian eects, giving further evidence to the claim that it is a useful tool for the modeling of open quantum system dynamics. 7 Chapter 2 Quantum Steganography over Noiseless Channels: Achievability and Bounds 2.1 Introduction The study of steganography is perhaps best motivated by considering an example. Suppose two political protestors Alice and Bob are arrested and put into two widely separated jail cells. The warden allows them to communicate with hand-written letters that he reads before delivering. However, if the warden reads anything in the letters that he nds suspicious (such as a possible escape plan), then he will not deliver the letter. Luckily, Alice and Bob exchanged a secret key before their incarceration. Can Alice and Bob communicate their escape plan to each other without arousing the warden's suspicions? This is where the study of steganography comes into play. The science of sending information through seemingly innocuous messages has a long history, dating back to at least 440 B.C.[27]. It is worth making clear its dierences from cryptography. In cryptography, a secret message (the plaintext) is encrypted using the shared secret key, and the resulting ciphertext is then sent to 8 the desired receiver to be decoded. If an eavesdropper (Eve) observes the ciphertext, she cannot decode it without the secret key. However, she will know that there is a secret message, since Alice is sending apparent gibberish to Bob. By contrast, if Alice uses a steganographic encoding, she hides the secret message (or stegotext) into a larger covertext, which appears to Eve as an innocuous message. The hidden message may or may not be encrypted itself, but the main line of defense is that the eavesdropper is unaware that a message is even being sent. During WWII, a Japanese spy named Velvalee Dickinson sent classied informa- tion to neutral South America. She was a dealer in dolls, and her letters discussed the quantity and type of doll to ship. The covertext was the doll orders, while the concealed stegotext was encoded information about battleship movements [23]. The quantum analogue of cryptography has been widely studied [22]. How- ever, the quantum analogue of steganography is still in a relatively early stage. There have been a number of dierent proposals for encoding quantum information steganographically, or encoding classical information into quantum states or chan- nels [35, 6]. In this paper we consider hiding secret messages as error syndromes of a quantum error-correcting code [25]. This approach to quantum steganography has been studied in detail by Shaw and Brun, with explicit encoding and decod- ing procedures and calculated rates of communication and secret key consumption [48, 47]. It was shown that such schemes can hide both quantum and classical in- formation, with a quantitative measure of secrecy, even in the presence of a noisy physical channel. When the error rate of the physical channel is lower than the eavesdropper's expectation, it is possible to achieve non-zero asymptotic rates of communication. (If the eavesdropper has exact knowledge of the channel, secret 9 communication may still be possible, but the amount of secret information that can be transmitted in general grows sublinearly with the number of channel uses.) More recently, a closely related idea has been studied under the name of quan- tum covert communication [7, 49, 58, 8, 4]. Many of the ideas in this paper are closely related to steganographic requirements, such as secrecy and recoverability. This is not surprising, since covert quantum communication can be seen as a special case of quantum steganography over noisy quantum channels in the case when the eavesdropper has exact knowledge of the channel, and where Eve assumes the chan- nel is idle (so only noise is being transmitted). Similarly, quantum steganography is a type of covert quantum communication where Eve knows about the covertext communication but not the hidden stegotext, and where Eve may not have perfect knowledge of the channel. The work on covert communication has generally found that, if Eve has exact knowledge of the channel, the amount of secret communication that can be done grows like the square root of the number of channel used. The goal of this paper is to formalize the assumptions and reasonable condi- tions of quantum steganography introduced in [48], and to give upper bounds on the achievable rates of quantum communication while remaining secure from an eavesdropper's suspicion, for the special case when the true underlying channel is noiseless. Our results include achievability results as well as converse proofs for quantum steganography. In Section 2.2 we formalize our notion of quantum steganography where secret messages are hidden in the syndromes of an error-correcting code, and outline a spe- cic steganographic encoding where Alice is able to emulate any general quantum channelN on her encoded secret message and covertext. We work out specic ex- amples for the bit- ip channel and the depolarizing channel, before giving the more 10 general result. In Section 2.3 we prove upper bounds on the amount of stegano- graphic communication possible, and show that these bounds are asymptotically equal to the rates achieved in the previous section. The assumption that the physical channel is noiseless greatly simplies the anal- ysis. However, we believe that the main intuition underlying this approach will apply equally well in the case of a noisy channel. We will end this paper with a dis- cussion of how to extend this work to the case where the physical channel between the two parties is noisy. 2.2 Quantum Steganography: Achievability As discussed in the introduction, there have been several approaches to generalizing steganography to the quantum setting. Here we will make explicit the notion of quantum steganography based on syndromes of quantum error-correcting codes. We assume that Eve expects to see quantum information passing through a noisy quantum channel. However, the actual physical channel is assumed to be noiseless. This is obviously an idealized assumption, which greatly simplies the analysis; we will discuss below how it might be justied at least as an approximation. Alice wants send a secret message steganographically to Bob. Using her shared secret key, she encodes the stego text into a codeword of a quantum error-correcting code (QECC) with errors applied to it, and sends it to Bob. The codeword encodes an innocent state; the stego text is conveyed in the errors. If Eve were to perform measurements on this codeword, it would be indistinguishable from an innocent encoded covertext that had passed through a given noisy quantum channel to Bob. 11 Before discussing how to quantify the security of a quantum steganographic protocol, let us make clear what Alice is trying to achieve. Alice wants to encode an innocent covertext state, together with her secret message, into an N-qubit codeword in such a way that it cannot be distinguished from the covertext alone encoded into a quantum error-correcting code that has undergone typical errors induced by the quantum channelN N . The steganographic encoding works by mapping all possible secret messages onto syndromes of the QECC. This encoding is not limited to classical messages: it is possible to encode a quantum state by preparing the codeword in a superposition of dierent error syndromes. In analyzing this quantum steganography protocol, we make the following as- sumptions. Alice is communicating with Bob by a quantum channel that is actually noiseless. But the eavesdropper, Eve, believes that this channel is noisy, perhaps because Alice and Bob have been systematically making the channel appear noisier than it actually is. Because Alice and Bob have been systematically deceiving Eve in this way, we assume that they know (at least fairly closely) what Eve's estimate of the channel is. Before the protocol began, Alice and Bob shared with each other a secret key: an arbitrarily long string of random bits. This key is known only to the two of them. But once the protocol begins, they cannot communicate except through channels that can be monitored by Eve. Alice sends an innocent-looking message to Bob over the channel. This is a covertext state c , encoded into an error-correcting code; it is assumed that the choice of code is known to Eve, and this code should be a plausible choice for the noisy channel that Eve believes exists. One important caveat for this section: we will be considering the case where the QECC that Alice uses is nondegenerate. That is, each typical error corresponds to a unique error syndrome. This allows Alice to communicate as much steganographic 12 information as possible, and it allows us to ignore the details of which QECC is being used. Methods similar to those in this section should also work for degenerate codes; but in that case, the encoding will be strongly dependent on the properties of the particular code, since the typical errors must rst be grouped into equivalent sets, and then the possible messages mapped into these sets. We also use this assumption in the next section to get specic expressions for the upper bound on the secret communication rate. To clarify how the encoding works, we start by considering two examples for relatively simple channels: rst, the case where Alice is emulating a bit ip channel N BF p on the codeword, and second, the case where she is emulating the depolarizing channel. Finally we consider a more general error mapN N . The message qubits are encoded into into the error syndromes of the codeword of the QECC she is using. 2.2.1 The Bit Flip Channel Suppose that Eve believes the channel connecting Alice and Bob to be a bit ip channel, with a probabilityp of error per qubit sent. (The actual physical channel is noiseless, as assumed above.) Alice sends a codeword of lengthN to Bob, encoding some \innocent" covertext state c . The errors in the codewords that Alice sends to Bob should be binomially distributed: pN is the mean number of errors of this distribution, and the variance is (1p)pN. The total probability that there is an error of weight w on the codeword should be p k = N w p w (1p) Nw : (2.1) 13 There are N w N! w!(Nw)! such errors, all with equal probability p w (1p) Nw . If N is large, then it is extremely likely that the number of bit ips will be a typical error|that is, an error of weight w within a narrow range about the mean pN. Alice's encoding will make use of these typical errors. For each w from Np(1) to Np(1 +), where p (1p)=pN 1, Alice chooses at random a set ofC w possible error strings of weightw. (An error string of weightw is a string ofN bits, with a 1 at every location with a bit ip and 0 at every location with no error.) This random choice is made using the shared secret key with Bob, so that Bob also knows which set of errors is being used to encode secret messages, but Eve (who does not share the key) could not know this. Let these sets of error strings of weight w be calledfS w g, and the set of all strings used in the encoding is S = [ w S w : (2.2) We sum up C = Np(1+) X w=Np(1) C w =jSj: (2.3) So the total number of strings in the setS isC. This numberC is the total number of possible distinct secret messages that Alice can send to Bob (though she may also send superpositions of these messages). We assume all these messages to be equally likely. So the message encodes M = log 2 C bits (or qubits) of information. Dene the probability q = 1=C. These error strings S are typical strings (us- ing the denition of weak typicality from information theory). Eve should not be 14 suspicious at seeing such an error string, since it matches a probable result for the channel that she expects. For this encoding to be indistinguishable from the bit ip channel, the probability of the message being an error string of weight w should equal the value from the distribution in Eq. (2.1) above. This means we want to satisfy qC w = C w C =p w : (2.4) Clearly we must have C w N w ; for all w in the typical range. This implies that: C w p w (1p) Nw N w p w (1p) Nw =C w q; )p w (1p) Nw q: (2.5) To communicate the maximum amount of information steganographically we want C to be as large as possible, which means we wantq to be as small as possible. The constraint in Eq. (2.5) then gives us q =p Np(1) (1p) N(1p+p) : (2.6) 15 So Alice can send M stego qubits to Bob, where M = log 2 C = log 2 1=q =N(p log 2 p (1p) log 2 (1p) +(p log 2 pp log 2 (1p))) =N(h(p)p log 2 ((1p)=p)) Nh(p); (2.7) whereh(p) =p log 2 p (1p) log 2 (1p) is the entropy of the bit ip channel on one qubit. Therefore, with this encoding Alice can send almost Nh(p) bits. In [48] it is shown that the diamond norm distance between the channel (N BF p ) N and Alice's encoding is exponentially small in N. This justies the claim that this protocol will not arouse suspicion from Eve. In section III we use a slightly modied denition of secrecy that allows us to prove the converse bound on this rate of stego communication by information theoretic techniques. That means that this encoding is essentially optimal: the maximum rate of steganographic communication for a nondegenerate code in the case of the bit ip channel is h(p). 2.2.2 Depolarizing Channel Here we will consider the scenario where the channel Alice is emulating is the depo- larizing channel. It turns out that due to the symmetric nature of the depolarizing channel the encoding looks quite similar to that of the bit ip channel. Recall that the depolarizing channel acting on a single qubit is given by N DC p () = (1p) + (p=3)(XX +YY +ZZ): 16 Applying this channel on N qubits, the total probability of all errors with exactly n 1 X, n 2 Y , and n 3 Z errors (and n 4 =Nn 1 n 2 n 3 identity \errors") is p(n 1 ;n 2 ;n 3 ;n 4 ) = N! n 1 !n 2 !n 3 !n 4 ! (p=3) n 1 +n 2 +n 3 (1p) n 4 : Notice that instead of specifying n 1 ;n 2 ; and n 3 exactly, we can instead talk about errors with weightw =n 1 +n 2 +n 3 . It follows by simple calculation that the total probability of all errors of weight w is p(w) = 3 w N w (p=3) w (1p) Nw = N w p w (1p) Nw ; which is just a binomial distribution in w. As in the bit ip case, we will need say what strings of errors are typical. There are a number of ways we could specify this, but for simplicity we will consider weights w that lie between Np(1) and Np(1 +) for p (1p)=pN 1. The astute reader will notice that this set includes some errors that are not typical: for instance, it includes errors of weight w where all (or most) of the errors are X's and none (or few) are Y 's or Z's. If such errors are used as codewords, they might make Eve suspicious. Still, the eect of this is not too large, because this set is still dominated by typical errors, and the probabilities of these strings are similar to the expected probabilities of atypical errors. With this denition of typicality, we can follow the exact same encoding given for the bit ip code using errors with weight w, except that the set of errors of weight w is now of size 0 @ N w 1 A 3 w ; 17 and errors of weight w have probability (p=3) w (1p) Nw . This then leads to the following encoding rate: M =N(p log 2 (p=3) (1p) log 2 (1p) +(p log 2 (p=3)p log 2 (1p))) =N(s(p) +(p log 2 (p=3)p log 2 (1p)) Ns(p) (2.8) where we have dened s(p) =p log 2 (p=3) (1p) log 2 (1p) to be the entropy of the depolarizing channel on one qubit. 2.2.3 General Channels Special case: random unitaries Consider a quantum channel acting on a single qubit of the form N () = k X i=1 p i U i U y i ; (2.9) where the operators U i are all unitary, so U i U y i = U y i U i = I. The set of Kraus operatorsf p p i U i g can be thought of as a set of possible single-qubit unitary errors U i that occur with probability p i . Note that both the bit- ip and depolarizing channels are special cases of the random unitary channel, as is any Pauli channel. The channel acts on an N-qubit encoded state asN N (). 18 The total probability of all errors with n 1 U 1 errors, n 2 U 2 errors, and so forth, is given by the multinomial distribution: p(n 1 ;:::;n k ) = N! n 1 !n k ! p n 1 1 p n k k : (2.10) Now consider weightsn j in the range fromNp j (1) toNp j (1+), where is large enough that this set includes all the typical strings. (This denition can be modied, but for simplicity we stick with it in this paper.) Randomly choose C n 1 ;:::;n k strings with weights n 1 ;n 2 ;:::;n k in this range such that n 1 +::: +n k =N. As with the bit ip and depolarizing channels, let these sets of strings be called S n 1 ;:::;n k and let S denote the union of all these sets of strings, which are a subset of the typical strings. For all weights n 1 ;:::;n k outside the typical set, we let C n 1 ;:::;n k = 0. The total number of strings in the set S is C: C = X n 1 ;:::;n k C n 1 ;:::;n k : (2.11) Dening q 1=C, we want to satisfy C n 1 ;:::;n k q =C n 1 ;:::;n k =C =p(n 1 ;:::;n k ) (2.12) 19 for all weightsn 1 ;:::;n k in the typical set, so that Eve does not become suspicious. Also, clearly C n 1 ;:::;n k must be less than N! n 1 !n k ! . This implies that: C n 1 ;:::;n k p n 1 1 p n k k N! n 1 !n k ! p n 1 1 p n k k C n 1 ;:::;n k p n 1 1 p n k k C n 1 ;:::;n k q p n 1 1 p n k k q: (2.13) Notice that this time we cannot simply plug in the lower bounds of the sums forn j , as we did for the depolarizing and bit ip channels, because we have the additional constraint that n 1 +::: +n k = N. However, the same general argument applies. Inside the set of typical weights, there is a string ~ n 1 ; ; ~ n k withj~ n j =Np j jp j for all j, that maximizes the probability: p max p ~ n 1 1 p ~ n 2 2 p ~ n k k : (2.14) We can chooseq =p max , and use this to put a bound on the number of stego qubits M Alice can send to Bob: M = log 2 C = log 2 (q) = log 2 p max =~ n 1 log 2 (p 1 )::: ~ n k log 2 (p k ) =N( ~ n 1 N log 2 (p 1 )::: ~ n k N log 2 (p k )) N(1)( k X i=1 p i log 2 (p i )) =N(1)H(p 1 ;:::;p k ): (2.15) 20 So in the limit of large N, we should approach a rate H(p 1 ;:::;p k ) with this en- coding. Encoding general channels across multiple code blocks This argument does not necessarily apply directly to a general quantum channel, since the probabilities of the dierent outcomes can be state dependent. However, we should be able to do a similar type of encoding for a general quantum channel N by encoding across multiple code blocks. Consider a general quantum channel acting on a single qubit as N () = k X i=1 A i A y i : (2.16) The channel acts on an N-qubit encoded state asN N (), where we will let N become large. For most states , we can well approximate this N-qubit channel by a sum over the typical errors [29, 30], N N () X i2T E i E y i ; (2.17) where is now the N-qubit codeword, the index is i =i 1 i 2 :::i N , the typical error E i is E i =A i 1 A i 2 A i N ; (2.18) andT is the set of typical sequences i [59]. We assume that the QECC Alice uses is one that can correct the typical errors of the channel. (Indeed, using a code that was not strong enough to correct the typical errors might well arouse Eve's suspicions.) We will also assume, for simplicity of this analysis, that the QECC is nondegenerate. This means that on a valid codeword 21 in the QECC, the typical errors E i all have distinct error syndromes, and act as unitaries that move the state to a distinct, orthogonal subspace labeled by i. This means that errorE i occurs with a xed probabilityp i for all valid codewords of the QECC. We can then essentially repeat the argument that leads to Eq. (2.15), but now using the probabilities p i . Note that we now need to take two limits: the limit of many blocks, and also the limit where the individual blocks are large. For this argument to apply, we need to rst go to the limit of many blocks, and then to the limit of large block size. In those limits, we can approach a rate 1 N X i p i log 2 p i H; (2.19) where H is an eective entropy per qubit from the channel. Note that there are some ambiguities in making this argument. The Kraus map in Eq. (2.16) is not unique. Choosing dierent sets of Kraus operators will lead to dierent sets of typical errors. However, these dierences should not lead to signicant changes to the eective entropy in the limit of large block size, so long as the code is nondegenerate on both sets of typical errors. 2.2.4 Secret key consumption For the above encodings, how much secret key must be consumed? In general, we can assume that all the details of the encoding, etc., have been decided between Alice and Bob ahead of time. So in the protocol as described above, the only place where secret key is consumed is to pick the subsets of errors used in the encoding. 22 Let's consider the bit ip channel as a simple example. The possible messages are mapped onto a set ofC error syndromes, representing errors of weights (1)Np w (1 +)Np. For each error weight w in that range, a subset of C w errors is chosen to represent possible messages. Alice and Bob can agree before the protocol begins to divide the set of errors of weight w inton w nonoverlapping subsets of C w errors each, where n w = 0 @ N w 1 A =C w = 1p p wNp(1) : (2.20) (Since this is unlikely to be an exact integer, one must generally round down, which means that a small fraction of possible errors will be omitted. This will slightly reduce the match between the steganographic encoding and the noisy channel being simulated, but for large N and p 1 the dierence will be small.) For each transmitted block, Alice and Bob must randomly choose one of these n w subsets for each weight w in the typical range. Choosing a subset requires log 2 n w random bits, which are drawn from their shared key. However, since any given message is encoded as an error of some specic weight w, Alice and Bob can reuse the same secret key bits to choose the subset for each error weight w. So the number of key bits consumed to transmit one block is equal to the maximum value of log 2 n w for (1)Npw (1 +)Np, which is K = max Np(1)wNp(1+) log 2 n w = max Np(1)wNp(1+) log 2 1p p wNp(1) = (2Np) log 2 1p p : (2.21) 23 How does this scale with N? Since this is a binomial distribution, will take the form =D s 1 N 1p p ; (2.22) where D is a xed constant determining what fraction of all errors are included in the typical set. The key consumption therefore is K = 2D s N 1p p log 2 1p p : (2.23) The key consumption scales sublinearly with N, and asymptotically the key con- sumption rate goes to zero. While the details will vary, we expect this kind of sublinear scaling of K with N to be generic. A few words more on secret key consumption are in order. In [48], Shaw and Brun make a distinction between the secrecy and the security of a steganographic protocol. A steganographic protocol is secret if an eavesdropper without the secret key cannot distinguish between an encoded message being sent and the noisy chan- nel being applied. It is secure if the eavesdropper cannot learn anything about the message, even if she knows that a message is begin sent. Using a sublinear amountK of shared secret key is sucient to make the stegano- graphic protocol secret, by this denition. However, it is not secure, in general. Since the number of qubits M transmitted is typically larger than the number of secret key bits K consumed, we would generically expect an eavesdropper to be able to learn on the order of MK bits of information about the message if she became aware of its existence. This can be prevented by rst encrypting the message before doing the stegano- graphic encoding. Encryption requiresM bits of secret key in the case of a classical 24 message (using a one-time pad), or 2M bits of secret key in the case of a quantum message (by twirling). In this case, the protocol is both secret and secure. However, there is a cost: the secret key is now consumed asymptotically at a linear rate. 2.3 Secrecy, Reliability, and Bounds 2.3.1 The information processing task Here we consider the steganographic scenario as outlined above where Alice is us- ing fake noise to hide her message from Eve, but the actual physical channel she is sending her information over is noiseless. We will consider the task known as entan- glement transmission. This notion of quantum communication encompasses other quantum information-processing tasks such as mixed-state transmission, pure-state transmission, and entanglement generation. We follow closely the discussion of quantum communication in [59]. The information processing task we are considering is visualized in Figure 2.1. Alice has a secret message of M = log 2 jA 1 j qubits, which is maximally entangled with a reference system R. She also prepares an innocent covertext c which will be encoded into theN-qubit quantum error-correcting code. Let us rst dene her encoded state, dependent on the secret key element k: ! k;A 0n R E k;A 1 C!A 0 n( c A 1 R ): (2.24) 25 Figure 2.1: The information processing task we consider for Alice sending M stego qubits to Bob over a quantum channel (which is identity for the noiseless case). Alice encodes her messageM and an innocent covertext c into a suitable quantum error-correcting code which has had typical errors applied to it, where the encoding depends on the secret keyk. She sends this to Bob, who then decodes the message and covertext using his copy of the shared secret keyk. Alice's message is entangled with a reference systemR. The ability to transmit entanglement implies the ability to do general quantum communication. 26 This dependence of the encoding on the secret key corresponds to choosing among the dierent sets of error strings S in the protocols from the previous section. To someone (like Eve) who does not know the secret key k, the state is eectively ! A 0n R X k p k ! k;A 0n R ; (2.25) where ! A 0n R is the state averaged over all possible values of the secret key k with probabilitiesp k . (We can choose this probability to be uniform for simplicity,p k =p for all k, if we so desire.) What is a good way to guarantee secrecy from Eve? We propose the following secrecy condition: 1 2 kTr R (! A 0n R )N N (V c V y )k 1 (2.26) whereN is whatever channel Alice is emulating, V is an isometry representing the encoding of the covertext into a suitably chosen codeword (one which can correct typical errors induced by the channelN ) and> 0 is some small parameter. What this condition says is that if Eve observes the quantum state, it will be eectively indistinguishable from an encoded covertext being sent through the noisy quantum channelN . We introduce another requirement which corresponds to a notion of recoverabil- ity. Once Bob receives the state, he applies his decoderD k;A 0n !B 1 C to obtain the original c B 1 R . We can relax this by only requiring that the input states and output states are close, that is: 1 2 kD k;A 0n !B 1 C (! k;A 0n R ) c B 1 R k 1 ;8k (2.27) 27 where > 0 is a small parameter. 2.3.2 Upper bound on steganographic rate With these two assumptions of secrecy and recoverability, we can now put a bound on the number of qubits M that can be sent reliably and stegonagraphically from Alice to Bob. Dening E N N (V c V y ) and applying the Fannes-Audeneart inequality to the secrecy condition we have: H(Tr R (! A 0 n R ))H( E ) +N +h 2 () (2.28) where h 2 is the binary entropy function. Furthermore, from the recoverability con- dition we have M = logjA 1 j =I(RiB 1 ) I(RiB 1 ) D k (!) +N + (1 +)h 2 (=[1 +]) I(RiA 0n ) ! k +f(N;) H(Tr R (! k;A 0 n R )) +f(N;): (2.29) The rst equality follows from the fact that the coherent information of a maximally entangled state is just the logarithm of the dimension of one of the subsystems. The rst inequality follows from the AFW inequality applied to (2.27). The second inequality is the data processing inequality. The last inequality follows from the denition of the coherent information. 28 The concavity of entropy implies that X k p k H(! k;A 0n)H X k p k ! k;A 0n ! =H(! A 0n): (2.30) The encodingsE k;A 1 C!A 0 n are isometries, which means thatH(! k;A 0n) has the same value for every k. We can therefore sum over the probabilities p k on the left-hand side of (2.30) to get H(Tr R (! k;A 0 n R ))H(Tr R (! A 0 n R )): (2.31) Now putting (2.28) and (2.29) together we arrive at our main result, which states that Alice can secretly and reliably send M stego qubits to Bob, whereM is bounded above by MH(Tr R (! RA 0 n)) +f(N;) H( E ) +g(N;) +f(N;); (2.32) whereg(N;)N +h 2 (). Thus, if we can compute a maximum for H(N N ()) when is pure (becauseV is an isometric encoding and c is pure), we have a tight upper bound on the number of qubits M that can be sent steganographically over a noiseless quantum channel. (Of course, if the actual quantum channel is noisy, then this bound will in general be changed. This is the topic of future work.) 29 2.3.3 Upper bounds for specic channels We will now apply our result (2.32) to the channels discussed in the previous section, where we make the implicit assumption that Alice is using a nondegenerate code. Though our result (2.32) is true in general, for a degenerate code the number of dis- tinct error syndromes is smaller (depending on the code), and the bounds discussed here and achievable rates discussed in the previous section would be adjusted. The bit ip channel For the bit ip channel, i.e.,N BF () = (1p)+pXX, the maximum ofH(N N ()) over allN-qubit pure states isNh(p) whereh(p) =p logp (1p) log(1p) is the entropy of a single qubit sent through a bit ip channel. To prove this, consider some pure state =j ih j. Then N N BF (j ih j) = X s p(s)X s j ih jX s (2.33) where we are summing over all binary strings s of length N; X s is the operator acting on N qubits with an X acting at every location where s has a 1 and an I where s has a 0. The probability p(s) is given by p(s) =p w(s) (1p) (Nw(s)) ; (2.34) where w(s) is the weight of string s. The Shannon entropy of this distribution is Nh(p), since it is a binomial distribution. The von Neumann entropy is the minimum Shannon entropy over all possible ensemble decompositions of the given state, and it is not hard to check that it is achieved whenj i is aZ eigenstate. Thus 30 the encoding described in the previous section for steganography with an simulated bit ip channel is essentially optimal. More general channels Unfortunately, for a more general quantum channelN we may not know, in general, what N-qubit pure state maximizes H(N N ()). However, we can still bound this quantity. First, consider a general quantum channelN that acts on anN qubit pure state as follows: N N () X j E j E y j (2.35) wherefE j g is the set of typical errors associated withN applications of the channel N . Recall that we are choosing our isometric encoding to correct for typical errors of whatever channelN it is we are emulating. Though the set of correctable errors fE j g need not act like unitaries on the codespace, we can always nd a set of correctable errorsf e E j g j that do [36]. To see this, rst consider the Knill-La amme condition: PE y i E j P = ij P (2.36) whereP is the codespace projector and is a Hermitian matrix. Thus, we can write e =U y U where U is a unitary matrix and e is diagonal. e E k = X j M jk E k (2.37) 31 where the unitary M is chosen in such a way as to diagonalize . That is P e E y k e E l P = X ij M ik M jl PE y i E j P = ( X ij M ik ij M jl )P =e kl P = kl e kk P: (2.38) Note that these errorsf e E j g act unitarily on the codespace. So long as the Knill- La amme condition is satised, we can always diagonalize in this way. Now going back to our expression for the channel action, X j E j E y j = X k;l;j M kj M lj e E k e E y l = X k e E k e E y k : (2.39) Because we have assumed that the typical errors are all correctable, and that the code is nondegenerate, the states e E k e E y k are all orthogonal to each other, and Trf e E k e E y k g = kk . The von Neumann entropy is the Shannon entropy minimized over all possible decompositions, so the entropy of this state is clearly H( E ) =H(N N (V c V y )) X k kk log 2 ( kk ): (2.40) By (2.32) we have shown that the amount of steganographic communication allowed for a quantum channelN emulation is upper bounded by this quantity. Applying this to the general channel discussed in section II.C above, we see that this quantity is equal toN H, where H is the eective entropy per qubit dened in Eq. (2.19). So this encoding approaches the maximum possible rate for the general channel, just as for the bit ip channel. 32 2.4 Conclusions and Future Work Quantum steganography is the study of secret quantum communication between two parties, Alice and Bob. We have shown that Alice and Bob are able to commu- nicate with each other secretly at a nonzero rate using a shared secret key, without arousing suspicion from a potential eavesdropper Eve. In this paper we gave explicit bounds on the number of stego qubits that Alice can send to Bob when Alice is simulating a general quantum channelN with her stego encoded message, as well as explicit encodings to that achieve these bounds, for the case when the actual physical channel is noiseless. The obvious next question is what if the channel shared between Alice and Bob (as is generally the case) is noisy? There is reason to believe that so long as Eve has some ignorance about the actual physical channel, then Alice will still be able to communicate steganographically to Bob. For instance, suppose the actual physical channel is a depolarizing channelN p wherep is the depolarizing parameter and the channel that Eve expects isN p+4p=3 for some small suitably chosen> 0. Then Alice can emulate a depolarizing channel N in such a way such that if Eve observes the state Alice is sending to Bob, it will look like an innocent encoded covertext passing throughN applications of a channel N p N (where N is the length of the codeword Alice is using). There should be elements of the encoding given in this paper that will generalize to the noisy case for general channelsN . In Ch. 3 we look at this case, and show that for the bit- ip and depolarizing channels this is indeed possible. Based on those examples, we conjecture a general formula for the steganographic rate. Proving a general formula is an interesting open question. 33 Chapter 3 Quantum Steganography over Noisy Channels: Achievability and Bounds 3.1 Introduction This chapter is based on joint work with Prof. Brun [55]. Suppose Alice and Bob are the respective leaders of two countries and they wish to communicate highly classied information with each other over a public channel. They do not want other countries to know they are communcating secret information, perhaps because they have a history of shady international political relations. Simple cryptography would not be good enough here beacause it would alert a potential eavesdropper (Eve) that secret communication is taking place, even if she cannot read it. Therefore, if Alice and Bob wish to keep their conversations secret, they must employ a steganographic protocol. Both cryptography and steganography are interesting and well-developed sub- jects, the studies of which date back millenia [27, 51]. In cryptography, a secret message is encrypted using a shared secret key between Alice and Bob, and Alice sends the resulting ciphertext across a channel to be decoded. Should Eve observe 34 this ciphertext, she would not be able to decode it without the secret key. How- ever, she would undoubtedly become suspicious if she weren't already, due to Alice sending encrypted messages to Bob. Steganography solves this problem of secrecy. Although cryptography allows for secure communication, in this paper we are interested in secret communication. In steganography, a secret message is hidden into a larger covertext, which appears to Eve as an innocuous message. This seeming innocuousness of the message is what makes the protocol secret. The hidden message may also be encrypted itself to make the protocol not only secret but secure, so that even if Eve were tipped o to there being secret communication between Alice and Bob, she would not be able to decode the hidden message. For example, digital audio, video, and pictures are increasingly furnished with distinguishing but imperceptible marks, which may contain a hidden copyright notice or encrypted serial number [38]. Ever since Shor's remarkable discovery that a quantum computer could solve the prime factorization problem eciently, hence cracking one of the internet's most common encryption schemes [50], interest in quantum cryptography has been intense. Quantum steganography is of more recent development [35, 6, 25]. The pro- tocol we will be considering is to encode quantum information steganographically as error syndromes of a quantum error-correcting code. This was detailed extensively by Shaw and Brun [47, 48], where it was shown that such schemes can hide both quantum and classical information, with a quantitative measure of secrecy, even in the presence of a noisy physical channel. A more precise analysis of this quantum steganographic protocol over noiseless channels was done by the present authors in terms of achievability and converse 35 proofs [54]. In this work, we treated the case where Eve believes the channel con- necting Alice and Bob to be some noisy quantum channel, but the actual physical channel is noiseless, and we gave optimal rates of steganographic communication. A related eld is known as covert quantum communication [7, 49, 58, 8, 4]. In covert communications, it is often assumed that the channel between Alice and Bob is a noisy optical channel, which is modelled by a beamsplitter with some trans- missivity parameter that characterizes how many photons are lost to Eve. Covert quantum communication can be seen as a special case of quantum steganography over noisy quantum channels in the case where the eavesdropper has exact knowl- edge of the channel, and where Eve assumes the channel is idle (so only noise is being transmitted). Similarily, quantum steganography is a type of covert quantum communication where Eve knows about the covertext communication but not the hidden stegotext, and where Eve may not have perfect knowledge of the channel. In covert quantum communication, it has been shown that in general one can secretly communicate an amount of classical information which scales like the square root of the number of channel uses. The goal of this paper is to extend the previous work on quantum steganography over noiseless channels in [54] to the scenario where the channel Alice and Bob share is noisy. We assume that Eve believes the channel to be noisier than it is, which allows Alice and Bob to communicate at a linear rate in the number of channel uses. This assumption is not unreasonable, especially when Alice and Bob have been systemetically deceiving Eve by adding extra noise. We also assume that Alice and Bob are using an error-correcting code powerful enough to correct errors induced by the channel Eve believes to be connecting them, or else she would become suspicious. Eve would also become suspicious if the pattern of errors Alice 36 uses to encode her secret information does not match the typical errors induced by the channel that Eve expects. In Section 3.2 we formalize our notion of quantum steganography where secret messages are hidden in the syndromes of an error-correcting code. We outline a specic steganographic encoding where Alice is able to emulate a bit ip or de- polarizing channelN p+p (the channel Eve believes to be connecting them) on her encoded secret message and covertext, where the actual physical channel isN p . We also calculate the amount of key consumed in our protocol. In Section 3.3 we prove upper bounds on the amount of steganographic communication possible over these channels, and show that these bounds are asymptotically equal to the rates achieved in the previous section. Finally, in Section 3.4 we summarize our results, and discuss quantum steganography for general quantum channels, conjecturing a capacity formula for general quantum steganographic communication. 3.2 Achievability 3.2.1 Bit Flip Channel Suppose that Alice wishes to communicate steganographically to Bob by secretly sending him a message m drawn from a set of possible messagesM, assumed to all be equally likely. Alice and Bob can communicate via a quantum channel, but the eavesdropper Eve can monitor their communications over this channel if she chooses. They cannot communicate clasically without Eve intercepting their communications, but before the protocol began they exchanged a secret key, in the form of an arbitrarily long string of random bits, unknown to Eve. We assume that 37 Eve believes the quantum channel shared between Alice and Bob to be a bit ip channel with error rate p +p, i.e., N BF p+p () = (1 (p +p)) + (p +p)XX: (3.1) However, the actual physical channel between Alice and Bob isN BF p . First, Alice encodes an innocent state, i.e. the covertext c , into a nondegenerate quantum error-correcting code (QECC) on N qubits. This code should be able to correct typical errors induced by the channel (N BF p+p ) N . Next, depending on the secret key k2K and the message m2M that she would like to send, she applies the error E N (k;m) =E 1 (k;m) ::: E N (k;m) (3.2) to her state. This produces the codeword corresponding to the message m. If her message to Bob is a quantum state, she can prepare the system in a superposition of these codewords. These codewords are generated by applying errors drawn ran- domly from the channel (N BF q ) N , using the shared secret key k as the source of randomness. That is, the errors X or I on each qubit are drawn from the product distribution p E N (e N ), where p E (e) is given by p E (X) =q; p E (I) = 1q; (3.3) where q = p=(1 2p). Since the set of errors is selected using the shared secret key k, Bob knows which codeword corresponds to each message m. 38 The errors given by Eq. (3.2) are typical errors associated with the channel (N BF q ) N [29, 30]. By the asymptotic equipartition theorem [59], for large enough N, it is highly likely that each of these codewords that Alice generates is a typical sequence with a sample entropy close toH(E) =(1q) log(1q)q logq =h(q). Furthermore, it follows from a simple calculation thatN p N q =N p+p if we set q =p=(1 2p). This will become important later when we discuss the secrecy of this protocol. Alice then sends her state through the channel (N BF p ) N . We are now essentially in the scenario of classical random coding over a classical bit- ip channel with parameter p. By the asymptotic equipartition theorem for conditionally typical sequences [59], for each input sequence (i.e., errorE N (k;m) applied to the encoded covertext) there is a corresponding conditionally typical set of errorsfF N (k;m)g which has the following properties: its total probability is close to 1, its size is 2 nH(FjE) , and the probability of each conditionally typical error given knowledge of the input error E N (k;m) is 2 nH(FjE) . With high probability, the error F N Bob observes will be a typical error of the channel (N BF p+p ) N . We know from Shannon's noisy channel coding theorem that if Alice and Bob set the number of messagesjMj = 2 NR such that 2 nR 2 NH(F ) 2 NH(FjE) = 2 N(H(F )H(FjE)) ; (3.4) then Bob is able to decode correctly with high probability [19, 36, 59] which error E N (k;m) was applied by Alice, as long as the code is nondegenerate. For our 39 protocol, it is straightforward to calculate that H(F ) =h(p +q 2pq) =h(p +p) for q =p=(1 2p), and H(FjE) =h(p). Hence Alice can communicate M = logjMjN(h(p +p)h(p)) (3.5) bits of information to Bob steganographically. Moreover, this protocol does not arouse suspicion from Eve. We say that this protocol is secret, because the state passing through the channel is to good approx- imation the state Eve would expect to see. To see this, note that X k2K X m2M p k (N BF p ) N (E N (k;m)V c V y E N (k;m)) = (N BF p ) N ( X k2K X m2M p k E N (k;m)V c V y E N (k;m)) (N BF p N BF p=(12p) ) N (V c V y ) = (N BF p+p ) N (V c V y ); (3.6) where V is the isometry corresponding to the QECC Alice and Bob are using. The rst equality follows from linearity of quantum operations. The approximate equality follows from the fact that when we average the transmitted codeword over the key and all possible messages, we are applying all the typical errors of the channel (N BF q ) N with their correct probabilities, and hence to good approximation [29, 30] we are simply applying the full channel. The nal equality follows from calculating the composition of these quantum operations. This is exactly the state Eve expects to observe, hence our steganographic protocol is secret to an arbitrarily good approximation. 40 As described above, this protocol allows Alice to transmit a classical message m secretly to Bob. But in fact, by making this protocol coherent Alice can equally well transmit a quantum state|that is, a superposition of possible messages m. So we see that this protocol can transmit either classical or quantum information at the same rate h(p +p)h(p). The one signicant dierence between these two cases is that if Eve actually carries out a measurement of the error on the transmitted state, this would destroy the superpositions of a quantum message, but not aect the ability to transmit classical messages. So this protocol works for secret quantum communication if it is assumed that Eve only sometimes checks the code blocks transmitted from Alice to Bob. As we did in the case of steganographic communication over a noiseless channel [54], we can show this by considering a protocol in which Alice sends a subsystemM to Bob which is maximally entangled with a reference subsystem R (see Figure 2.1). 3.2.2 Depolarizing Channel Suppose that Eve believes the quantum channel shared between Alice and Bob is a depolarizing channel with error rate p +p, i.e., N DC p+p () = (1 (p +p)) + p +p 3 (XX +YY +ZZ); (3.7) where the actual physical channel between Alice and Bob isN DC p . The protocol for Alice and Bob to communicate steganographically in this scenario is nearly identical to the protocol described in the previous subsection. First, Alice encodes 41 an innocent state, i.e,. the covertext c , into a nondegenerate quantum error- correcting code (QECC) on N qubits. This code should be able to correct typical errors induced by the channel (N DC p+p ) N . Next, depending on the secret keyk2K and the message m2M that she would like to send, she applies the error G N (k;m) =G 1 (k;m) ::: G N (k;m) (3.8) to her state. This produces the codeword corresponding to the message m. If her message to Bob is a quantum state, she can prepare the system in a superposition of these codewords. These codewords are generated by applying errors drawn ran- domly from the channel (N DC q ) N , using the shared secret key k as the source of randomness. That is, the errors X, Y , Z or I on each qubit are drawn from the product distribution p G N (g N ), where p G (g) is given by p G (X) =p G (Y ) =p G (Z) =q=3; p G (I) = 1q; (3.9) and q =p=(1 4p=3). Since the errors are selected using the shared secret key k, Bob knows what codeword corresponds to each message m. The errors given by Eq. (3.8) are typical errors associated with the channel (N DC q ) N [29, 30]. By the asymptotic equipartition theorem [59], for large enough N, it is highly likely that each of these codewords that Alice generates is a typical sequence with a sample entropy close to H(G) =(1q) log(1q)q log(q=3) s(q), where s(q) is the entropy of a qubit passed through the depolarizing channel with error parameter q. Furthermore, it follows from a simple calculation that 42 N p N q =N p+p if we set q = p=(1 4p=3), which is important for secrecy as discussed in the previous subsection. Alice then sends her state through the channel (N DC p ) N . Following the same random coding argument described for the bit- ip channel, with high probability, the error J N Bob observes will be a typical error of the channel (N DC p+p ) N . We know from Shannon's noisy channel coding theorem that if Alice and Bob set the number of messagesjMj = 2 NR such that 2 NR 2 NH(J) 2 NH(JjG) = 2 N(H(J)H(JjG)) ; (3.10) then Bob is able to decode correctly with high probability [19, 36, 59] which error G N (k;m) was applied by Alice, as long as the code is nondegenerate. For our protocol, it is straightforward to calculate thatH(J) =s(p+q4qp=3) =s(p+p) for q =p=(1 4p=3), and H(JjG) =s(p). Hence Alice can communicate M = logjMjN(s(p +p)s(p)) (3.11) classical or quantum bits of information to Bob steganographically. The proof of secrecy of this protocol is nearly identical to the one given in Eq. (3.6). Note that the assumption of a nondegenerate code is quite natural in the case of the bit- ip channel, which is essentially classical; but not as much so for the depolarizing channel, where the errors do not commute. We believe that this general procedure for encoding will work for degenerate codes as well, but the achievable rate may be lower, and will require an analysis specic to the code in question. We will return to this point at the end of the paper, where we conjecture a general 43 formula for the steganographic rate of a quantum channel using general quantum codes. 3.2.3 Secret key consumption For the above encodings, how much secret key must be consumed? In general, we assume that the details of the encoding|that is, how each messagem is mapped to a codeword for a particular key element k|have been decided between Alice and Bob ahead of time. So in the protocol as described above, the only place where secret key is consumed is in picking the subsets of errors used in the encoding. Before the protocol begins, Alice and Bob divide the set of typical errors of the channel (N BF p=(12p) ) N into n nonoverlapping subsets of sizejMj = 2 N(h(p+p)h(p)) each, where n = 2 Nh(p=(12p)) 2 N(h(p+p)h(p)) : (3.12) For each transmitted block, Alice and Bob must randomly choose one of these n subsets to encode her messages. This requires a number of bits K of secret key, K = log 2 n =N(h(p=(1 2p))h(p +p) +h(p)); (3.13) which is positive for p +p < 0:5. Therefore the key consumption scales linearly with N. Notice in the limit where the physical channel is noiseless i.e., p = 0, we have thatK = 0, which agrees with our result in [54] where it was shown that only a sublinear amount of key is needed for encoding across noiseless channels. As discussed in [54], using this amount K of shared secret is key is sucient to make the steganographic protocol secret, but not necessarily secure. That is, Eve should not become suspicious if she observes the state passing through the channel. 44 However, if for some reason she knew a message was being sent, she would be able to deduce signicant information about the message. This can be prevented by rst encrypting the message before doing the stegano- graphic encoding. Encryption requiresM bits of secret key in the case of anM-bit classical message (using a one-time pad), or 2M bits of secret key in the case of anM-qubit quantum message (by twirling). In this case, the protcol is both secret and secure. However, now the rate of key consumption would be increased by R (classical) or 2R (quantum, where R is the steganographic rate. 3.3 Secrecy, Reliability, and Bounds 3.3.1 The information processing task Here we consider the steganographic scenario outlined above where Alice is using Eve's ignorance of the actual noise rate of the physical channel to hide her mes- sage. We will consider the task known as entanglement transmission, because this corresponds to the strongest denition of quantum capacity [59]. The information processing task we are considering is visualized in Figure 3.1. Alice has a secret message of M = log 2 jA 1 j qubits, which is maximally entangled with a reference system R. She also prepares an innocent covertext c , which will be encoded into the N-qubit quantum error-correcting code. Let us rst dene her encoded state, dependent on the secret key element k: ! k;A 0 n R E k;A 1 C!A 0 n( c A 1 R ): (3.14) 45 Figure 3.1: The information processing task we consider, of Alice sending M stego qubits to Bob over a quantum channelN p (which is either the quantum bit ip channel or depolarizing channel), which Eve believes to be noisier. Eve's ignorance of the channel is characerized by the parameter p. Dependent on the secret key k, Alice encodes her message subsystem M and an innocent covertext c into a suitable quantum error-correcting code in such a way that once passed through the physical channel, it looks as though typical errors of the channelN p+p have been applied. Bob then decodes the message and covertext using his copy of the shared secret keyk. Alice's message is entangled with a reference systemR. The ability to transmit entanglement implies the ability to do general quantum communication. 46 The dependence of the encoding on the secret key corresponds to choosing among the dierent sets of typical errors of the channelN q in the protocols from the previous section. To someone (like Eve) who does not know the secret key k, the state is eectively X k p k N N p (! k;A 0 n R ) =N N p (! A 0 n R ); (3.15) where we have used linearity of quantum operations and ! A 0 n R P k p k ! k;A 0 n R is the state averaged over all possible values of the secret key k with probabilities p k . (We can choose this probability to be uniform for simplicity, if we so desire.) What is a good way to guarentee secrecy from Eve? We propose the following secrecy condition: 1 2 jjN N p (Tr R [! A 0 n R ])N N p+p (V c V y )jj; (3.16) whereN p+p is what Eve believes the physical channel to be, V is an isometry representing the encoding of the covertext into a suitably chosen codeword (one which can correct typical errors induced by the channelN p+p ) and > 0 is some small parameter. This condition means that if Eve observes the quantum state, it will be eectively indistinguishable from an encoded covertext being sent through the noisy quantum channelN p+p . We introduce another requirement which corresponds to a notion of recoverabil- ity. Once Bob receives the state, he applies his decoderD k;A 0 n !B 1 C to obtain the original state c B 1 R . We can relax this by only requiring that the input states and output states are close, that is: 1 2 jjD k;A 0 n !B 1 C (N N p I R (! k;A 0 n R )) c B 1 R jj 1 (3.17) 47 for all k, where > 0 is a small parameter. 3.3.2 Upper bound on steganographic rate With these two assumptions of secrecy and recoverability, we can now put a bound on the number of qubits M that can be sent reliably and steganographically from Alice to Bob. Dening E =N N p+p (V c V y ) and applying the Fannes-Audeneart inequality [5] to the secrecy condition in Eq. (3.16) we have: H N N p (Tr R [! A 0 n R ]) H( E ) +g(N;) (3.18) whereg(N;)N +h 2 (), andh 2 () is the binary entropy function. Furthermore, we have M = logjA 1 j =I(RiB 1 ) B 1 R I(RiB 1 ) D k (N N p (! k )) +N + (1 +)h 2 (=[1 +]) I(RiA 0 n ) N N p (! k ) +f(N;) =H N N p (Tr R [! k;A 0 n R ]) H N N p I R (! k;A 0 n R ) +f(N;); (3.19) where f(N;)N + (1 +)h 2 (=[1 +]). The rst equality follows from the fact that the coherent information of a maximally entangled state is just the logarithm of the dimension of one of the subsystems. The rst inequality follows from the Alicki-Fannes-Audeneart inequality [1] applied to the recoverability condition given in Eq. (3.17). The second inequality is a quantum data processing inequality [59]. The last equality follows from the denition of the coherent information. 48 Furthermore, using the concavity of the von Neumann entropy and linearity of quantum operations we have that min k2K H N N p (Tr R [! k;A 0 n R ]) X k p k H N N p (Tr R [! k;A 0 n R ]) H N N p (Tr R [ X k p k ! k;A 0 n R ]) =H N N p (Tr R [! A 0 n R ]) ; (3.20) and for many cases we expect H(N N p (Tr R [! k;A 0 n R ])) to be roughly the same for every k (see Sec. 3.3.3 and 3.3.3). Thus H N N p (Tr R [! k;A 0 n R ]) H N N p (Tr R [! A 0 n R ]) (3.21) for allk. Now putting Eq. (3.18), (3.19), and (3.21) together we arrive at our main result for this section, which states that Alice can secretly and reliably send M stego qubits to Bob, where M is bounded above by MH( E ) H (N N p I R )(! k;A 0 n R ) +g(N;) +f(N;): (3.22) Thus, if we can compute a maximum for H( E ) = H(N N p+p ()) where is pure (because V is an isometric encoding and c is pure), and also compute a lower bound for H (N N p I R )(! k;A 0 n R ) (or compute it explictly, recalling that ! k;A 0 n R 49 is a pure state), then we have a tight upper bound on the number of qubitsM that can be sent steganographically over a noisy quantum channelN p . 3.3.3 Upper bounds for specic channels We will now apply our result given in Eq. (3.22) to the channels discussed in the previous section, where we make the implicit assumption that Alice is using a nondegenerate code. Though our result given by Eq. (3.22) is true in general, for a degenerate code the number of distinct error syndromes is smaller (depending on the code), and the bounds discussed here and achievable rates discussed in the previous section would be adjusted. The bit ip channel For the bit ip channel with parameter p +p given by Eq. (3.1), the maximum of H((N BF p+p ) N ()) over all N-qubit pure states is Nh(p +p) where h(p +p) = (p +p) log(p +p) (1 (p +p)) log(1 (p +p)) is the entropy of a single qubit sent through this bit ip channel. To prove this, consider some pure state =j ih j. Then (N BF p+p ) N (j ih j) = X s p(s)X s j ih jX s (3.23) where we are summing over all binary strings s of length N; X s is the operator acting on N qubits with an X acting at every location where s has a 1 and an I where s has a 0. The probability p(s) is given by p(s) =p w(s) (1p) Nw(s) ; (3.24) 50 where w(s) is the weight of string s. The Shannon entropy of this distribution is Nh(p +p) since it is a binomial distribution. The von Neumann entropy is the minimum Shannon entropy over all possible ensemble decompositions of the given state, and it is not hard to check that it is achieved whenj i is a Z eigenstate. Thus we have computed a maximum for the rst term on the right hand side of Eq. (3.22). Now we compute a lower bound for the second term on the right hand side of Eq. (3.22) in the case of the bit ip channel, i.e.,H((N BF p ) N I R )(! k;A 0 n R )). First note that we can write ((N BF p ) N I R )(! k;A 0 n R ) = X i2T (E i I R )! k;A 0 n R (E y i I R ) + X i= 2T (E i I R )! k;A 0 n R (E y i I R ); (3.25) where the index is i =i 1 i 2 :::i N , the errors E i are given by E i =A i 1 ::: A i N ; (3.26) andT is the set of typical sequences i corresponding to the typical errors of the channel (N BF p ) N [29, 30]. Recall we are making the assumption that the QECC Alice is using to correct the typical errors of the channel (N BF p+p ) N is nondegenerate. Because of this, we can infer that for each k, her encoded state ! k;A 0 n R forms a nondegenerate code for the channel (N BF p ) N . This follows from the discussion in Section 3.2.1. This means that on a valid codeword in the QECC, for i2T the typical errors E i all 51 have distinct error syndromes, and act as unitaries that move the code space to a distinct, orthogonal subspace labeled by i. So an error E i occurs with a xed probability p i for all valid codewords of the QECC. Recall also that since these errors are typical, they have almost all the probability, i.e. X i2T p i = 1 (3.27) for arbitrarily small > 0 (in the limit of large N). From Eq. (3.25) we have that H ((N BF p ) N I R )(! k;A 0 n R ) (1)H 1 1 X i2T (E i I R )! k;A 0 n R (E y i I R ) +H 1 X i= 2T (E i I R )! k;A 0 n R (E y i I R ) (1)H 1 1 X i2T (E i I R )! k;A 0 n R (E y i I R ) (3.28) where the rst inequality follows from the concavity of the von Neumann entropy and Eq. (3.27) , and the second inequality follows because the term proportional to is positive. Continuing, we have (1)H 1 1 X i2T (E i I R )! k;A 0 n R (E y i I R ) = X i2T p i log p i 1 = X i2T p i log(1)p i logp i = (1) log(1) X i2T p i logp i Nh(p)O(): (3.29) 52 The rst equality follows from the denition of the von Neumann entropy and Eq. (3.25). The inequality follows from performing a Taylor expansion on log(1) and from the fact that for the bit ip channel (N BF p ) N : X i2T p i logp i X i2T 2 Nh(p) log 2 Nh(p) =Nh(p) X i2T 2 Nh(p) Nh(p)2 Nh(p) 2 Nh(p) =Nh(p); (3.30) where the approximate equalities follow directly from the theory of typical se- quences. Thus H ((N BF p ) N I R )(! k;A 0 n R ) Nh(p)O(); (3.31) and combining this with the upper bound we computed for the rst term in Eq. (3.22) gives MN(h(p +p)h(p)) (3.32) for a suciently reliable and secret protocol and large enough block size N. Com- paring this to Eq. (3.5), we have that the encoding described in the previous section for steganography over the channelN BF p where Eve expects the channel to beN BF p+p is essentially optimal. The depolarizing channel Unfortunately, for the depolarizing channelN DC we do not know whatN-qubit pure state maximizesH((N DC p+p ) N ()). However, we can still bound this quantity, i.e., 53 give an upper bound on the rst term in Eq. (3.22). Consider the action of this channel on an N qubit pure state as follows: (N DC p+p ) N () X j E j E y j (3.33) wherefE j g is the set of typical errors associated withN applications of the channel (N DC p+p ) N . Recall that we are choosing our isometric encoding to correct for typ- ical errors of the channel Eve beleives to be connecting Alice and Bob, i.e. N DC p+p . Furthermore, we're assuming that our code is nondegenerate. Therefore the states E j E y j are all orthogonal to each other for in the codespace, and Tr[E j E y j ] =p j where p j are the typical probabilities associated with the errors E j . The von Neu- mann entropy is the Shannon entropy minimized over all possible decompositions, so the entropy of this state is clearly H( E ) =H((N DC p+p ) N (V c V y )) X j p j log(p j ) = X j 2 Ns(p) log 2 Ns(p) Ns(p)2 Ns(p) 2 Ns(p) =Ns(p) (3.34) In a similar argument to the one given in Sec. 3.3.3 , we can give a lower bound for the second term on the right hand side of Eq. (3.22) in the case of the depolarizing channel. We get that H(((N DC p ) N I R )(! k;A 0 n R ))Ns(p)O(); (3.35) 54 where becomes arbitrarily small for large N. Combining this with the upper bound given in Eq. (3.34) we have that MN(s(p +p)s(p)) (3.36) for a suciently reliable and secret protocol and large enough block size N. Com- paring this to Eq. (3.11) we have that the encoding described in the previous section for steganography over the channelN DC p where Eve expects the channel to beN DC p+p is essentially optimal, at least when we restrict ourselves to nondegenerate codes. 3.4 Conclusions and Future Work Quantum steganography is the study of secret quantum communication between two parties, Alice and Bob. We have shown that Alice and Bob are able to communicate with each other secretly at a nonzero rate over a bit- ip or a depolarizing channelN p using a shared secret key, without arousing suspicion from a potential eavesdropper Eve, so long as Eve believes the channel to be noisier than it really is. Eve can be made to believe this through Alice and Bob systematically adding extra noise to the channel prior to secret communication. In this paper we gave explicit bounds on the number of stego qubits that Alice can send to Bob by hiding her secret message in the syndromes of a nondegenerate error-correcting code designed to correct the typical errors of the channel Eve believes: N p+p . We also gave explicit encodings that achieve these bounds. Interesting future work should include a generalization of these results to steganog- raphy over general quantum channelsN . It is possible that in order to achieve the 55 maximum possible rates in this scenario that degenerate codes are needed. For example, it is likely that the steganographic capacity we calculated for the depolar- izing channel in this paper could be improved in this way. It is also possible that coding across multiple codeblocks using degenerate quantum codes could increase the steganographic capacity. If the actual physical channel shared between Alice and Bob isN , and the channel Eve believes isM, then what is the quantum steganographic capacity? In this paper we proved that for the bit- ip channel, the rate is the dierence of quantum capacities, i.e.,Q(N BF p+p )Q(N BF p ) =N(h(p+p)h(p)). Also, allowing for our restriction to nondegenerate codes, this is true for the depolarizing channel as well. We conjecture that one might be able to prove that the steganographic rate in general will be Q(M)Q(N ). This will require proof methods that go beyond those of the current paper, but we believe that this wll be an area of fruitful future study. 56 Chapter 4 Non-Markovianity of the Post Markovian Master Equation 4.1 Introduction This chapter is based on joint work done with Prof. Brun and Prof. Lidar [56]. The study of open quantum systems leads to a theoretically rich and experimentally useful theory [12, 2]. It allows us to make concrete predictions about a quan- tum system of interest that is interacting with its environment. One of the most commonly used equations to model open quantum system dynamics is the Gorini- Kossakowski-Sudarshan-Lindblad (GKSL) master equation [26, 33], mainly due to its easily solvable nature and the fact that it satises the condition of complete positivity. One of the main assumptions that goes into its derivation, and that makes it easily solvable is the assumption of Markovianity or being `memoryless'. This means that although the quantum system is interacting with a bath, no infor- mation about past states of the system ows back from the bath; the bath `forgets' about earlier states of the system in a very short time. That is, the evolution of a 57 quantum system at time t depends only on its density matrix (t) and not on its state (t 0 ) at earlier times t 0 <t. Unfortunately, although the assumption of Markovianity allows for a pleasing simplication, the Lindblad master equation is only an approximation, and non- Markovian eects are often too important to neglect. At the other extreme, the formally exact Nakajima-Zwanzig master equation [60] is too hard to solve. Hence compromises leading to master equations that are both easily solvable and account for non-Markovian eects are desirable. This is particularly true in light of re- cent developments in quantum computation and quantum annealing, where non- Markovian eects often play an important role [53]. There has already been much work on this problem, e.g., Gaussian [24], quantum collisional models [14], and time- convolutionless master equations [52]. Here we focus on the post-Markovian master equation (PMME) [45]. The PMME was derived via an interpolation between the generalized measurement interpretation of the exact Kraus operator sum map [31] and the continuous measurement interpretation of Markovian-limit dynamics [9]. Previous work implied that the PMME was essentially Markovian [34]. This claim was subsequently countered in [15]. Our goal in this work is to revisit the question of the (non-)Markovianity of the PMME. We conrm that the PMME can describe non-Markovian dynamics, and provide a simple example to illustrate this. The structure of this paper is as follows. In Section 4.2 we outline the deni- tions and measures of non-Markovianity that we use here to study the PMME. We describe what it means for quantum dynamics to be non-CP-divisible and how that relates to quantum non-Markovianity, and explain why an increase of distinguishability between two distinct initial quantum states is a witness of non- Markovianity [10]. We also brie y explain how the PMME is derived. In Section 58 4.3, we brie y summarize the reasoning behind the work which stated that the PMME is essentially Markovian [34], and the more recent work oering evidence to the contrary [15]. Then, in Section 4.4, we describe the simple physical scenario of a qubit dephasing with a bath and show how the PMME accounts for non-Markovian eects that are not captured by the Lindblad equation. Finally, in Section 4.5 we give a mathematical condition on the memory kernel used in the PMME which guarantees non-CP-divisible dynamics. Supporting information for Section 4.3 is given in the Appendix. 4.2 Quantum non-Markovianity and the Post Marko- vian master equation 4.2.1 Quantum non-Markovianity The problem of quantifying and describing quantum non-Markovianity has been the subject of deep study in recent years. Several key measures, witnesses, and denitions of quantum non-Markovianity are now well accepted [42, 17, 57]. Here we give a brief summary of the approach contained in [11]. Quite generally, quantum dynamics is described by quantum dynamical maps: t = t ( 0 ); (4.1) where t is a completely-positive trace-preserving (CPTP), time dependent map with 0 =I (for a more general class see [20, 21]). The Markovian quantum master 59 equation is a special case of this, where the quantum dynamical map (superoperator) t can be written as t =T exp h Z t 0 dL i ; (4.2) where the T denotes time ordering andL is the GKSL generator [26, 33], L t =i[H;] + X k k L k L y k 1 2 fL y k L k ;g : (4.3) Therefore, Eq. (4.1) becomes _ =L t : (4.4) For this process to represent a completely positive one-parameter semigroup, the coecients must satisfy k 0. This is known as the GKSL theorem. ∗ An interesting class of dynamical maps t are those for which the inverse process 1 t exists. Then for t 2 t 1 0, one can dene a two-parameter family of maps given by t 2 ;t 1 = t 2 1 t 1 ; (4.5) such that t 2 ;0 = t 2 and t 2 ;0 = t 2 ;t 1 t 1 ;0 : (4.6) ∗ This is actually a pair of theorems that were discovered independently and nearly simultane- ously, for the nite dimensional case by Gorini, Kossakowski, and Sudarshan [26], and the innite dimensional case by Lindblad [33]; for a detailed account of this history see [18]. 60 In this case one can always write down a time-local quantum master equation with the following form: _ =K t =i[H(t);] + X k k (t) L k (t)L y k (t) 1 2 fL y k (t)L k (t);g : (4.7) Note the explicit time dependence in the Hamiltonian, Lindblad operators, and the rates. The k (t) coecients in Eq. (4.7) need not be positive. The two-parameter family of quantum dynamical maps t 2 ;t 1 is said to be P- divisible or CP-divisible if t 2 ;t 1 is positive or completely positive, respectively, for all t 2 > t 1 . It turns out that the master equation (4.7) leads to CP-divisible dynamics if and only if all rates are positive for all times, that is k (t) 0 which follows from a straightforward extension of the GKSL theorem [16]. The notions of P- and CP-divisibility are intimately related to the notion of quantum non-Markovianity. For our purposes, the relationship between CP-divisibility and quantum non-Markovianity that was rst suggested in [41] will suce (for a more detailed description of the relationship between the two see the reviews [42, 11]). Essentially, the condition (4.6) is the quantum counterpart of the classical Chapman-Kolmogorov equation, and one can make the relationship between clas- sical Markovianity and quantum Markovianity quite precise. One important way to detect quantum non-Markovianity in an open quantum system is to measure how the distinguishability of quantum states changes over time. For a non-Markovian process, quantum states should at some times become more distinguishable due to a reverse ow of information from the environment to 61 the open system [10]. Recall that the trace-norm distance between two quantum states 1 and 2 is given by D( 1 ; 2 ) = 1 2 Trj 1 2 j; (4.8) wherejAj = p A y A, and is contractive for any positive and trace-preserving map [44] (in particular for any CPTP map), i.e., D(( 1 ); ( 2 ))D( 1 ; 2 ): (4.9) Suppose that Alice prepares a quantum system in either the state 1 or 2 . She then hands the system to Bob, and Bob measures the system and decides whether the system was in the state 1 or 2 . The probability that Bob can successfully identify the state of the system is given by 1 2 (1 +D( 1 ; 2 )). Thus we can interpret the trace-norm distance between two quantum states as a measure of distinguishability between the two. Because we are interested in the change of distinguishability over time, the following quantity is of particular interest: (t; 1;2 (0)) = d dt D( 1 (t); 2 (t)); (4.10) where 1;2 (0) denotes the initial states. Following [10], we will say a process is Markovian if for all pairs of initial states (t; 1;2 (0)) 0 for all times. Therefore, we will say a process is non-Markovian if there exists any pair of initial states 1;2 (0) and a time t for which (t; 1;2 (0)) > 0. However, there is some ambiguity as to whether the process is necessarily Markovian when (t; 1;2 (0)) 0. For instance, an example exists, which we review in the Appendix, where (t; 1;2 (0)) 0 for 62 all t and 1;2 (0), but the quantum dynamical map associated with the evolution is non-divisible [34]. We note that very recent work [3] uses an updated witness of information back- ow instead of Eq. (4.10). It is calculated by considering the trace of the so-called Helmstrom matrix (essentially a weighted average between two dierent dynami- cally evolved initial states), which was shown to also admit an information back ow interpretation. However, for our purposes, the measure given by Eq. (4.10) suces. 4.2.2 The post-Markovian master equation Here we give a brief review of the quantum master equation that is the focus of this paper. Recall that for a quantum system S coupled to a bath B evolving unitarily under the total system-bath Hamiltonian H SB , the dynamics of the quantum sys- tem have a measurement picture interpretation [45]. Essentially, the exact system dynamics (t) = Tr B [U(t) SB (0)U y (t)] (4.11) can be derived by performing a single projective measurement on identical en- sembles initially prepared in the state SB (0). In the Markovian case where the quantum dynamics can be written in the form (4.3) with k 08k, there again exists a measurement interpretation. In the Markovian case the bath functions as a probe coupled to the system while being subjected to a continuous series of mea- surements at an innitesimal time interval . This is related to the well known quantum jump process [39]. The PMME interpolates between these two measurement pictures. In the sin- gle measurement picture, exact dynamics are seen as an evolution of the coupled 63 system-bath followed by a single generalized measurement at time t. The Marko- vian dynamics are represented as a series of measurements interrupting the joint evolution after each short time interval . The idea is that by relaxing the many- measurements process one is led to a less restricted approximation than the Marko- vian one. We skip to the conclusion of the derivation contained in [45] and give the nal form of the PMME: d dt =L Z t 0 k(t 0 ) exp(Lt 0 )(tt 0 )dt 0 ; (4.12) where k(t) is the memory kernel andL is the Markovian generator. In the deriva- tion, the memory kernel k(t) is introduced phenomenologically to assign weights to dierent measurements performed on the bath. While k(t) is left unspecied, it can in principle be determined by an appropriate quantum state tomography experiment. Another important feature of the PMME is the dynamical map t :(0)!(t) that governs it. The quantum map corresponding to Eq. (4.12) is (t) :X! X i i (t)Tr[L i X]R i (4.13) where the left and right eigenoperatorsfL i g andfR i g of the generatorL are known as the damping basis [13] ofL, and i (t) = Lap 1 [ 1 s i ~ k(s i ) ]; (4.14) 64 where i are the eigenvalues from solvingL = and the tilde over the kernel rep- resents its Laplace transform. The following condition ensures complete positivity of this dynamical map [45]: X k k (t)L T k R k 0; (4.15) which results in a condition on the memory kernel k(t). Also, we can expand (t) in the damping basis as (t) = X i i (t) i R i ; (4.16) where thef i g can be obtained by expanding(0) in the basisfR i g. It was shown in the original derivation that for a qubit dephasing with a bath, the solution of this equation indeed interpolates between the exact and Markovian solutions [45]. 4.3 Previous Examples of the non-Markovianity of the PMME The non-Markovianity of the PMME and memory kernel master equations more generally were studied by Mazzola et al. [34]. They detailed a specic example (re-derived in the Appendix) where the PMME leads to non-divisible quantum dynamics, yet has zero measure for non-Markovianity under Eq. (4.10). They thus included these nondivisible processes which have unidirectional information ow into the class of time-dependent Markovian processes. This, of course, does not rule out the possibility of the PMME including non-Markovian eects, given that this analysis was done for a specic example. In fact, in later work by Budini [15] 65 it was shown that for a similar example to the one in Ref. [34], if the system Hamiltonian does not commute with the dissipative term in Eq. (4.3) then there indeed is back ow of information, and thus the PMME includes non-Markovian eects. More concretely, there is indeed information back ow in the PMME for the generator L t () =i 2 [ x ;] +C() =i 2 [ x ;] + 2 ([ ; + ] + [ ; + ]); (4.17) with exponential memory kernels. The analysis in Ref. [15] is particularly useful because it also gives rst-principles derivations for the memory kernels used. Also, in Ref. [14] it was shown that for so-called collisional models, i.e., scenarios where the dissipative termC can be written as C() = X V V y I; (4.18) that using an approximate version of the PMME can lead to non-Markovian eects in the case of qubit dephasing. These two examples give a clear indication for non-Markovian eects present in the PMME. We provide an additional perspective in the next section, where we study qubit dephasing with the PMME given by Eq. (4.12), for two choices of memory kernels. We show that the solutions obtained include non-Markovian eects by analyzing both nondivisibility and information back ow. 66 Figure 4.1: The non-Markovianity measure given by Eq. (4.27) corresponding to the two dierent choices of kernelsk 1 (t) (top) andk 2 (t) (bottom). Onlyf 1;2 (t) d dt f 1;2 (t) is plotted since it characterizes the behaviour of 1;2 and we can ignore discontinuities induced by the denominator. Note the non-Markovianity regions shown by the shaded regions in both plots. Both solutions exhibit damped oscillations, and hence an innite number of non-Markovianity regions, however the plot parameters are chosen so that only one such region is displayed. For 1 , the zeroes are given by r n = 2 ! (n arctan[!=(a + )]) and s n = n ! where ! = p 4aA (a + ) 2 . For 2 the zeroes are given byt n =n= where = p 2 +Aa. The parametersA;a, and are chosen so that ! is real or else no non-Markovian eects would be present. The values used for these plots are A = 6;a = 1; = 1:1; and =. 67 4.4 Non-Markovianity of the PMME via qubit dephasing Borrowing the example from Ref. [45], let us consider the problem of single-qubit dephasing. The GKSL generator is L = a 2 [ z ; [ z ;]]; (4.19) wherea> 0. Using the damping basis method [13], we have the following eigenval- ues and left and right eigenoperators for the generatorL: f i g 3 i =f0;a;a; 0g; (4.20) fR i g 3 i=0 =fL i g 3 i=0 = 1 p 2 fI; x ; y ; z g: (4.21) It follows immediately from Eq. (4.14) that 0 (t) = z (t) = Lap 1 [1=s] = 1 and x (t) = y (t) f(t) where f(t) can be given explicitly once we have chosen a kernel. From Eq. (4.16) we have (t) = 1 2 I +f(t) x (0) x +f(t) y (0) y + z (0) z : (4.22) To proceed we must make a choice for the kernel function. Consider the following simple memory kernels [45]: k 1 (t) =Ae t ; (4.23) k 2 (t) =Ae ( a)t [cos(t) sin(t)]: (4.24) 68 The rather specic form ofk 2 (t) is because the associated solution is known to lead to damped oscillatory dynamics with a non-zero asymptotic coherence, which is a feature of the exact solution for a single qubit dephasing in the presence of a boson bath. We are now in a position to analyze these solutions with the non-Markovianity measure given by Eq. (4.10). To do so, we rst expand two evolved density matrices u (t) and v (t) as in Eq. (4.22) and take their dierence to obtain the matrix A = u v = 0 @ (u;v) z ( (u;v) x i (u;v) y )f(t) ( (u;v) x +i (u;v) y )f(t) (u;v) z 1 A (4.25) where we use (u;v) i = (u) i (0) (v) i (0),i =x;y;z to denote the dierence of Bloch vector coecients between u (0) and v (0). Computing the trace of 1 2 j u v j = 1 2 p A y A = 1 2 p A 2 gives 1 2 Trj u v j = q f(t) 2 [( (u;v) x ) 2 + ( (u;v) y ) 2 ] + ( (u;v) z ) 2 : (4.26) Finally, by taking the time derivative derivative of this quantity, we arrive at 1;2 (t; u;v (0)) = ( 2 x + 2 y )f 1;2 (t) d dt f 1;2 (t) q ( 2 x + 2 y )(f 1;2 (t)) 2 + 2 z ; (4.27) 69 where f 1 (t) =e t(a+ )=2 [cos(!t) + sin(!t)(a + )=2!]; (4.28) f 2 (t) = 1 Aa 2 + 2 [1e t (cos t + sin t)]; (4.29) ! = p 4aA ( +a) 2 =2, = p 2 +Aa, and (u;v) i = i to simplify notation. Note that the condition for complete positivity [Eq. (4.15)] results injf 1;2 (t)j 1 [45], which imposes restrictions on the allowed values of the various parameters appearing here, but we are mainly interested in the damped oscillatory nature of the solutions. We choose the various parameters such that! is real, which ensures that f 1 (t) is oscillatory. To demonstrate non-Markovianity from information back ow from the bath to the system it is sucient to show that f 1;2 (t) d dt f 1;2 (t) can become positive. This is illustrated in Fig. 4.1. We can go further by explicitly checking whether the dynamical map t associ- ated with this evolution is CP-divisible or not. Upon inspection of Eq. (4.22), we see that for any t 1 > 0, t 1 must have the following form: (t 1 ) = t 1 ((0)) =a 1 (0) +b 1 z (0) z (4.30) where a 1 > 0 and b 1 > 0 satisfy a 1 +b 1 = 1. By comparing the Bloch vector expansions of both (0) and (t) we can solve for the coecients a 1 ;b 1 : a 1 = 1 +f(t 1 ) 2 ; b 1 = 1f(t 1 ) 2 : (4.31) 70 Now consider the map t 2 ;t 1 for t 2 >t 1 . This map acting on (t 1 ) yields (t 2 ) =a 2 (t 1 ) +b 2 z (t 1 ) z = (a 2 a 1 +b 2 b 1 )(0) + (a 2 b 1 +b 2 a 1 ) z (0) z : (4.32) Therefore a 2 a 1 +b 2 b 1 = 1 +f(t 2 ) 2 ; a 2 b 1 +b 2 a 1 = 1f(t 2 ) 2 ; (4.33) and by plugging this solution into Eq. (4.31) and solving for a 2 ;b 2 we arrive at a 2 = 1 2 1 + f(t 2 ) f(t 1 ) ; b 2 = 1 2 1 f(t 2 ) f(t 1 ) : (4.34) Now consider the case wheret 2 =t 1 +h whereh> 0 is small. Then we can rewrite f(t 2 ) =f(t 1 +h)f(t 1 ) + d dt f(t 1 )h, and Eq. (4.34) becomes a 2 = 1 + 1 2 d dt f(t 1 ) f(t 1 ) h; b 2 = 1 2 d dt f(t 1 ) f(t 1 ) h: (4.35) From this we see that if d dt [f(t 1 )]=f(t 1 ) > 0, then t 2 ;t 1 is not a valid Kraus map. Indeed, we can infer from Fig. 4.1 that there exist time intervals for which this inequality holds for both choices of kernel. Hence we do not have CP-divisible dynamics, as expected from our analysis of 1;2 . Furthermore, we can write down a quantum master equation of the form given by Eq. (4.7) for these dynamics generated by the PMME. We follow the method 71 given in Ref. [37] for how to do this in general when the solution to the system evolution is known. From Eq. (4.22), we see that _ (t) = 1 2 d dt f(t)( x (0) x + y (0) y ): (4.36) If we denote the vector of Bloch coecients of (t) by ~ (t), then it can be easily checked that d dt ~ (t) =Q~ (t); (4.37) where Q = 0 B B B B B B B @ 0 0 0 0 0 d dt [f(t)]=f(t) 0 0 0 0 d dt [f(t)]=f(t) 0 0 0 0 0 1 C C C C C C C A : (4.38) Now we can nd the superoperator corresponding toQ. Note that there are 16 basis elements i [] j for the superoperator which we denote by s ij . The basis elements s zz = 0 B B B B B B B @ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 C C C C C C C A ; s 00 = 0 B B B B B B B @ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 C C C C C C C A (4.39) allow us to write Q = 1 2 d dt [f(t)]=f(t)(s zz s 00 ). Therefore the quantum master equation of the form given by Eq. (4.7) for these dynamics is _ = (t) 2 ( z z ); (4.40) 72 where (t) = d dt [f(t)]=f(t) (note the negative sign). As expected, there exist time intervals where (t)< 0 for both memory kernels (as can be inferred from Fig. 4.1), which is consistent with the extended GKSL theorem [16] and our non-Markovian analysis thus far. 4.5 Which kernels give rise to CP-divisible dy- namics? A natural question to ask is what are the classes of kernels k(t) that give rise to CP-divisible dynamics for the PMME? Given a general Markovian generator L (and hence the eigenoperators fL i ;R i g and eigenvalues f i g), the condition on the kernel k(t) for the associated dynamics to be CP divisible can be de- rived from Eq. (4.15). First we derive the quantum map which maps (t) to (t +dt). Multiplying both sides of Eq. (4.13) by L j and taking the trace, we have Tr[L j (t)] = P i i (t)Tr[L i (0)]Tr[L j R i ] = j (t)Tr[L j (0)], where in the sec- ond equality we used the fact that Tr[L j R i ] = ij . Therefore Tr[L j (0)] = Tr[L j (t)] j (t) (4.41) for all t where j (t)6= 0. Note that from Eq. (4.13) we also have (t +dt) = X i i (t +dt)Tr[L i (0)]R i ; (4.42) 73 so combining this with Eq. (4.41) gives (t +dt) = X i i (t +dt) i (t) Tr[L i (t)]R i : (4.43) For CP-divisible dynamics, this map must be completely positive. By applying Eq. (4.15) to Eq. (4.43), we arrive at the condition for CP-divisible dynamics: X i i (t +dt) i (t) L T i R i 0: (4.44) Because the functions i are given in terms of the memory kernel k(t) through Eq. (4.14), this inequality gives a condition on k(t) that can be checked to verify that the given kernel produces CP divisible dynamics. Given our analysis so far, we expect Eq. (4.44) to be violated for the qubit dephasing example studied in Section 4.4. The left-hand side of Eq. (4.44) for this example becomes 1 2 I + (1 + d dt f(t) f(t) ) x x + T y y + z z : (4.45) The eigenvectors of this operator are given by the Bell basis, and from a straight- forward calculation we see that the eigenvalues are i 4 i=1 =f0; 0; (t); 2 + (t)g (4.46) where (t) = d dt [f(t)]=f(t). Since 3 < 0 when (t) < 0 (which corresponds precisely to the non-Markovianity region in Fig. 4.1), the CP-divisibility condition given by Eq. (4.44) is violated for both memory kernels, as expected. 74 4.6 Conclusions and Future Work In conclusion, we have shown through a simple example that the post Markovian master equation (PMME) can describe non-Markovian eects. We did this by analyzing the change of distinguishability of quantum states, and by checking the divisibility of the associated quantum maps. This complements the much more general non-Markovian analysis of the PMME given in Ref. [15]. Quantum non-Markovianity is a good metric for judging whether solutions ob- tained from the PMME really do interpolate between the Markovian master equa- tion and the exact (Nakajima-Zwanzig) equation. Ultimately, being able to describe as many dierent physical scenarios as possible, with dierent bath characteriza- tions (memory kernels) and an easily solvable master equation is an important step towards modeling of open system quantum dynamics that is both rich and tractable. Future work should will this analysis in more physically motivated scenarios. 75 Chapter 5 Conclusions In this dissertation, we studied the quantum information ow in quantum steganog- raphy as well as in the post-Markovian master equation (PMME). First, in Ch. 2, we studied an explicit steganographic encoding for a sender, Alice, to hide her secret message in the syndromes of an error-correcting code, so that the encoding simulates a given noisy quantum channel. We calculated achiev- able rates of steganographic communication over noiseless quantum channels using this encoding. We gave denitions of secrecy and reliability for the communica- tion process, and with these assumptions derived upper bounds on the amount of steganographic communication possible, and showed that these bounds match the communication rates achieved with our encoding. This gives a steganographic capacity for a noiseless channel emulating a given noisy channel. Second, in Ch. 3, we again studied an explicit steganographic encoding for Alice to hide a secret message in the syndromes of an error-correcting code, so that the encoding simulates a given noisy quantum channel that Eve believes to connect Alice and Bob. Here, however, the actual physical channel connecting Alice and Bob is noisy, but less noisy than Eve believes. We showed that for the bit- ip and 76 depolarizing channels Alice can use Eve's lack of knowledge of the channel parameter to encode quantum information steganographically. We gave an explicit encoding procedure and calculated the rate at which Alice and Bob can communicate secretly. We also showed that our encoding is optimal. We calculated the rate at which secret key must be consumed. Furthermore, we discussed the possiblity of steganographic communication over more general quantum channels, and conjectured a general formula for the steganographic rate. Finally, in Ch. 4, we brie y reviewed the PMME, which is relatively easy to solve, and analyzed a simple example where solutions obtained exhibit non-Markovianity. We applied the distinguishability measure introduced by Breuer et al. which en- codes a notion of information ow between system and bath, and we also explicitly analyzed the divisibility of the associated quantum dynamical maps. We also gave a mathematical condition on the memory kernel used in the PMME that guarantees non-CP-divisible dynamics. These projects all involve quantifying the ow of information in quantum me- chanics, which has many interesting and unintuitive aspects. Many interesting open problems remain. These should lead to more research results in the future. 77 Appendix A An example with no information back ow but with a non-divisible map We review an example due to Ref. [34] that illustrates the subtlety of dening non-Markovianity purely via information back ow from the environment. Consider the dynamics of a spin-1=2 particle interacting with a bosonic reservoir at temperature T . The Markovian generator associated with this process is L = 0 2 (N + 1)(2 + + + ) + 0 2 N(2 + + + ); (A.1) where 0 is the dissipation rate,N is the mean number of excitations of the reservoir, and are the raising and lowering operators. Using the kernel function k(t) = 78 e t we can solve the PMME by following the procedure given in Sec. 4.2.2. The result is: (t) = N 1 + 2N 0 @ 1+N N 0 0 1 1 A +(A;B;t) b 0 @ 0 0 1 0 1 A +b 0 @ 0 1 0 0 1 A + (1a N 1 + 2N ) 0 @ 1 0 0 1 1 A (A.2) where the initial state is (0) = 0 @ a b b 1a 1 A , where A = (1 + 2N) 0 , B = + (1 + 2N) 0 , and where (A;B;t) =e Bt 2 cosh t 2 p B 2 4A + B p B 2 4A sinh t 2 p B 2 4A : (A.3) We can now compute the non-Markovianity measure given by Eq. (4.10). For initial states 1 (0) = 0 @ a 0 0 1a 1 A and 2 (0) = 0 @ c 0 0 1c 1 A we have (t; 1;2 (0)) = 4Ae Bt 2 jacj sinh( t 2 p B 2 4A) p B 2 4A (A.4) which is always negative since B 2 4A = ( + (1 + 2N) 0 ) 2 is a perfect square. It is straightforward to verify that this is in fact true for all initial states 1 and 2 . Hence the solution contains no information back ow from the bath to the system, and so one might be tempted to conclude that it is Markovian. However, as shown in Ref. 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Abstract (if available)

Abstract In this thesis, by applying ideas and results from quantum information theory, I study the fields of quantum steganography and the post-Markovian master equation (PMME), which is used in the field of open quantum systems to model system-bath dynamics. ❧ Quantum steganography is the study of hiding secret quantum information by encoding it into what an eavesdropper would perceive as an innocent-looking message. In the first part of this thesis, I study an explicit steganographic encoding for Alice to hide her secret message in the syndromes of an error-correcting code, so that the encoding simulates a given noisy quantum channel. I calculate achievable rates of steganographic communication over noiseless quantum channels using this encoding. I give definitions of secrecy and reliability for the communication process, and with these assumptions derive upper bounds on the amount of steganographic communication possible, and show that these bounds match the communication rates achieved with our encoding. This gives a steganographic capacity for a noiseless channel emulating a given noisy channel. I then go on to study the case where the actual physical channel shared between Alice and Bob is noisy. I study in particular the bit-flip and depolarizing channels, and derive results that generalize from the noiseless scenario. Finally, I discuss the possibility of steganographic communication over more general quantum channels, and conjecture a general formula for the steganographic rate. ❧ In the second part of this thesis, I analyze an easily solvable quantum master equation known as the PMME that takes into account memory effects induced on the system by the bath, i.e., non-Markovian effects. I briefly review the PMME, and analyze a simple example where solutions obtained exhibit non-Markovianity. I apply the distinguishability measure introduced by Breuer et al. which encodes a notion of information flow between the system and bath, and I also explicitly analyse the divisibility of the associated quantum dynamical maps. I give a mathematical condition on the memory kernel used in the PMME that guarantees non-CP-divisible dynamics.

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Asset Metadata

Creator Sutherland, Christopher (author)

Core Title Quantum information flow in steganography and the post Markovian master equation

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 07/25/2018

Defense Date 06/12/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag OAI-PMH Harvest,quantum computing,quantum information,quantum physics

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Brun, Todd (committee chair), Haas, Stephan (committee member), Jonckheere, Edmond (committee member), Lidar, Daniel (committee member), Pilch, Krzysztof (committee member)

Creator Email chrisjgsutherland@gmail.com,cjsuther@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-27238

Unique identifier UC11669135

Identifier etd-Sutherland-6461.pdf (filename),usctheses-c89-27238 (legacy record id)

Legacy Identifier etd-Sutherland-6461.pdf

Dmrecord 27238

Document Type Dissertation

Format application/pdf (imt)

Rights Sutherland, Christopher

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA