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1 Quantum rate distortion, reverse Shannon theorems, and source-channel separation Nilanjana Datta, Min-Hsiu Hsieh, and Mark M. Wilde Abstract—We derive quantum counterparts of two key the- of the original data, then the compression is said to be orems of classical information theory, namely, the rate distor- lossless. The simplest example of an information source is tion theorem and the source-channel separation theorem. The a memoryless one. Such a source can be characterized by a rate-distortion theorem gives the ultimate limits on lossy data random variable U with probability distribution pU (u) and compression, and the source-channel separation theorem implies f g that a two-stage protocol consisting of compression and channel each use of the source results in a letter u being emitted with coding is optimal for transmitting a memoryless source over probability pU (u). Shannon’s noiseless coding theorem states a memoryless channel. In spite of their importance in the P that the entropy H (U) u pU (u) log2 pU (u) of such classical domain, there has been surprisingly little work in these an information source is≡ the − minimum rate at which we can areas for quantum information theory. In the present paper, we prove that the quantum rate distortion function is given in compress signals emitted by it [49], [21]. terms of the regularized entanglement of purification. We also The requirement of a data compression scheme being loss- determine a single-letter expression for the entanglement-assisted less is often too stringent a condition, in particular for the quantum rate distortion function, and we prove that it serves case of multimedia data, i.e., audio, video and still images as a lower bound on the unassisted quantum rate distortion or in scenarios where insufficient storage space is available. function. This implies that the unassisted quantum rate distortion function is non-negative and generally not equal to the coherent Typically a substantial amount of data can be discarded before information between the source and distorted output (in spite the information is sufficiently degraded to be noticeable. of Barnum’s conjecture that the coherent information would A data compression scheme is said to be lossy when the be relevant here). Moreover, we prove several quantum source- decompressed data is not required to be identical to the original channel separation theorems. The strongest of these are in the one, but instead recovering a reasonably good approximation entanglement-assisted setting, in which we establish a necessary and sufficient codition for transmitting a memoryless source over of the original data is considered to be good enough. a memoryless quantum channel up to a given distortion. The theory of lossy data compression, which is also referred Index Terms—quantum rate distortion, reverse Shannon the- to as rate distortion theory, was developed by Shannon [50], orem, quantum Shannon theory, quantum data compression, [11], [21]. This theory deals with the tradeoff between the source-channel separation rate of data compression and the allowed distortion. Shannon proved that, for a given memoryless information source and a distortion measure, there is a function R(D), called the I. INTRODUCTION rate-distortion function, such that, if the maximum allowed Two pillars of classical information theory are Shannon’s distortion is D then the best possible compression rate is given data compression theorem and his channel capacity theorem by R(D). He established that this rate-distortion function is [49], [21]. The former gives a fundamental limit to the equal to the minimum of the mutual information I(U; U^) := compressibility of classical information, while the latter deter- H (U) + H(U^) H(U; U^) over all possible stochastic maps − mines the ultimate limit on classical communication rates over p ^ (^u u) that meet the distortion requirement on average: UjU j a noisy classical channel. Modern communication systems ^ arXiv:1108.4940v3 [quant-ph] 19 Aug 2012 exploit these ideas in order to make the best possible use of R(D) = min I(U; U): (1) ^ communication resources. p(^uju): Efd(U;U)}≤D Data compression is possible due to statistical redundancy in ^ the information emitted by sources, with some signals being In the above d(U; U) denotes a suitably chosen distortion measure between the random variable U characterizing the emitted more frequently than others. Exploiting this redun- ^ dancy suitably allows one to compress data without losing source and the random variable U characterizing the output essential information. If the data which is recovered after of the stochastic map. the compression-decompression process is an exact replica Whenever the distortion D = 0, the above rate-distortion function is equal to the entropy of the source. If D > 0, then Nilanjana Datta and Min-Hsiu Hsieh are with the Statistical Laboratory, the rate-distortion function is less than the entropy, implying University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United that fewer bits are needed to transmit the source if we allow Kingdom. The contribution of M.-H. H. was mainly done when he was with the Statistical Laboratory, University of Cambridge. Now he is with for some distortion in its reconstruction. Centre for Quantum Computation and Intelligent Systems (QCIS), Faculty of Alongside these developments, Shannon also contributed the Engineering and Information Technology (FEIT), University of Technology, theory of reliable communication of classical data over clas- Sydney (UTS), PO Box 123, Broadway NSW 2007, Australia. Mark M. Wilde is with the School of Computer Science, McGill University, Montreal,´ Quebec,´ sical channels [49], [21]. His noisy channel coding theorem Canada H3A 2A7. gives an explicit expression for the capacity of a memoryless 2 classical channel, i.e., the maximum rate of reliable communi- both necessary and sufficient for the reliable transmission of cation through it. A memoryless channel is one for which an information source over a noisy channel, up to some amount there is no correlation in the noise acting onN successive inputs, of distortion D [21]. Thus, we can consider the problems of and it can be modelled by a stochastic map pY jX (y x). lossy data compression and channel coding separately, and the Shannon proved that the capacity of such aN channel ≡ is givenj two-stage concatenation of the best lossy compression code by with the best channel code is optimal. C ( ) = max I (X; Y ) : Considering the importance of all of the above theorems N pX (x) for classical information theory, it is clear that theorems in Any scheme for error correction typically requires the use of this spirit would be just as important for quantum information redundancy in the transmitted data, so that the receiver can theory. Note, however, that in the quantum domain, there perfectly distinguish the received signals from one another in are many different information processing tasks, depending the limit of many uses of the channel. on which type of information we are trying to transmit and Given all of the above results, we might wonder whether which resources are available to assist the transmission. For it is possible to transmit an information source U reliably example, we could transmit classical or quantum data over a over a noisy channel , such that the output of the infor- quantum channel, and such a transmission might be assisted N mation source is recoverable with an error probability that is by entanglement shared between sender and receiver before asymptotically small in the limit of a large number of outputs communication begins. of the information source and uses of the noisy channel. An There have been many important advances in the above immediate corollary of Shannon’s noiseless and noisy channel directions (some of which are summarized in the recent coding theorems is that reliable transmission of the source text [57]). Schumacher proved the noiseless quantum coding is possible if the entropy of the source is smaller than the theorem, demonstrating that the von Neumann entropy of capacity of the channel: a quantum information source is the ultimate limit to the compressibility of information emitted by it [45]. Hayashi H (U) C ( ) : (2) ≤ N et al. have also considered many ways to compress quantum The scheme to demonstrate sufficiency of (2) is for the sender information, a summary of which is available in Ref. [30]. to take the length n output of the information source, compress Quantum rate distortion theory, that is the theory of lossy it down to nH (U) bits, and encode these nH (U) bits into a quantum data compression, was introduced by Barnum in length n sequence for transmission over the channel. As long 1998. He considered a symbol-wise entanglement fidelity as as H (U) C ( ), Shannon’s noisy channel coding theorem a distortion measure [4] and, with respect to it, defined the guarantees≤ that itN is possible to transmit the nH (U) bits over quantum rate distortion function as the minimum rate of data the channel reliably such that the receiver can decode them, compression, for any given distortion. He derived a lower and Shannon’s noiseless coding theorem guarantees that the bound on the quantum rate distortion function, in terms of decoded nH (U) bits can be decompressed reliably as well in well-known entropic quantity, namely the coherent informa- order to recover the original length n output of the information tion. The latter can be viewed as one quantum analogue of source (all of this is in the limit as n ). Given that mutual information, since it is known to characterize the the condition in (2) is sufficient for reliable! 1 communication quantum capacity of a channel [38], [52], [23], just as the of the information source, is it also necessary? Shannon’s mutual information characterizes the capacity of a classical source-channel separation theorem answers this question in channel.

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