We consider a rate-distortion version of the quantum state redistribution task, where the error of the decoded state is judged via an additive distortion measure; it thus constitutes a quantum generalisation of the classical Wyner-Ziv problem. The quantum source is described by a tripartite pure state shared between Alice (A, encoder), Bob (B, decoder) and a reference (R). Both Alice and Bob are required to output a system (ÃandB̃, respectively), and the distortion measure is encoded in an observable onÃB̃R. It includes as special cases most quantum rate-distortion problems considered in the past, and in particular quantum data compression with the fidelity measured per copy; furthermore, it generalises the well-known state merging and quantum state redistribution tasks for a pure state source, with per-copy fidelity, and a variant recently considered by us, where the source is an ensemble of pure states [ZBK & AW, Proc. ISIT 2020, pp. 1858-1863 and ZBK, PhD thesis, UAB 2020, arXiv:2012.14143]. We derive a single-letter formula for the rate-distortion function of compression schemes assisted by free entanglement. A peculiarity of the formula is that in general it requires optimisation over an unbounded auxiliary register, so the rate-distortion function is not readily computable from our result, and there is a continuity issue at zero distortion. However, we show how to overcome these difficulties in certain situations.

This paper is concerned with quantum data compression of asymptotically many independent and identically distributed copies of ensembles of mixed quantum states. The encoder has access to a side information system. The figure of merit is per-copy or local error criterion. Rate-distortion theory studies the trade-off between the compression rate and the per-copy error. The optimal trade-off can be characterized by the rate-distortion function, which is the best rate given a certain distortion. In this paper, we derive the rate-distortion function of mixed-state compression. The rate-distortion functions in the entanglement-assisted and unassisted scenarios are in terms of a single-letter mutual information quantity and the regularized entanglement of purification, respectively. For the general setting where the consumption of both communication and entanglement are considered, we present the full qubit-entanglement rate region. Our compression scheme covers both blind and visible compression models (and other models in between) depending on the structure of the side information system.

In this paper we consider the compression of asymptotically many i.i.d. copies of ensembles of mixed quantum states where the encoder has access to a side information system. This source is equivalently defined as a classical-quantum state, namely, a quantum system correlated with a classical system playing the role of an inaccessible reference system. The figure of merit is evaluated based on per-copy or local error criterion. The rate-distortion theory aims to reveal the trade-off between the compression rate and the per-copy error. The optimal trade-off can be characterized by the rate-distortion function, which is the best rate given a certain distortion. In this paper, we analyze the rate-distortion functions of mixed-state compression. We find the rate-distortion functions in the entanglement-assisted and unassisted scenarios, in terms of a single-letter mutual information quantity and the regularized entanglement of purification, respectively.

We consider the most general finite-dimensional quantum mechanical information source, which is given by a quantum system$A$that is correlated with a reference system$R$. The task is to compress$A$in such a way as to reproduce the joint source state$\rho ^{AR}$at the decoder with asymptotically high fidelity. This includes Schumacher’s original quantum source coding problem of a pure state ensemble and that of a single pure entangled state, as well as general mixed state ensembles. Here, we determine the optimal compression rate (in qubits per source system) in terms of the Koashi-Imoto decomposition of the source into a classical, a quantum, and a redundant part. The same decomposition yields the optimal rate in the presence of unlimited entanglement between compressor and decoder, and indeed the full region of feasible qubit-ebit rate pairs.

We consider the most general (finite-dimensional) quantum mechanical information source, which is given by a quantum system A that is correlated with a reference system R. The task is to compress A in such a way as to reproduce the joint source state ρARat the decoder with asymptotically high fidelity. This includes Schumacher's original quantum source coding problem of a pure state ensemble and that of a single pure entangled state, as well as general mixed state ensembles. Here, we determine the optimal compression rate (in qubits per source system) in terms of the Koashi-Imoto decomposition of the source into a classical, a quantum, and a redundant part. The same decomposition yields the optimal rate in the presence of unlimited entanglement between compressor and decoder, and indeed the full region of feasible qubitebit rate pairs. Full version at arXiv:1912.08506 [1].

We consider a generalization of the quantum state redistribution task, where pure multipartite states from an ensemble source are distributed among an encoder, a decoder and a reference system. The encoder, Alice, has access to two quantum systems: system A which she compresses and sends to the decoder, Bob, and the side information system C which she wants to keep at her site. Bob has access to quantum side information in a system B, wants to decode the compressed information in such a way to preserve the correlations with the reference system on average. As figures of merit, we consider both block error (which is the usual one in source coding) and per-copy error (which is more akin to rate-distortion theory), and find the optimal compression rate for the second criterion, and achievable and converse bounds for the first. The latter almost match in general, up to an asymptotic error and an unbounded auxiliary system; for so-called irreducible sources they are provably the same. Full paper forthcoming [1].

We resume the investigation of the problem of independent local compression of correlated quantum sources, the classical case of which is covered by the celebrated Slepian-Wolf theorem. We focus specifically on classical-quantum (cq) sources, for which one edge of the rate region, corresponding to the compression of the classical part, using the quantum part as side information at the decoder, was previously determined by Devetak and Winter [Phys. Rev. A 68, 042301 (2003)]. Whereas the Devetak-Winter protocol attains a rate-sum equal to the von Neumann entropy of the joint source, here we show that the full rate region is much more complex, due to the partially quantum nature of the source. In particular, in the opposite case of compressing the quantum part of the source, using the classical part as side information at the decoder, typically the rate sum is strictly larger than the von Neumann entropy of the total source. We determine the full rate region in the generic case, showing that, apart from the Devetak-Winter point, all other points in the achievable region have a rate sum strictly larger than the joint entropy. We can interpret the difference as the price paid for the quantum encoder being ignorant of the classical side information. In the general case, we give an achievable rate region, via protocols that are built on the decoupling principle, and the protocols of quantum state merging and quantum state redistribution. Our achievable region is matched almost by a single-letter converse, which however still involves asymptotic errors and an unbounded auxiliary system.

Ask how the quantum compression of ensembles of pure states is affected by the availability of entanglement, and in settings where the encoder has access to side information. We find the optimal asymptotic quantum rate and the optimal tradeoff (rate region) of quantum and entanglement rates. It turns out that the amount by which the quantum rate beats the Schumacher limit, the entropy of the source, is precisely half the entropy of classical information that can be extracted from the source and side information states without disturbing them at all ("reversible extraction of classical information").In the special case that the encoder has no side information, or that she has access to the identity of the states, this problem reduces to the known settings of blind and visible Schumacher compression, respectively, albeit here additionally with entanglement assistance. We comment on connections to previously studied and further rate tradeoffs when also classical information is considered.

We resume the investigation of the problem of independent local compression of correlated quantum sources, the classical case of which is covered by the celebrated Slepian-Wolf theorem. We focus specifically on classical-quantum (cq) sources, for which one edge of the rate region, corresponding to the compression of the classical part, using the quantum part as side information at the decoder, was previously determined by Devetak and Winter [Phys. Rev. A 68, 042301 (2003)]. Whereas the Devetak-Winter protocol attains a rate-sum equal to the von Neumann entropy of the joint source, here we show that the full rate region is much more complex, due to the partially quantum nature of the source. In particular, in the opposite case of compressing the quantum part of the source, using the classical part as side information at the decoder, typically the rate sum is strictly larger than the von Neumann entropy of the total source.We determine the full rate region in the generic case, showing that, apart from the Devetak-Winter point, all other points in the achievable region have a rate sum strictly larger than the joint entropy. We can interpret the difference as the price paid for the quantum encoder being ignorant of the classical side information. In the general case, we give an achievable rate region, via protocols that are built on the decoupling principle, and the principles of quantum state merging and quantum state redistribution. Our achievable region is matched almost by a single-letter converse, which however still involves asymptotic errors and an unbounded auxiliary system.

We propose a distributed opportunistic medium access control (MAC) scheme for maximizing the expected aggregate throughput in a multichannel wireless network such as a clustered orthogonal frequency-division multiple access (OFDMA) network. In our proposed scheme, each user attempts to send only on its best channel and transmits if the best-channel gain is higher than a given threshold, which is dynamically updated depending on previous idle and collision situations. In this way, with our proposed scheme, in a homogeneous system where the channel fading distribution is identical for all users, the best user for each channel is obtained in a distributed manner. We also obtain the optimal values of the thresholds so that the probability of successful transmission is maximized and a minimal number of transmission opportunities are wasted (e.g., due to collision or idle transmissions). In the asymptotic limit of a large number of users and sufficiently long transmission slot duration, we show that, in comparison with the optimal centralized scheme, the throughput loss for our proposed scheme goes to zero. Furthermore, we extend our distributed opportunistic MAC scheme for a homogeneous system to that for a heterogeneous system where the channel fading distribution is heterogeneous across users. Throughput performances and signaling overhead are analyzed for the proposed distributed MAC schemes and compared with those of the existing schemes. Simulation results show that our proposed schemes significantly improve the average aggregate throughput when compared with the existing schemes.