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IEEE International Symposium on Information Theory (ISIT)

Rate-Distortion Theory for Mixed States

Zahra Baghali Khanian, Kohdai Kuroiwa, Debbie Leung

Published Jun 25, 2023• 3 citations
Publisher

Abstract

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.

DOI
10.1109/isit54713.2023.10206960

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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.
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.
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.
We derive quantum counterparts of two key theorems of classical information theory, namely, the rate-distortion theorem and the source-channel separation theorem. The rate-distortion theorem gives the ultimate limits on lossy data compression, and the source-channel separation theorem implies that a two-stage protocol consisting of compression and channel coding is optimal for transmitting a memoryless source over a memoryless channel. In spite of their importance in the classical domain, there has been surprisingly little work in these areas for quantum information theory. In this paper, we prove that the quantum rate-distortion function is given in terms of the regularized entanglement of purification. We also determine a single-letter expression for the entanglement-assisted quantum rate-distortion function, and we prove that it serves as a lower bound on the unassisted quantum rate-distortion function. This implies that the unassisted quantum rate-distortion function is nonnegative and generally not equal to the coherent information between the source and distorted output (in spite of Barnum's conjecture that the coherent information would be relevant here). Moreover, we prove several quantum source-channel separation theorems. The strongest of these are in the entanglement-assisted setting, in which we establish a necessary and sufficient condition for transmitting a memoryless source over a memoryless quantum channel up to a given distortion.
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.
We derive quantum counterparts of two key theorems of classical information theory, namely, the rate-distortion theorem and the source-channel separation theorem. The rate-distortion theorem gives the ultimate limits on lossy data compression, and the source-channel separation theorem implies that a two-stage protocol consisting of compression and channel coding is optimal for transmitting a memoryless source over a memoryless channel. In spite of their importance in the classical domain, there has been surprisingly little work in these areas for quantum information theory. In this paper, we prove that the quantum rate-distortion function is given in terms of the regularized entanglement of purification. We also determine a single-letter expression for the entanglement-assisted quantum rate-distortion function, and we prove that it serves as a lower bound on the unassisted quantum rate-distortion function. This implies that the unassisted quantum rate-distortion function is nonnegative and generally not equal to the coherent information between the source and distorted output (in spite of Barnum's conjecture that the coherent information would be relevant here). Moreover, we prove several quantum source-channel separation theorems. The strongest of these are in the entanglement-assisted setting, in which we establish a necessary and sufficient condition for transmitting a memoryless source over a memoryless quantum channel up to a given distortion. © 2012 IEEE.
Rate-distortion theory provides bounds for compressing data produced by an information source to a specified encoding rate that is strictly less than the source's entropy. This necessarily entails some loss, or distortion, between the original source data and the best approximation after decompression. The so-called Information Bottleneck Method is designed to compress only “relevant” information. Which information is relevant is determined by the correlation between the data being compressed and a variable of interest, so-called side information. In this paper, an Information Bottleneck Method is introduced for the compression of quantum data. The channel communication picture is used for compression and decompression. The rate of compression is derived using an entanglement-assisted protocol with classical communication, and under an unproved conjecture that the rate function is convex in the distortion parameter. The optimum channel achieving this rate for a given input state is characterized. The conceptual difficulties arising due to differences in the mathematical formalism between quantum and classical probability theory are discussed and solutions are presented.