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arXiv:2112.11952 · IEEE Transactions on Information Theory

A Rate-Distortion Perspective on Quantum State Redistribution

Zahra Baghali Khanian, Andreas Winter

Published Dec 12, 2024• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1109/tit.2024.3516505

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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 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 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].
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.
In this work, we address the lossy quantum-classical source coding with the quantum side-information (QC-QSI) problem. The task is to compress the classical information about a quantum source, obtained after performing a measurement while incurring a bounded reconstruction error. Here, the decoder is allowed to use the side information to recover the classical data obtained from measurements on the source states. We introduce a new formulation based on a backward (posterior) channel, replacing the single-letter distortion observable with a single-letter posterior channel to capture reconstruction error. Unlike the rate-distortion framework, this formulation imposes a block error constraint. An analogous formulation is developed for lossy classical source coding with classical side information (C-CSI) problem. We derive an inner bound on the asymptotic performance limit in terms of single-letter quantum and classical mutual information quantities of the given posterior channel for QC-QSI and C-CSI cases, respectively. Furthermore, we establish a connection between rate-distortion and rate-channel theory, showing that a rate-channel compression protocol attains the optimal rate-distortion function for a specific distortion measure and level.