Compression of sources of probability distributions and density operators

Abstract

We study the problem of efficient compression of a stochastic source of probability distributions. It can be viewed as a generalization of Shannon's source coding problem. It has relation to the theory of common randomness, as well as to channel coding and rate--distortion theory: in the first two subjects ``inverses'' to established coding theorems can be derived, yielding a new approach to proving converse theorems, in the third we find a new proof of Shannon's rate--distortion theorem. After reviewing the known lower bound for the optimal compression rate, we present a number of approaches to achieve it by code constructions. Our main results are: a better understanding of the known lower bounds on the compression rate by means of a strong version of this statement, a review of a construction achieving the lower bound by using common randomness which we complement by showing the optimal use of the latter within a class of protocols. Then we review another approach, not dependent on common randomness, to minimizing the compression rate, providing some insight into its combinatorial structure, and suggesting an algorithm to optimize it. The second part of the paper is concerned with the generalization of the problem to quantum information theory: the compression of mixed quantum states. Here, after reviewing the known lower bound we contribute a strong version of it, and discuss the relation of the problem to other issues in quantum information theory.

12 pages, 1 eps figure, IEEEtran.cls

References in corpus (5)

• Coding Theorem and Strong Converse for Quantum Channels
• A classical analogue of entanglement
• Visible compression of commuting mixed states
• Quantum Data Compression of Ensembles of Mixed States with Commuting Density Operators
• Compressing Mixed-State Sources by Sending Classical Information

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• Distributed Channel Synthesis
• Quantum Reverse Shannon Theorem
• A Resource Framework for Quantum Shannon Theory
• Sample-optimal tomography of quantum states
• Generalized remote state preparation: Trading cbits, qubits and ebits in quantum communication
• The information-theoretic costs of simulating quantum measurements
• Applications of coherent classical communication and the Schur transform to quantum information theory
• One-shot lossy quantum data compression
• Quantum-to-classical rate distortion coding
• Strong converse for the classical capacity of the pure-loss bosonic channel
• Quantum Rate-Distortion Coding of Relevant Information
• Strong coordination of signals and actions over noisy channels with two-sided state information
• Quantum Side Information: Uncertainty Relations, Extractors, Channel Simulations
• Multiple Access Channel Simulation
• Non-Asymptotic and Second-Order Achievability Bounds for Coding With Side-Information
• Strong Converse and Second-Order Asymptotics of Channel Resolvability
• On Exact and $\infty$-Rényi Common Informations
• Exact Channel Synthesis
• Channel simulation with quantum side information