Compression of sources of probability distributions and density operators

TL;DR

This paper explores efficient compression methods for probabilistic sources, extending classical information theory to quantum states, and offers new bounds, constructions, and algorithms for optimal compression in both classical and quantum contexts.

Contribution

It provides a stronger understanding of lower bounds, reviews existing and new compression schemes, and extends the theory to quantum states with novel bounds and insights.

Findings

Improved lower bounds on compression rates for classical sources.

A review of schemes achieving optimal compression using common randomness.

Extension of compression bounds and methods to quantum mixed states.

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.