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IEEE International Symposium on Information Theory (ISIT)

Quantum data compression with commuting density operators

Gerhard G. Kramer, Serap A. Savari

Published Jun 25, 2003• 8 citations
Publisher

Abstract

The problem of compressing strings of quantum states whose density operators commute is considered. Two versions of the problem are discussed. In the visible case arbitrary operations on the strings are permitted and in the blind case only quantum operations on the density operator of the strings are permitted. Both cases have been reformulated as classical information-theoretic problems, and these problems are solved.

DOI
10.1109/isit.2002.1023316

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The problem of compressing strings of quantum states whose density operators commute is considered. Two versions of the problem are discussed. In the visible case arbitrary operations on the strings are permitted and in the blind case only quantum operations on the density operator of the strings are permitted. Both cases have been reformulated as classical information-theoretic problems, and these problems are solved.
We provide a rate distortion interpretation of the problem of quantum data compression of ensembles of mixed states with commuting density operators. There are two versions of this problem. In the visible case the sequence of states is available to the encoder and in the blind or hidden case the encoder may access only a sequence of measurements. We find the exact optimal compression rates for both the visible and hidden cases. Our analysis includes the scenario in which asymptotic reconstruction is imperfect.
We consider visible compression for discrete memoryless sources of mixed quantum states when only classical information can be sent from Alice to Bob. We assume that Bob knows the source statistics, and that Alice and Bob have access to the same source of random numbers. We put in an information-theoretic framework some previous results on visible compression for sources of states with commuting density operators, and remove the commutativity requirement. We derive a general achievable compression rate, which is for the noncommutative case still higher than the known lower bound. We also present several related problems of classical information theory, and show how they can be used to answer some questions of the mixed-state compression problem.
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.
We consider visible compression for discrete memoryless sources of mixed quantum states when only classical information can be sent from Alice to Bob. We assume that Bob knows the source statistics, and that Alice and Bob have identical random number generators. We put in an information theoretic framework some recent results on visible compression for sources of states with commuting density operators, and remove the commutativity requirement. We derive a general achievable compression rate, which is for the noncommutative case still higher than the known lower bound. We also present several related problems of classical information theory, and show how they can be used to answer some questions of the mixed state compression problem.
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.