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arXiv:quant-ph/9905058 · Physical Review A

Optimal compression for mixed signal states

Michał Horodecki

Published Apr 11, 2000• 24 citations
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Abstract

We consider the problem of the optimal compression rate in the case of a source producing mixed signal states within the visible scheme (where Alice, who is to compress the signal, can know the identities of the produced states). We show that a simple strategy based on replacing the signal states with their extensions gives optimal compression. As a result we obtain a considerable simplification of the formula for an optimal compression rate within a visible scheme.

DOI
10.1103/physreva.61.052309

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We consider the problem of the optimal compression rate in the case of a source producing mixed signal states within the visible scheme (where Alice, who is to compress the signal, can know the identities of the produced states). We show that a simple strategy based on replacing the signal states with their extensions gives optimal compression. As a result we obtain a considerable simplification of the formula for an optimal compression rate within a visible scheme.
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 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.
We consider the problem of compression of the quantum information carried by ensemble of mixed states. We prove that for arbitrary coding schemes the least number of quantum bits (qubits) per message needed to convey the signal states asymptotically faithfully is bounded from below by the Holevo function $S(\ensuremath{\varrho})\ensuremath{-}{\ensuremath{\sum}}_{i}{p}_{i}S({\ensuremath{\varrho}}_{i})$. We also show that a compression protocol can be composed with another one, provided that the latter offers perfect transmission. Such a compound protocol is applied to the case of binary source. It is conjectured to reach the obtained bound. Finally, we point out that in the case of mixed signal states there could be a difference between the maximal compression rates at the coding schemes that are ``blind'' to the signal and the ones that assume the knowledge about the identities of the signal states.
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.
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.