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Quantum-to-classical rate distortion coding

### Abstract

We establish a theory of quantum-to-classical rate distortion coding. In this setting, a sender Alice has many copies of a quantum information source. Her goal is to transmit a classical description of the source, obtained by performing a measurement on it, to a receiver Bob, up to some specified level of distortion. We derive a single-letter formula for the minimum rate of classical communication needed for this task. We also evaluate this rate in the case in which Bob has some quantum side information about the source. Our results imply that, in general, Alice's best strategy is a non-classical one, in which she performs a collective measurement on successive outputs of the source.

© 2013 American Institute of Physics

Received 19 January 2013
Accepted 11 March 2013
Published online 02 April 2013

Acknowledgments:
We acknowledge useful discussions Patrick Hayden. M.M.W. acknowledges support from the Centre de Recherches Mathématiques at the University of Montreal. M.-H.H. received support from the Chancellor's postdoctoral research fellowship, University of Technology Sydney (UTS) and was also partly supported by the National Natural Science Foundation of China (Grant No. 61179030) and the Australian Research Council (Grant No. DP120103776). AW was supported by the Royal Society, the Philip Leverhulme Trust, EC integrated project QAP (contract IST-2005-15848), the STREPs QICS and QCS, and the ERC Advanced Grant “IRQUAT”.

Article outline:

I. INTRODUCTION
II. NOTATIONS AND DEFINITIONS
III. DISTORTION OBSERVABLES
IV. QUANTUM-TO-CLASSICAL RATE-DISTORTION CODING
V. QUANTUM-TO-CLASSICAL RATE-DISTORTION CODING WITH QUANTUM SIDE INFORMATION
VI. CONCLUSIONS AND DISCUSSIONS

/content/aip/journal/jmp/54/4/10.1063/1.4798396

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7. P. Cuff, “Communication requirements for generating correlated random variables,” in Proceedings of the 2008 International Symposium on Information Theory, Toronto, Ontario, Canada, July 2008 (Stanford Univ., Stanford, CA), pp. 1393–1397;

http://dx.doi.org/10.1109/ISIT.2008.4595216
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8. N. Datta, M.-H. Hsieh, and M. M. Wilde, “Quantum rate distortion, reverse Shannon theorems, and source-channel separation,” IEEE Trans. Inf. Theory 59, 615–630 (2013);

http://dx.doi.org/10.1109/TIT.2012.2215575
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From Classical to Quantum Shannon Theory,” preprint

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2011).

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Compression of sources of probability distributions and density operators,” preprint

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http://dx.doi.org/10.1007/s00220-003-0989-z
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27.We note that a similar perspective was used to justify the development of quantum-to-classical randomness extractors.^{4}

28.

28.A “classical strategy” here corresponds to a measurement in the eigenbasis of ρ, followed by classical post-processing.

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2013-04-02

2015-11-26

### Abstract

We establish a theory of quantum-to-classical rate distortion coding. In this setting, a sender Alice has many copies of a quantum information source. Her goal is to transmit a classical description of the source, obtained by performing a measurement on it, to a receiver Bob, up to some specified level of distortion. We derive a single-letter formula for the minimum rate of classical communication needed for this task. We also evaluate this rate in the case in which Bob has some quantum side information about the source. Our results imply that, in general, Alice's best strategy is a non-classical one, in which she performs a collective measurement on successive outputs of the source.

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