^{1}, Min-Hsiu Hsieh

^{2}, Mark M. Wilde

^{3}and Andreas Winter

^{4}

### 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.

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”.

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

### Key Topics

- Quantum information
- 63.0
- Quantum measurement theory
- 42.0
- Information and communication theory
- 27.0
- Chaos
- 24.0
- Inequalities
- 17.0

## Figures

The most general protocol for quantum-to-classical rate-distortion coding. Alice has many copies of the quantum information source, on which she performs a collective measurement with classical output *L*. She sends the variable *L* over noiseless classical bit channels to Bob. Bob then performs a classical decoding map on *L* that outputs the classical sequence *X* ^{ n }. The average deviation of this sequence from the quantum source, according to some distortion observable, provides a measure of the distortion caused by this protocol.

The most general protocol for quantum-to-classical rate-distortion coding. Alice has many copies of the quantum information source, on which she performs a collective measurement with classical output *L*. She sends the variable *L* over noiseless classical bit channels to Bob. Bob then performs a classical decoding map on *L* that outputs the classical sequence *X* ^{ n }. The average deviation of this sequence from the quantum source, according to some distortion observable, provides a measure of the distortion caused by this protocol.

A plot of compression rate vs. distortion for the quantum information source ρ given by (31) and the rate distortion observable given by (32) . It was obtained by randomly sampling 250 000 two-outcome POVMs, and (for those POVMs which satisfy the distortion criterion *D* ⩽ 1/4) plotting the mutual information *I*(*X*; *R*)_{σ} for the resulting state σ_{ RX } (defined by (16) ) against the corresponding value of the distortion. The boundary of the shaded region defines the rate-distortion trade-off curve.

A plot of compression rate vs. distortion for the quantum information source ρ given by (31) and the rate distortion observable given by (32) . It was obtained by randomly sampling 250 000 two-outcome POVMs, and (for those POVMs which satisfy the distortion criterion *D* ⩽ 1/4) plotting the mutual information *I*(*X*; *R*)_{σ} for the resulting state σ_{ RX } (defined by (16) ) against the corresponding value of the distortion. The boundary of the shaded region defines the rate-distortion trade-off curve.

The most general protocol for quantum-to-classical rate-distortion coding with quantum side information. Alice and Bob share many copies of a quantum state ρ_{ AB }, which is purified by an inaccessible reference system. We also allow them access to common randomness *M* before the protocol begins. Alice first performs a collective measurement on her systems, producing a classical output *L*. She then transmits *L* over noiseless classical bit channels to Bob. Bob performs a collective measurement on his quantum systems, depending on what he receives from Alice and his share of the common randomness. This measurement produces a classical sequence *X* ^{ n } and has quantum outputs as well. The protocol is deemed successful if the classical sequence *X* ^{ n } is not distorted on average from the quantum source more than a specified amount according to a suitable distortion observable. We also demand that the disturbance caused by the protocol to the joint state of the reference and Bob's systems is asymptotically negligible. This in turn implies that quantum side information suffers a negligible disturbance and hence is available to Bob for future use.

The most general protocol for quantum-to-classical rate-distortion coding with quantum side information. Alice and Bob share many copies of a quantum state ρ_{ AB }, which is purified by an inaccessible reference system. We also allow them access to common randomness *M* before the protocol begins. Alice first performs a collective measurement on her systems, producing a classical output *L*. She then transmits *L* over noiseless classical bit channels to Bob. Bob performs a collective measurement on his quantum systems, depending on what he receives from Alice and his share of the common randomness. This measurement produces a classical sequence *X* ^{ n } and has quantum outputs as well. The protocol is deemed successful if the classical sequence *X* ^{ n } is not distorted on average from the quantum source more than a specified amount according to a suitable distortion observable. We also demand that the disturbance caused by the protocol to the joint state of the reference and Bob's systems is asymptotically negligible. This in turn implies that quantum side information suffers a negligible disturbance and hence is available to Bob for future use.

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