Non-Asymptotic Classical Data Compression with Quantum Side Information

TL;DR

This paper provides finite blocklength bounds and asymptotic analysis for classical data compression with quantum side information, focusing on large and moderate deviation regimes, extending understanding beyond traditional asymptotic limits.

Contribution

It derives finite blocklength bounds on the error exponent function and characterizes the convergence rates in large and moderate deviation regimes for the protocol.

Findings

Finite blocklength bounds on error exponents

Exact asymptotic value of the strong converse exponent

Convergence speed of error probability in moderate deviations

Abstract

In this paper, we analyze classical data compression with quantum side information (also known as the classical-quantum Slepian-Wolf protocol) in the so-called large and moderate deviation regimes. In the non-asymptotic setting, the protocol involves compressing classical sequences of finite length $n$ and decoding them with the assistance of quantum side information. In the large deviation regime, the compression rate is fixed, and we obtain bounds on the error exponent function, which characterizes the minimal probability of error as a function of the rate. Devetak and Winter showed that the asymptotic data compression limit for this protocol is given by a conditional entropy. For any protocol with a rate below this quantity, the probability of error converges to one asymptotically and its speed of convergence is given by the strong converse exponent function. We obtain finite blocklength bounds on this function, and determine exactly its asymptotic value. In the moderate deviation regime for the compression rate, the latter is no longer considered to be fixed. It is allowed to depend on the blocklength $n$, but assumed to decay slowly to the asymptotic data compression limit. Starting from a rate above this limit, we determine the speed of convergence of the error probability to zero and show that it is given in terms of the conditional information variance. Our results complement earlier results obtained by Tomamichel and Hayashi, in which they analyzed the so-called small deviation regime of this protocol.