Achievable error exponents of data compression with quantum side information and communication over symmetric classical-quantum channels

TL;DR

This paper establishes a lower bound on the error exponent for data compression with quantum side information over symmetric classical-quantum channels, connecting privacy amplification bounds to classical information compression and matching known sphere-packing bounds.

Contribution

It links privacy amplification error exponents to classical information compression with quantum side information, providing bounds that align with sphere-packing limits for symmetric channels.

Findings

Lower bound on error exponent matches sphere-packing upper bound for certain rates.

Reproduces classical results for symmetric channels.

Sharpened bounds for linear randomness extractors in privacy amplification.

Abstract

A fundamental quantity of interest in Shannon theory, classical or quantum, is the optimal error exponent of a given channel W and rate R: the constant E(W,R) which governs the exponential decay of decoding error when using ever larger codes of fixed rate R to communicate over ever more (memoryless) instances of a given channel W. Here I show that a bound by Hayashi [CMP 333, 335 (2015)] for an analogous quantity in privacy amplification implies a lower bound on the error exponent of communication over symmetric classical-quantum channels. The resulting bound matches Dalai's [IEEE TIT 59, 8027 (2013)] sphere-packing upper bound for rates above a critical value, and reproduces the well-known classical result for symmetric channels. The argument proceeds by first relating the error exponent of privacy amplification to that of compression of classical information with quantum side information, which gives a lower bound that matches the sphere-packing upper bound of Cheng et al. [IEEE TIT 67, 902 (2021)]. In turn, the polynomial prefactors to the sphere-packing bound found by Cheng et al. may be translated to the privacy amplification problem, sharpening a recent result by Li, Yao, and Hayashi [arXiv:2111.01075 [quant-ph]], at least for linear randomness extractors.