Finite blocklength and moderate deviation analysis of hypothesis testing of correlated quantum states and application to classical-quantum channels with memory

TL;DR

This paper develops finite blocklength and moderate deviation bounds for quantum hypothesis testing of correlated states, improving existing bounds and applying results to classical-quantum channels with memory.

Contribution

It introduces tighter finite blocklength bounds for quantum hypothesis testing of correlated states and extends moderate deviation analysis to non-i.i.d. quantum sequences.

Findings

Tighter bounds on type II errors for quantum hypothesis testing.

Finite blocklength bounds for correlated quantum states.

Application to classical-quantum channels with memory.

Abstract

Martingale concentration inequalities constitute a powerful mathematical tool in the analysis of problems in a wide variety of fields ranging from probability and statistics to information theory and machine learning. Here we apply techniques borrowed from this field to quantum hypothesis testing, which is the problem of discriminating quantum states belonging to two different sequences $\{\rho_n\}_{n}$ and $\{\sigma_n\}_n$. We obtain upper bounds on the finite blocklength type II Stein- and Hoeffding errors, which, for i.i.d. states, are in general tighter than the corresponding bounds obtained by Audenaert, Mosonyi and Verstraete [Journal of Mathematical Physics, 53(12), 2012]. We also derive finite blocklength bounds and moderate deviation results for pairs of sequences of correlated states satisfying a (non-homogeneous) factorization property. Examples of such sequences include Gibbs states of spin chains with translation-invariant finite range interaction, as well as finitely correlated quantum states. We apply our results to find bounds on the capacity of a certain class of classical-quantum channels with memory, which satisfy a so-called channel factorization property- both in the finite blocklength and moderate deviation regimes.