A smooth entropy approach to quantum hypothesis testing and the classical capacity of quantum channels

TL;DR

This paper employs smooth entropy techniques to analyze quantum hypothesis testing and classical communication capacity of quantum channels, establishing bounds and recovering key asymptotic theorems with a unified approach.

Contribution

It introduces a smooth entropy framework to derive bounds and recover fundamental quantum information theorems, unifying hypothesis testing and channel capacity analysis.

Findings

Bounds on quantum hypothesis testing error probabilities

Recovery of the strong converse rate for i.i.d. hypothesis testing

Bounds on one-shot classical capacity of quantum channels

Abstract

We use the smooth entropy approach to treat the problems of binary quantum hypothesis testing and the transmission of classical information through a quantum channel. We provide lower and upper bounds on the optimal type II error of quantum hypothesis testing in terms of the smooth max-relative entropy of the two states representing the two hypotheses. Using then a relative entropy version of the Quantum Asymptotic Equipartition Property (QAEP), we can recover the strong converse rate of the i.i.d. hypothesis testing problem in the asymptotics. On the other hand, combining Stein's lemma with our bounds, we obtain a stronger ($\ep$-independent) version of the relative entropy-QAEP. Similarly, we provide bounds on the one-shot $\ep$-error classical capacity of a quantum channel in terms of a smooth max-relative entropy variant of its Holevo capacity. Using these bounds and the $\ep$-independent version of the relative entropy-QAEP, we can recover both the Holevo-Schumacher-Westmoreland theorem about the optimal direct rate of a memoryless quantum channel with product state encoding, as well as its strong converse counterpart.