Operational interpretation of the reverse sandwiched Renyi divergences in composite quantum hypothesis testing

TL;DR

This paper establishes the operational significance of reverse sandwiched Renyi divergences in composite quantum hypothesis testing, revealing their role in determining optimal discrimination exponents.

Contribution

It provides the first direct operational interpretation of reverse sandwiched Renyi divergence for composite hypotheses, contrasting with previous i.i.d. results.

Findings

Reverse sandwiched Renyi divergence governs Hoeffding exponents in composite hypothesis testing.

Stein regime is governed by reverse quantum relative entropy, giving it an operational meaning.

Changing from simple to composite hypotheses alters which quantum divergence determines discrimination limits.

Abstract

We study the Hoeffding regime of composite quantum hypothesis testing, in which each hypothesis is specified by a sequence of sets of quantum states. We establish quantum Hoeffding bounds under a set of structural assumptions, orthogonal to those of our previous framework. A notable consequence is the direct operational interpretation of the reverse sandwiched Renyi divergence for $\alpha \in (0,1)$: for the task of discriminating a thermal equilibrium state from a probe state subject to unknown dephasing in the energy eigenbasis, with free Hamiltonian evolution as a special case, the optimal Hoeffding exponent is given exactly by this divergence evaluated on a single copy of the system. The same task in the Stein regime is governed by the reverse quantum relative entropy, providing its operational interpretation as well. This behavior contrasts both with the simple independent and identically distributed (i.i.d.) setting, where the Petz Renyi divergence and the Umegaki relative entropy govern the Hoeffding and Stein exponents, respectively, and with many composite settings, where only regularized many-copy formulas are available. This finding reveals that passing from simple to composite hypotheses can fundamentally change which quantum divergence determines the operational limits of discrimination, and suggests a new avenue for seeking operational interpretations of quantum divergences by lifting simple hypotheses to richer composite scenarios.