On Composite Quantum Hypothesis Testing

TL;DR

This paper generalizes quantum hypothesis testing to composite states, establishing a regularized quantum relative entropy formula for the asymptotic error exponent, with implications for quantum information measures.

Contribution

It extends quantum Stein's lemma to composite hypotheses, providing a regularized formula and conditions for single-letter expressions, including operational interpretations.

Findings

Asymptotic error exponent expressed as regularized quantum relative entropy.

Single-letter formulas obtained in specific settings.

Improved recoverability bounds for quantum mutual information.

Abstract

We extend quantum Stein's lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states $\rho^{\otimes n}$ against convex combinations of quantum states $\sigma^{\otimes n}$ can be written as a regularized quantum relative entropy formula. We prove that in general such a regularization is needed but also discuss various settings where our formula as well as extensions thereof become single-letter. This includes an operational interpretation of the relative entropy of coherence in terms of hypothesis testing. For our proof, we start from the composite Stein's lemma for classical probability distributions and lift the result to the non-commutative setting by using elementary properties of quantum entropy. Finally, our findings also imply an improved recoverability lower bound on the conditional quantum mutual information in terms of the regularized quantum relative entropy -- featuring an explicit and universal recovery map.