Generalized quantum Stein's lemma for mixed sources

TL;DR

This paper extends the quantum Stein's lemma to mixed sources, characterizing the optimal error exponents in quantum hypothesis testing with composite hypotheses, especially for non-IID null hypotheses.

Contribution

It develops techniques for non-commutative quantum settings to characterize error exponents for mixed quantum sources, extending classical results to quantum hypothesis testing.

Findings

Characterizes the optimal type-II error exponent for mixed quantum sources.

Shows the characterization does not hold for fixed nonzero type-I error thresholds.

Provides a counterexample illustrating limitations of the generalized lemma.

Abstract

The generalized quantum Stein's lemma characterizes the optimal asymptotic exponent of the type-II error in quantum hypothesis testing for an independent and identically distributed (IID) null hypothesis against a composite alternative hypothesis. Classically, a probabilistic mixture of IID sources arises as a natural generalization of IID sources, and, in the non-composite setting, the optimal type-II error exponent in hypothesis testing for such classical mixed sources is known to be characterized concisely by the worst-case component of the mixture. In this work, we extend these foundational results to composite quantum hypothesis testing where the null hypothesis is a mixed source, i.e., a probabilistic mixture of IID quantum states, and the alternative hypothesis is composite as in the generalized quantum Stein's lemma. When the type-I error vanishes asymptotically, we characterize the optimal type-II error exponent of this composite quantum hypothesis testing problem in terms of the worst-case component of the mixture, by developing techniques for the non-commutative quantum setting inspired by the classical information-spectrum analysis. We also show that the analogous characterization does not hold in general for a fixed nonzero type-I error threshold, by providing a counterexample beyond the vanishing type-I error regime. These results clarify the applicability of the generalized quantum Stein's lemma to highly non-IID null hypotheses arising from arbitrary finite probabilistic mixtures of IID quantum states.