Generalised quantum Sanov theorem revisited
Ludovico Lami
TL;DR
This work generalises quantum hypothesis testing by deriving a comprehensive formula for the Stein exponent when the null hypothesis is a general, weakly constrained set and the alternative is either composite i.i.d. or arbitrarily varying from a base set. The authors achieve a quantum-to-classical reduction via minimax measurements and leverage classical doubly-classical Stein results together with new quantum techniques to obtain a regularised, two-sided expression for the exponent. They prove a central result, Theorem "stronger_genq_Sanov_thm", and derive consequential corollaries for entanglement (SEP) and magic (STAB) testing, along with refinements for composite i.i.d. and arbitrarily varying hypotheses (Theorem "q_both_composite_iid_or_av_thm"). While the general formula involves regularisation, the work also clarifies the equivalence between arbitrarily varying and convex-hull composite i.i.d. models and highlights the broader operational relevance to resource testing in quantum information. Overall, the paper extends and simplifies previous generalised quantum Sanov results and provides a robust framework for analysing Stein exponents in highly general, physically relevant quantum-hypothesis-testing scenarios.
Abstract
Given two families of quantum states $A$ and $B$, called the null and the alternative hypotheses, quantum hypothesis testing is the task of determining whether an unknown quantum state belongs to $A$ or $B$. Mistaking $A$ for $B$ is a type I error, and vice versa for the type II error. In quantum Shannon theory, a fundamental role is played by the Stein exponent, i.e. the asymptotic rate of decay of the type II error probability for a given threshold on the type I error probability. Stein exponents have been thoroughly investigated -- and, sometimes, calculated. However, most currently available solutions apply to settings where the hypotheses simple (i.e. composed of a single state), or else the families $A$ and $B$ need to satisfy stringent constraints that exclude physically important sets of states, such as separable states or stabiliser states. In this work, we establish a general formula for the Stein exponent where both hypotheses are allowed to be composite: the alternative hypothesis $B$ is assumed to be either composite i.i.d. or arbitrarily varying, with components taken from a known base set, while the null hypothesis $A$ is fully general, and required to satisfy only weak compatibility assumptions that are met in most physically relevant cases -- for instance, by the sets of separable or stabiliser states. Our result extends and subsumes the findings of [BBH, CMP 385:55, 2021] (that we also simplify), as well as the 'generalised quantum Sanov theorem' of [LBR, arXiv:2408.07067]. The proof relies on a careful quantum-to-classical reduction via measurements, followed by an application of the results on classical Stein exponents obtained in [Lami, arXiv:today]. We also devise new purely quantum techniques to analyse the resulting asymptotic expressions.