Multiple Quantum Hypothesis Testing: One-Shot Pairwise Bounds and Sharp Asymptotics

2026-06-04

AI Summary

This study addresses the problem of minimizing error probability in multi-hypothesis quantum state discrimination, with a focus on asymptotic performance in infinite-dimensional Hilbert spaces. By introducing a single-letter upper bound based on pairwise error sums and leveraging tools from quantum information theory—including trace harmonic means, Nussbaum–Szkola distributions, and both non-asymptotic and asymptotic analyses—the authors establish dimension-independent, tight upper bounds. Key contributions include confirming the conjecture by Audenaert and Mosonyi, removing the dimension-dependent factor in Li’s multi-quantum Chernoff bound, establishing for the first time the attainability of the multi-quantum Chernoff distance in infinite dimensions, and proving that the ratio between quantum and classical error probabilities in binary hypothesis testing does not exceed two.

Abstract

We consider Bayesian discrimination among multiple quantum states and establish a dimension-free one-shot upper bound on the minimum probability of error in terms of the sum of pairwise errors. This resolves a conjecture of Audenaert and Mosonyi [J. Math. Phys. 55 (2014)] and improves the multiple quantum Chernoff bound of Li [Ann. Statist. 44 (2016)] by removing its dimension-dependent prefactor. In the asymptotic many-copy regime, our bound proves the achievability of the multiple quantum Chernoff distance for arbitrary separable Hilbert spaces, thereby settling the previously open infinite-dimensional case, and further yields constant-factor sharp asymptotics for the optimal error probability. In binary quantum hypothesis testing, we prove that the minimum error probability is characterized, up to universal constants, by a trace harmonic-mean quantity. Consequently, the optimal binary quantum error probability is within a factor of two of the optimal classical error probability for the associated Nussbaum-Szkoła distributions, complementing the lower bound of Nussbaum and Szkoła [Ann. Statist. 37 (2009)].

Problem

• quantum hypothesis testing
• multiple quantum states
• error probability
• Chernoff bound
• asymptotics

Innovation

• quantum hypothesis testing
• multiple Chernoff bound
• dimension-free bound
• one-shot error probability
• sharp asymptotics