Optimal Error Exponents for Composite Sequential Quantum Hypothesis Testing

TL;DR

This paper introduces an adaptive mixture-sequential quantum hypothesis test that optimally distinguishes a null state from a set of alternatives, achieving the best possible error exponents under sample size constraints.

Contribution

It proposes a novel mixture-sequential quantum test, proves its optimality in error exponents, and establishes the fundamental sample complexity limits for composite SQHT.

Findings

Achieves optimal Type-I and Type-II error exponents characterized by minimal measured relative entropies.

Provides a matching converse, establishing the optimal error exponent region.

Shows that vanishing error probabilities require sample complexity at least as large as fixed-state sequential testing.

Abstract

We study the composite sequential quantum hypothesis testing (SQHT) problem, where the objective is to distinguish a null quantum state from a set of alternative quantum states. We propose a mixture-sequential quantum probability ratio test that adaptively selects measurements based on the current mixture estimate of the alternative set, and stops upon the first threshold crossing of the mixture log-likelihood ratio. Under an expected sample size constraint, we show that our proposed strategy simultaneously achieves the Type-I and (worst-case) Type-II error exponents, characterized by the minimal measured relative entropies between the null state and the alternative set. We further establish a matching converse, thereby characterizing the optimal error exponent region. Finally, our results show that achieving vanishing error probabilities in composite SQHT requires an expected sample complexity at least as large as that of sequential testing between two fixed states.