An asymptotic error bound for testing multiple quantum hypotheses

TL;DR

This paper extends the understanding of quantum hypothesis testing by establishing an asymptotic error bound for discriminating multiple quantum states, including mixed states, and proposes a universal detector nearly attaining this bound.

Contribution

It generalizes the attainability of the multiple quantum Chernoff bound to a broader class of states and introduces a universal detector with near-optimal performance.

Findings

Asymptotic error bound is attainable for pairwise linearly independent mixed states.

Constructed a universal detector that nearly attains the multiple quantum Chernoff bound.

Extended previous results from pure states to a larger class of mixed states.

Abstract

We consider the problem of detecting the true quantum state among $r$ possible ones, based of measurements performed on $n$ copies of a finite-dimensional quantum system. A special case is the problem of discriminating between $r$ probability measures on a finite sample space, using $n$ i.i.d. observations. In this classical setting, it is known that the averaged error probability decreases exponentially with exponent given by the worst case binary Chernoff bound between any possible pair of the $r$ probability measures. Define analogously the multiple quantum Chernoff bound, considering all possible pairs of states. Recently, it has been shown that this asymptotic error bound is attainable in the case of $r$ pure states, and that it is unimprovable in general. Here we extend the attainability result to a larger class of $r$-tuples of states which are possibly mixed, but pairwise linearly independent. We also construct a quantum detector which universally attains the multiple quantum Chernoff bound up to a factor 1/3.