Discriminating quantum states: the multiple Chernoff distance

TL;DR

This paper proves that the error exponent for discriminating multiple quantum states is equal to the minimum pairwise quantum Chernoff distance, solving a long-standing open problem in quantum hypothesis testing.

Contribution

It establishes the exact error exponent for multiple quantum hypothesis testing as the minimum pairwise Chernoff distance, confirming a conjecture by Nussbaum and Szkoła.

Findings

Proved the conjecture that the error exponent equals the minimum pairwise Chernoff distance.

Developed a new upper bound for the average error probability in quantum state discrimination.

Provided an alternative proof for the binary quantum Chernoff distance's achievability.

Abstract

We consider the problem of testing multiple quantum hypotheses $\{\rho_1^{\otimes n},\ldots,\rho_r^{\otimes n}\}$, where an arbitrary prior distribution is given and each of the $r$ hypotheses is $n$ copies of a quantum state. It is known that the average error probability $P_e$ decays exponentially to zero, that is, $P_e=\exp\{-\xi n+o(n)\}$. However, this error exponent $\xi$ is generally unknown, except for the case that $r=2$. In this paper, we solve the long-standing open problem of identifying the above error exponent, by proving Nussbaum and Szko\l a's conjecture that $\xi=\min_{i\neq j}C(\rho_i,\rho_j)$. The right-hand side of this equality is called the multiple quantum Chernoff distance, and $C(\rho_i,\rho_j):=\max_{0\leq s\leq 1}\{-\log\operatorname{Tr}\rho_i^s\rho_j^{1-s}\}$ has been previously identified as the optimal error exponent for testing two hypotheses, $\rho_i^{\otimes n}$ versus $\rho_j^{\otimes n}$. The main ingredient of our proof is a new upper bound for the average error probability, for testing an ensemble of finite-dimensional, but otherwise general, quantum states. This upper bound, up to a states-dependent factor, matches the multiple-state generalization of Nussbaum and Szko\l a's lower bound. Specialized to the case $r=2$, we give an alternative proof to the achievability of the binary-hypothesis Chernoff distance, which was originally proved by Audenaert et al.