Discriminating quantum states: The multiple Chernoff distance

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 minimal 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 Szkoła’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 $C(\rho_{i},\rho_{j}):=\max_{0\leq s\leq1}\{-\log\operatorname{Tr}\rho_{i}^{s}\rho_{j}^{1-s}\}$ 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 Szkoła’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.