Discriminating quantum states: the multiple Chernoff distance

We consider the problem of testing multiple quantum hypotheses $\{ρ_1^{\otimes n},\ldots,ρ_r^{\otimes n}\}$, 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, $P_e=\exp\{-ξn+o(n)\}$. However, this error exponent $ξ$ 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 $ξ=\min_{i\neq j}C(ρ_i,ρ_j)$. The right-hand side of this equality is called the multiple quantum Chernoff distance, and $C(ρ_i,ρ_j):=\max_{0\leq s\leq 1}\{-\log\operatorname{Tr}ρ_i^sρ_j^{1-s}\}$ has been previously identified as the optimal error exponent for testing two hypotheses, $ρ_i^{\otimes n}$ versus $ρ_j^{\otimes n}$. 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.