Discriminating quantum states: the multiple Chernoff distance

We consider the problem of testing multiple quantum hypotheses \{ρ₁^{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_{ineq 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_{0leq sleq 1}\{-logoperatorname{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.