Discriminating quantum states: The multiple Chernoff distance

We consider the problem of testing multiple quantum hypotheses \ρ₁⊗ n,…,ρᵣ⊗ 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 Pₑ decays exponentially to zero, that is, Pₑ=exp\-ξ n+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=mini≠ jC(ρᵢ,ρⱼ). The right-hand side of this equality is called the multiple quantum Chernoff distance, and C(ρᵢ,ρⱼ):=max0≤ s≤1\-ρᵢˢρⱼ¹⁻ˢ\ has been previously identified as the optimal error exponent for testing two hypotheses, ρᵢ⊗ n versus ρⱼ⊗ 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.