Discriminating quantum states: the multiple Chernoff distance

We consider the problem of testing multiple quantum hypotheses {ρ 1 , . . . , ρ r }, 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{−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 Szko la’s conjecture that ξ = mini6=j C(ρi, ρj). The right-hand side of this equality is called the multiple quantum Chernoff distance, and C(ρi, ρj) := max0≤s≤1{− logTr ρ s iρ 1−s j } has been previously identified as the optimal error exponent for testing two hypotheses, ρ i versus ρ ⊗n j . 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 la’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. Email: [email protected] Supported by NSF grants CCF-1110941 and CCF-1111382. AMS 2000 subject classifications. 62P35, 62G10.