Discriminating quantum states: the multiple Chernoff distance

Ke Li

doi:10.1214/16-AOS1436

paper

Discriminating quantum states: the multiple Chernoff distance

arXiv:1508.06624 · doi:10.1214/16-AOS1436

Abstract

We consider the problem of testing multiple quantum hypotheses ${ρ_{1}^{\otimes n}, \dots, ρ_{r}^{\otimes n}}$ , 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, $P_{e} = exp {- ξ n + o (n)}$ . 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 $ξ = min_{i \neq = j} C (ρ_{i}, ρ_{j})$ . The right-hand side of this equality is called the multiple quantum Chernoff distance, and $C (ρ_{i}, ρ_{j}) := max_{0 \leq s \leq 1} {- lo g Tr ρ_{i}^{s} ρ_{j}^{1 - s}}$ 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.

v2: minor changes

Discriminating quantum states: the multiple Chernoff distance

Abstract

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