Discriminating quantum states: the multiple Chernoff distance

Understand

We consider the problem of testing multiple quantum hypotheses $\{\rho_1^{\otimes n},\ldots,\rho_r^{\otimes n}\}$, 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, $P_e=\exp\{-\xi n+o(n)\}$.
• However, this error exponent $\xi$ 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\l a's conjecture that $\xi=\min_{i\neq j}C(\rho_i,\rho_j)$.