Asymptotic quantum Chernoff bound

Let $\rho$ and $\sigma$ be quantum states on a finite-dimensional system. Denote by $P_e(\rho^{\otimes n}, \sigma^{\otimes n})$ the minimum error probability for distinguishing their $n$-fold tensor powers under equal priors. Then \[ \lim_{n\to\infty} -\frac{1}{n} \log P_e(\rho^{\otimes n}, \sigma^{\otimes n}) = -\log \min_{0 \le s \le 1} \operatorname{Tr}(\rho^s \sigma^{1-s}). \]