A lower bound of Chernoff type for symmetric quantum hypothesis testing

We consider symmetric hypothesis testing, where the hypotheses are allowed to be arbitrary density operators in a finite dimensional unital $C^{*}$-algebra capturing the classical and quantum scenarios simultaneously. We prove a Chernoff type lower bound for the asymptotically achievable error exponents. In the case of commuting density operators it coincides with the classical Chernoff bound. Moreover, the bound turns out to be tight in some non-commutative special cases, too. The general attainability of the bound is still an open problem