Asymptotic error rates in quantum hypothesis testing

We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the specified error probability tends to zero. This leads to the identification of the quantum gener-alisation of the classical Chernoff distance, which is the corresponding quantity in classical symmetric hypothesis testing, thereby solving a long standing open problem. The proof relies on a new trace inequality for pairs of positive operators as well as on a special mapping from pairs of density operators to pairs of probability distributions. These two new techniques have been introduced in [quant-ph/0610027] and [quant-ph/0607216], respectively. They are also well suited to prove the quantum generalisation of the Hoeffding bound, which is a modification of the Chernoff distance and specifies the optimal achievable asymptotic error rate in the contex