We investigate the ability of a quantum measurement device to discriminate two states or, generically, two hypothesis. In full generality, the measurement can be performed a number $n$ of times, and arbitrary pre-processing of the states and post-processing of the obtained data is allowed. Even if the two hypothesis correspond to orthogonal states, perfect discrimination is not always possible. There is thus an intrinsic error associated to the measurement device, which we aim to quantify, that limits its discrimination power. We minimize various error probabilities (averaged or constrained) over all pairs of $n$-partite input states. These probabilities, or their exponential rates of decrease in the case of large $n$, give measures of the discrimination power of the device. For the asymptotic rate of the averaged error probability, we obtain a Chernoff-type bound, dual to the standard Chernoff bound for which the state pair is fixed and the optimization is over all measurements. The key point in the derivation is that i.i.d. states become optimal in asymptotic settings. Minimum asymptotic rates are also obtained for constrained error probabilities, dual to Stein's Lemma and Hoeffding's bound. We further show that adaptive protocols where the state preparer gets feedback from the measurer do not improve the asymptotic rates. These rates thus quantify the ultimate discrimination power of a measurement device.