  Quantum detectors provide information about quantum systems by establishing
correlations between certain properties of those systems and a set of
macroscopically distinct states of the corresponding measurement devices. A
natural question of fundamental significance is how much information a quantum
detector can extract from the quantum system it is applied to. In the present
paper we address this question within a precise framework: given a quantum
detector implementing a specific generalized quantum measurement, what is the
optimal performance achievable with it for a concrete information readout task,
and what is the optimal way to encode information in the quantum system in
order to achieve this performance? We consider some of the most common
information transmission tasks - the Bayes cost problem (of which minimal error
discrimination is a special case), unambiguous message discrimination, and the
maximal mutual information. We provide general solutions to the Bayesian and
unambiguous discrimination problems. We also show that the maximal mutual
information has an interpretation of a capacity of the measurement, and derive
various properties that it satisfies, including its relation to the accessible
information of an ensemble of states, and its form in the case of a
group-covariant measurement. We illustrate our results with the example of a
noisy two-level symmetric informationally complete measurement, for whose
capacity we give analytical proofs of optimality. The framework presented here
provides a natural way to characterize generalized quantum measurements in
terms of their information readout capabilities.
