Tradeoff relations between accessible information, informational power, and purity

The accessible information and the informational power quantify the maximum amount of information that can be extracted from a quantum ensemble and by a quantum measurement, respectively. Here, we investigate the tradeoff between the accessible information (informational power, respectively) and the purity of the states of the ensemble (the elements of the measurement, respectively). Under any given lower bound on the purity, i) we compute the minimum informational power and show that it is attained by the depolarized uniformly-distributed measurement; ii) we give a lower bound on the accessible information. Under any given upper bound on the purity, i) we compute the maximum accessible information and show that it is attained by an ensemble of pairwise commuting states with at most two distinct non-null eigenvalues; ii) we give a lower bound on the maximum informational power. The present results provide, as a corollary, novel sufficient conditions for the tightness of the Jozsa-Robb-Wootters lower bound to the accessible information.