Quantum accessible information and classical entropy inequalities

Computing accessible information for an ensemble of quantum states is a basic problem in quantum information theory. We show that the optimality criterion recently obtained in [7], when applied to specific ensembles of states, leads to nontrivial tight lower bounds for the Shannon entropy that are discrete relatives of the famous log-Sobolev inequality. In this light, the hypothesis of globally information-optimal measurement for an ensemble of equiangular equiprobable states (quantum pyramids) put forward and numerically substantiated in [2] is reconsidered and the corresponding tight entropy inequalities are proposed. Via the optimality criterion, this suggests also a proof of the conjecture concerning globally information-optimal observables for quantum pyramids put forward in [2].