Tight lower bounds for Shannon entropy from “quantum pyramids”

Computing accessible information for an ensemble of quantum states is a basic problem in quantum information theory coming back to its origin and important in particular for applications in quantum cryptography. The problem is solved in some special cases but in general remains open. In this talk we show that recently obtained optimality criterion when applied to specific ensembles of states leads to nontrivial entropy inequalities that are discrete relatives of the log-Sobolev inequality. In this light, the hypothesis of an information-optimal measurement for an ensemble of equiangular equiprobable states – the quantum pyramids – is reconsidered and the corresponding tight entropy inequalities are proposed.