Matrices of fidelities for ensembles of quantum states and the Holevo quantity

The entropy of the Gram matrix of a joint purification of an ensemble of K mixed states yields an upper bound for the Holevo information χ of the ensemble. In this work we combine geometrical and probabilistic aspects of the ensemble in order to obtain meaningful bounds for χ. This is done by constructing various correlation matrices involving fidelities between every pair of states from the ensemble. For K = 3 quantum states we design a matrix of root fidelities that is positive and the entropy of which is conjectured to upper bound χ. Slightly weaker bounds are established for arbitrary ensembles. Finally, we investigate correlation matrices involving multi-state correlations in relation to the Holevo quantity.