Matrices of fidelities for ensembles of quantum states and the Holevo quantity
M. Fannes, Fernando de Melo, Wojciech Roga, Karol Życzkowski · Quantum Information and Computation · 2012
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 $\chi$ of the ensemble. In this work we combine geometrical and probabilistic aspects of the ensemble in order to obtain meaningful bounds for $\chi$. 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 $\chi$. Slightly weaker bounds are established for arbitrary ensembles. Finally, we investigate correlation matrices involving multi-state correlations in relation to the Holevo quantity.