A note on continuous ensemble expansions of quantum states

Abstract

Generalizing the notion of relative entropy, the difference between a priori and a posteriori relative entropy for quantum systems is drawn. The former, known as quantum relative entropy, is associated with quantum states recognition. The latter -- a posteriori relative quantum entropy is shown to be related with state reconstruction due to the following property: given a density operator $\rho$, ensembles of pure states with Gibbs distribution with respect to the defined distance are proved to represent the initial state $\rho$ up to an amount of white noise (completely mixed state) which can be made arbitrary small.