A note on continuous ensemble expansions of quantum states

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 introduced and shown to be related with state reconstruction due to the following property: given a density operator ρ, ensembles of pure states with Gibbs distribution with respect to the defined distance are proved to represent the initial state ρ up to an amount of white noise (completely mixed state) which can be made arbitrary small. In classical probability the relative entropy (or Kullback-Leibler distance) S(ρ||σ) of a distribution ρ = {p1, . . . , pn} with respect to another distribution σ = {q1, . . . , qn} is defined as S(ρ||σ) = { − ∑ k pk log ( pk qk ) if supp ρ ⊆ suppσ +∞ otherwise (1) Usually, this notion is generalized in quantum information theory (see, e.g. [5] for a review) by analogy with von Neumann entropy, namely, the sum is replaced by operator trace. For two density operators ρ and σ the quantum relative entropy S(ρ||σ) reads: S(ρ||σ) = 