A note on continuous ensemble expansions of quantum states

TL;DR

This paper explores a generalized form of relative entropy for quantum states, linking it to state recognition and reconstruction, and demonstrates how ensembles of pure states can approximate a given quantum state with minimal noise.

Contribution

It introduces a new concept of a posteriori relative quantum entropy and shows its connection to quantum state reconstruction using ensemble expansions.

Findings

Ensembles of pure states with Gibbs distribution can approximate any quantum state.

The approximation can be made arbitrarily close by reducing white noise.

The paper generalizes the notion of relative entropy for quantum systems.

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.