On quantum states with a finite-dimensional approximation property

TL;DR

This paper introduces the FA-property for quantum states, enabling finite-dimensional approximation of entropic and informational characteristics, with conditions for finiteness of von Neumann entropy and uniform continuity results.

Contribution

It defines the FA-property for quantum states, provides a simple sufficient condition for it, and demonstrates its implications for entropy finiteness and approximation of quantum information characteristics.

Findings

FA-property implies finiteness of von Neumann entropy

Uniform approximation results for quantum characteristics

Continuity of characteristics w.r.t. channel convergence

Abstract

We consider a class (convex set) of quantum states containing all finite rank states and infinite rank states with the sufficient rate of decreasing of eigenvalues (in particular, all Gaussian states). Quantum states from this class are characterized by the property (called the FA-property) that allows to obtain several results concerning finite-dimensional approximation of basic entropic and information characteristics of quantum systems and channels. We obtain a simple sufficient condition of the FA-property. We show that this property implies finiteness of the von Neumann entropy, but leave unsolved the question concerning the converse implication. We obtain uniform approximation results for characteristics depending on a pair (channel, input state) and for characteristics depending on a pair (channel, input ensemble). We establish the uniform continuity of the above characteristics as functions of a channel w.r.t. the strong convergence provided that the FA-property holds either for the input state or for the average state of input ensemble.