Renormalized von Neumann entropy with application to entanglement in genuine infinite dimensional systems

TL;DR

This paper introduces a renormalized von Neumann entropy suitable for infinite-dimensional quantum systems, ensuring finiteness and continuity, and extends entanglement comparison results from finite to infinite dimensions.

Contribution

It proposes a new renormalized entropy measure for infinite-dimensional systems using Fredholm determinants, preserving majorization properties and enabling extension of LOCC entanglement results.

Findings

Renormalized entropy is finite and continuous in infinite dimensions.

Majorization theory features are preserved under the new renormalization.

Extension of LOCC entanglement comparison results to infinite-dimensional systems.

Abstract

A renormalized version of the von Neumann quantum entropy (which is finite and continuous in general, infinite dimensional case) and which obeys several of the natural physical demands (as expected for a "good" measure of entanglement in the case of general quantum states describing bipartite and infinite-dimensional systems) is proposed. The renormalized quantum entropy is defined by the explicit use of the Fredholm determinants theory. To prove the main results on continuity and finiteness of the introduced renormalization the fundamental Grothendick approach, which is based on the infinite dimensional Grassmann algebra theory, is applied. Several features of majorization theory are preserved under then introduced renormalization as it is proved in this paper. This fact enables us to extend most of the known (mainly, in the context of two-partite, finite-dimensional quantum systems) results of the LOCC comparison theory to the case of genuine infinite-dimensional, two-partite quantum systems.