Continuity of the relative entropy of resource
Ludovico Lami, Maksim E. Shirokov
TL;DR
This work tackles the local continuity of the relative entropy of resource in infinite-dimensional quantum systems. It introduces a general convergence criterion (Theorem 1) for the relative entropy distance to a convex free set, requiring a weak*-closed cone and a convergent sequence of free states achieving a finite limit, and it derives corollaries that ensure stability under mixtures and quantum channels. The authors then apply this framework to multipartite entanglement, proving that the continuity of the relative entropy of entanglement and its pi-variants can be ensured under conditions tied to the continuity of the quantum mutual information $I(A_1:\cdots:\!A_m)_{\rho}$. The results provide practical, verifiable criteria for the continuity of resource measures in infinite dimensions and reveal a fundamental link between mutual information continuity and entanglement measure continuity.
Abstract
A criterion of local continuity of the relative entropy of resource -- the relative entropy distance to the set of free states -- is obtained. Several basic corollaries of this criterion are presented. Applications to the relative entropy of entanglement in multipartite quantum systems are considered. It is shown, in particular, that local continuity of any relative entropy of multipartite entanglement follows from local continuity of the quantum mutual information.