Approximation of multipartite quantum states and the relative entropy of entanglement

TL;DR

The paper develops a rigorous framework to extend multipartite entanglement measures to infinite-dimensional quantum systems. It introduces the FA-property-based finite-dimensional approximation and proves that many functionals, including the relative entropy of entanglement $E_R$ and its regularization $E_R^{\infty}$, behave well (lower semicontinuity and uniform continuity under energy constraints) on a large class of states 
$\mathfrak{S}_*(\mathcal{H}_{A_1...A_n})$. It also constructs a universal extension $\widehat{E}$ that yields a lower semicontinuous, convex entanglement monotone coinciding with $E$ on finite-dimensional marginals and aligns with $E_R$ on the infinite-dimensional subset; further, a finite-dimensional approximation property (FDA) is established, enabling transfer of finite-dimensional results to infinite dimensions. The paper further analyzes energy-constrained variants of $E_R$, providing conditions under which energy-limited definitions coincide with the standard ones, thereby offering practical tools for assessing entanglement under physically relevant energy limits.

Abstract

Special approximation technique for analysis of different characteristics of states of multipartite infinite-dimensional quantum systems is proposed and applied to study of the relative entropy of entanglement and its regularisation. We prove several results about analytical properties of the multipartite relative entropy of entanglement and its regularization (the lower semicontinuity on wide class of states, the uniform continuity under the energy constraints, etc.). We establish a finite-dimensional approximation property for the relative entropy of entanglement and its regularization that allows to generalize to the infinite-dimensional case the results proved in the finite-dimensional settings.