Attainability and Lower Semi-continuity of the Relative Entropy of Entanglement and Variations on the Theme

Abstract

The relative entropy of entanglement $$E_R$$ E R is defined as the distance of a multipartite quantum state from the set of separable states as measured by the quantum relative entropy. We show that this optimisation is always achieved, i.e. any state admits a closest separable state, even in infinite dimensions; also, $$E_R$$ E R is everywhere lower semi-continuous. We use this to derive a dual variational expression for $$E_R$$ E R in terms of an external supremum instead of infimum. These results, which seem to have gone unnoticed so far, hold not only for the relative entropy of entanglement and its multipartite generalisations, but also for many other similar resource quantifiers, such as the relative entropy of non-Gaussianity, of non-classicality, of Wigner negativity—more generally, all relative entropy distances from the sets of states with non-negative $$\lambda $$ λ -quasi-probability distribution. The crucial hypothesis underpinning all these applications is the weak*-closedness of the cone generated by free states, and for this reason, the techniques we develop involve a bouquet of classical results from functional analysis. We complement our analysis by giving explicit and asymptotically tight continuity estimates for $$E_R$$ E R and closely related quantities in the presence of an energy constraint.

On Lower Semicontinuity of the Quantum Conditional Mutual Information and Its Corollaries

“…By Lemma 1 the operator H 2 satisfies condition (10). Since (14) implies that TrH 2 ρ < +∞, the state ρ has the FA-property by the Theorem in [32, Section 2]. ✷ Remark 1.…”

“…It is shown in [14] that this infimum is always attained and that the state at which it is attained is unique provided that ρ is faithful state. If π is the finest partition {{1}, .…”

“…, n} described in Section 4. The faithful II n−1 n -type continuity bound for the function E ∞ R,π is presented in Proposition 39 in [14]. 9 3 Approximation of multipartite quantum states…”

“…valid for any states ρ and σ in S(H A 1 ...An ) and any p ∈ [0, 1], where h 2 is the binary entropy [41,14]. It is bounded above by the sum of any n − 1 marginal entropies, i.e.…”

“…The above result and its modifications are used in this article for analysis of the relative entropy of π-entanglement and its regularization in infinite-dimensional n-partite quantum systems (π is a partition of {1, 2, .., n}) [11,22,25,36,37]. In contrast to the recent articles [14,15] devoting mainly to the relative entropy of π-entanglement, in this article we focused on the properties of the regularized relative entropy of π-entanglement.…”