Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme

• URL: http://arxiv.org/abs/2105.08091v3
• Date: Tue, 18 Jul 2023 09:06:24 GMT
• Title: Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme
• Authors: Ludovico Lami and Maksim E. Shirokov
• Abstract summary: The relative entropy of entanglement $E_Rite is defined as the distance of a multi-part quantum entanglement from the set of separable states as measured by the quantum relative entropy. We show that this state is always achieved, i.e. any state admits a closest separable state, even in dimensions; also, $E_Rite is everywhere lower semi-negative $lambda_$quasi-probability distribution.
• Score: 8.37609145576126
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The relative entropy of entanglement $E_R$ is defined as the distance of a multi-partite 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, $E_R$ is everywhere lower semi-continuous. We use this to derive a dual variational expression for $E_R$ 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 multi-partite generalisations, but also for many other similar resource quantifiers, such as the relative entropy of non-Gaussianity, of non-classicality, of Wigner negativity $\unicode{8212}$ more generally, all relative entropy distances from the sets of states with non-negative $\lambda$-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 $E_R$ and closely related quantities in the presence of an energy constraint.

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