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

Understand

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.