New additivity properties of the relative entropy of entanglement and its generalizations

TL;DR

This paper establishes new additivity properties of the relative entropy of entanglement when at least one state belongs to specific classes, extending previous results and providing explicit formulas for various states.

Contribution

It proves additivity of the relative entropy of entanglement for certain classes of states and extends these results to related entanglement monotones based on the $ extit{α}$-$z$ Rénnyi relative entropy.

Findings

Additivity holds when at least one state is pure, maximally correlated, GHZ, Bell diagonal, isotropic, or Dicke.

Explicit formulas are derived for several classes of states.

Monotones based on quantum relative entropy are generally not additive for arbitrary states.

Abstract

We prove that the relative entropy of entanglement is additive when \emph{at least one of the two states} belongs to some specific class. We show that these classes include bipartite pure, maximally correlated, GHZ, Bell diagonal, isotropic, and generalized Dicke states. Previously, additivity was established only if \textit{both} states belong to the same class. Moreover, we extend these results to entanglement monotones based on the $\alpha$-$z$ R\'enyi relative entropy. Notably, this family of monotones includes also the generalized robustness of entanglement and the geometric measure of entanglement. In addition, we prove that any monotone based on a quantum relative entropy is not additive for general states. We also compute closed-form expressions of the monotones for bipartite pure, Bell diagonal, isotropic, generalized Werner, generalized Dicke, and maximally correlated Bell diagonal states. Our results rely on developing a method that allows us to recast the initial convex optimization problem into a simpler linear one. Even though we mostly focus on entanglement theory, we expect that some of our technical results could be useful in investigating more general convex optimization problems.