A smallest computable entanglement monotone

TL;DR

This paper establishes that the Rains relative entropy is the smallest computable selective entanglement monotone, strengthening its physical interpretation and providing bounds on distillable entanglement.

Contribution

It proves the Rains relative entropy is a selective entanglement monotone under certain operations, enhancing its theoretical significance and practical computability.

Findings

Rains relative entropy is monotone under positivity-preserving operations.

It is the smallest conservative computable entanglement monotone.

Provides upper bounds on probabilistic distillable entanglement.

Abstract

The Rains relative entropy of a bipartite quantum state is the tightest known upper bound on its distillable entanglement -- which has a crisp physical interpretation of entanglement as a resource -- and it is efficiently computable by convex programming. It has not been known to be a selective entanglement monotone in its own right. In this work, we strengthen the interpretation of the Rains relative entropy by showing that it is monotone under the action of selective operations that completely preserve the positivity of the partial transpose, reasonably quantifying entanglement. That is, we prove that Rains relative entropy of an ensemble generated by such an operation does not exceed the Rains relative entropy of the initial state in expectation, giving rise to the smallest, most conservative known computable selective entanglement monotone. Additionally, we show that this is true not only for the original Rains relative entropy, but also for Rains relative entropies derived from various R\'enyi relative entropies. As an application of these findings, we prove, in both the non-asymptotic and asymptotic settings, that the probabilistic approximate distillable entanglement of a state is bounded from above by various Rains relative entropies.