# Strict entanglement monotonicity under local operations and classical communication

## TL;DR

This paper establishes conditions under which certain entanglement measures strictly decrease on average under LOCC, highlighting the fundamental nature of entanglement reduction in quantum systems.

## Contribution

It proves that convex roof extended entanglement monotones with strictly concave functions, as well as negativity and relative entropy of entanglement, are strict entanglement monotones.

## Key findings

- Convex roof extended entanglement monotones with strictly concave functions are SEMs.
- Negativity and relative entropy of entanglement are SEMs.
- If squashed entanglement can be obtained by an optimal extension, it is a SEM.

## Abstract

Entanglement monotone is defined as a convex measure of entanglement that does not increase on average under local operations and classical communication (LOCC). Here we call an entanglement monotone a strict entanglement monotone (SEM) if it decreases strictly on average under LOCC. We show that, for any convex roof extended entanglement monotone that on pure states is given by a function of the reduced states, if the function is strictly concave, then it is a SEM. Moreover, we prove that the negativity and the relative entropy of entanglement, which are not defined by the convex roof structure, are also SEMs. In addition, if the squashed entanglement could be obtained by some optimal extension, then it is a SEM as well. Our results imply that entanglement is strictly decreasing on average under LOCC.