Tripartite entanglement measure under local operations and classical communication

TL;DR

This paper investigates the properties of the concurrence fill as a measure of tripartite entanglement, demonstrating it is not an entanglement monotone under LOCC, and proposes a new reliable monotone for genuine tripartite entanglement.

Contribution

It reformulates the concurrence fill using known entanglement measures, shows its non-monotonicity under LOCC, and introduces a new monotone for genuine tripartite entanglement.

Findings

Concurrence fill can increase under LOCC on average.

Three-tangle is LOCC-monotone, bipartite concurrence may not be.

Proposes a new reliable monotone for tripartite entanglement.

Abstract

Multipartite entanglement is an indispensable resource in quantum communication and computation, however, it is a challenging task to faithfully quantify this global property of multipartite quantum systems. In this work, we study the concurrence fill, which admits a geometric interpretation to measure genuine tripartite entanglement for the three-qubit system in [S. Xie {\it et al.}, Phys. Rev. Lett. \textbf{127}. 040403 (2021)]. First, we use the well-known three-tangle and bipartite concurrence to reformulate this quantifier for all pure states. We then construct an explicit example to conclusively show the concurrence fill can be increased under local operation and classical communications (LOCCs) {\it on average}, implying it is not an entanglement monotone. Moreover, we give a simple proof of the LOCC-monotonicity of three-tangle and find that the bipartite concurrence and the squared can have distinct performances under the same LOCCs. Finally, we propose a reliable monotone to quantify genuine tripartite entanglement, which can also be easily generalised to the multipartite system. Our results shed light on studying genuine entanglement and also reveal the complex structure of multipartite systems.