Tripartite entanglement transformations and tensor rank

TL;DR

This paper explores the complexity of transforming tripartite entangled states, revealing NP-hardness, the non-additivity of Schmidt rank, and connections to tensor rank and matrix multiplication algorithms.

Contribution

It establishes the NP-hardness of tripartite entanglement transformations, links tensor rank to entanglement, and improves known transformation rates between specific states.

Findings

Tripartite entanglement transformation is NP-hard.

Schmidt rank is not an additive entanglement measure.

Optimal matrix multiplication algorithms relate to entanglement conversion rates.

Abstract

Understanding the nature of multipartite entanglement is a central mission of quantum information theory. To this end, we investigate the question of tripartite entanglement convertibility. We find that there exists no easy criterion to determine whether a general tripartite transformation can be performed with a nonzero success probability and in fact, the problem is NP-hard. Our results are based on the connections between multipartite entanglement and tensor rank (also called Schmidt rank), a key concept in algebraic complexity theory. Not only does this relationship allow us to characterize the general difficulty in determining possible entanglement transformations, but it also enables us to observe the previously overlooked fact that {\em the Schmidt rank is not an additive entanglement measure}. As a result, we improve some best known transformation rates between specific tripartite entangled states. In addition, we find obtaining the most efficient algorithm for matrix multiplication to be precisely equivalent to determining the optimal rate of conversion between the Greenberger-Horne-Zeilinger state and a triangular distribution of three Einstein-Podolsky-Rosen states.