Transformations among Pure Multipartite Entangled States via Local Operations Are Almost Never Possible

TL;DR

This paper shows that almost all pure multipartite entangled states cannot be transformed into each other via local operations and classical communication, indicating that most such states are isolated and cannot be deterministically converted.

Contribution

It provides a comprehensive characterization of LOCC transformations among pure multipartite states, revealing their near impossibility for generic, fully entangled states.

Findings

Most multipartite pure states are isolated and cannot be transformed into each other.

Nontrivial LOCC transformations among generic states are almost never possible.

States generally have no nontrivial local symmetries.

Abstract

Local operations assisted by classical communication (LOCC) constitute the free operations in entanglement theory. Hence, the determination of LOCC transformations is crucial for the understanding of entanglement. We characterize here almost all LOCC transformations among pure multipartite multilevel states. Combined with the analogous results for qubit states shown by Gour \emph{et al.} [J. Math. Phys. 58, 092204 (2017)], this gives a characterization of almost all local transformations among multipartite pure states. We show that nontrivial LOCC transformations among generic, fully entangled, pure states are almost never possible. Thus, almost all multipartite states are isolated. They can neither be deterministically obtained from local-unitary-inequivalent (LU-inequivalent) states via local operations, nor can they be deterministically transformed to pure, fully entangled LU-inequivalent states. In order to derive this result, we prove a more general statement, namely, that, generically, a state possesses no nontrivial local symmetry. We discuss further consequences of this result for the characterization of optimal, probabilistic single copy and probabilistic multi-copy LOCC transformations and the characterization of LU-equivalence classes of multipartite pure states.