Local Transformations of Multiple Multipartite States

TL;DR

This paper explores multi-state LOCC transformations, revealing a richer structure in entanglement manipulation for multipartite and bipartite states, including state conversion, catalysis, and non-additivity of entanglement.

Contribution

It introduces multi-state LOCC as an extension of LOCC, demonstrating its ability to change SLOCC classes, enable entanglement transfer, and uncover non-trivial transformations.

Findings

Multi-state LOCC can change SLOCC classes via local unitaries.

Entanglement transfer allows state conversions not possible in single-copy scenarios.

Source entanglement is non-additive in bipartite multi-state transformations.

Abstract

Understanding multipartite entanglement is vital, as it underpins a wide range of phenomena across physics. The study of transformations of states via Local Operations assisted by Classical Communication (LOCC) allows one to quantitatively analyse entanglement, as it induces a partial order in the Hilbert space. However, it has been shown that, for systems with fixed local dimensions, this order is generically trivial, which prevents relating multipartite states to each other with respect to any entanglement measure. In order to obtain a non-trivial partial ordering, we study a physically motivated extension of LOCC: multi-state LOCC. Here, one considers simultaneous LOCC transformations acting on a finite number of entangled pure states. We study both multipartite and bipartite multi-state transformations. In the multipartite case, we demonstrate that one can change the stochastic LOCC (SLOCC) class of the individual initial states by only applying Local Unitaries (LUs). We show that, by transferring entanglement from one state to the other, one can perform state conversions not possible in the single copy case; provide examples of multipartite entanglement catalysis; and demonstrate improved probabilistic protocols. In the bipartite case, we identify numerous non-trivial LU transformations and show that the source entanglement is not additive. These results demonstrate that multi-state LOCC has a much richer landscape than single-state LOCC.