Approximate and ensemble local entanglement transformations for multipartite states

TL;DR

This paper investigates approximate and ensemble local entanglement transformations in multipartite quantum states, revealing that multipartite entanglement exhibits richer transformation hierarchies and cannot always be deterministically achieved, unlike bipartite cases.

Contribution

It introduces a hierarchy of approximate transformations for multipartite states, showing that these are more complex and richer than bipartite cases, with new insights into ensemble and deterministic transformations.

Findings

Approximate transformations can outperform deterministic ones within the same SLOCC class.

Multipartite entanglement transformations form a non-collapsing hierarchy, unlike bipartite cases.

Optimal approximate transformations are not always deterministic, highlighting the complexity of multipartite entanglement.

Abstract

Understanding multipartite entanglement is a key goal in quantum information. Entanglement in pure states can be characterised by considering transformations under Local Operations assisted by Classical Communication (LOCC). However, it has been shown that, for $n\ge5$ parties, multipartite pure states are generically isolated, i.e., they can neither be reached nor transformed under LOCC. Nonetheless, in any real lab, one never deterministically transforms a pure initial state exactly to a pure target state. Instead, one transforms a mixed state near the initial state to an ensemble that is on average close to the target state. This motivates studying approximate LOCC transformations. After reviewing in detail the known results in the bipartite case, we present the gaps that remain open in the multipartite case. While the analysis of the multipartite setting is much more technically involved due to the existence of different SLOCC classes, certain features simplify in the approximate setting. In particular, we show that it is sufficient to consider pure initial states, that it is sufficient to consider LOCC protocols with finitely-many rounds of communication and that approximate transformations can be approximated by ensemble transformations within an SLOCC class. Then, we formally define a hierarchy of different forms of approximate transformations that are relevant from a physical point of view. Whereas this hierarchy collapses in the bipartite case, we show that this is not the case for the multipartite setting, which is fundamentally richer. To wit, we show that optimal multipartite approximate transformations are not generally deterministic, that ensemble transformations within an SLOCC class can achieve a higher fidelity than deterministic transformations within an SLOCC class, and that there are approximate transformations with no deterministic transformations nearby.