A resource theory of entanglement with a unique multipartite maximally entangled state

TL;DR

This paper develops alternative resource theories for multipartite entanglement, establishing a meaningful partial order and identifying a unique maximally entangled state, the generalized GHZ state, unlike traditional LOCC-based theories.

Contribution

It introduces new resource theories relaxing LOCC constraints, demonstrating they are non-trivial and possess a unique maximally entangled state in the multipartite setting.

Findings

Both theories are non-trivial with meaningful partial orders.

Existence of a unique maximally entangled state, the generalized GHZ state.

Any state can be transformed into any other within these theories.

Abstract

Entanglement theory is formulated as a quantum resource theory in which the free operations are local operations and classical communication (LOCC). This defines a partial order among bipartite pure states that makes it possible to identify a maximally entangled state, which turns out to be the most relevant state in applications. However, the situation changes drastically in the multipartite regime. Not only do there exist inequivalent forms of entanglement forbidding the existence of a unique maximally entangled state, but recent results have shown that LOCC induces a trivial ordering: almost all pure entangled multipartite states are incomparable (i.e. LOCC transformations among them are almost never possible). In order to cope with this problem we consider alternative resource theories in which we relax the class of LOCC to operations that do not create entanglement. We consider two possible theories depending on whether resources correspond to multipartite entangled or genuinely multipartite entangled (GME) states and we show that they are both non-trivial: no inequivalent forms of entanglement exist in them and they induce a meaningful partial order (i.e. every pure state is transformable to more weakly entangled pure states). Moreover, we prove that the resource theory of GME that we formulate here has a unique maximally entangled state, the generalized GHZ state, which can be transformed to any other state by the allowed free operations.