The maximally entangled set of 4-qubit states

TL;DR

This paper characterizes the Maximally Entangled Set (MES) of 4-qubit states, revealing diverse transformation properties across classes and providing insights into LOCC entanglement manipulation for multipartite quantum systems.

Contribution

It extends the understanding of the MES to non-generic 4-qubit states, identifying classes with distinct LOCC transformation behaviors and deepening the theoretical framework of multipartite entanglement.

Findings

Most SLOCC classes behave similarly to generic states.

Identified three classes with unique properties: GHZ/W, isolated states, and a drastically different class.

Provides new insights into LOCC transformations among four-qubit states.

Abstract

Entanglement is a resource to overcome the natural restriction of operations used for state manipulation to Local Operations assisted by Classical Communication (LOCC). Hence, a bipartite maximally entangled state is a state which can be transformed deterministically into any other state via LOCC. In the multipartite setting no such state exists. There, rather a whole set, the Maximally Entangled Set of states (MES), which we recently introduced, is required. This set has on the one hand the property that any state outside of this set can be obtained via LOCC from one of the states within the set and on the other hand, no state in the set can be obtained from any other state via LOCC. Recently, we studied LOCC transformations among pure multipartite states and derived the MES for 3- and generic 4-qubit states. Here, we consider the non-generic 4-qubit states and analyze their properties regarding local transformations. We prove that most SLOCC classes show a similar behavior as the generic states, however we also identify three classes with very distinct properties. The first consists of the GHZ and W class, where any state can be transformed into some other state non-trivially. In particular, there exists no isolation. On the other hand, there also exist classes where all states are isolated. Last but not least we identify an additional class of states, whose transformation properties differ drastically from all the other classes. Our investigations do not only identify the most relevant classes of states for LOCC entanglement manipulation, but also reveal new insight into the similarities and differences between separable and LOCC transformations and enable the investigation of LOCC transformations among arbitrary four qubit states.