# Characterizing multipartite entanglement classes via higher-dimensional embeddings

## TL;DR

This paper establishes a correspondence between SLOCC witnesses and entanglement witnesses in an extended space, enabling new methods for classifying and detecting multipartite entanglement.

## Contribution

It introduces a one-to-one relation between SLOCC witnesses and entanglement witnesses, facilitating the derivation and analysis of entanglement classes in multipartite systems.

## Key findings

- Derived SLOCC witnesses from separability criteria
- Calculated overlaps between different SLOCC classes
- Classified entanglement in 2x3x3 systems

## Abstract

Witness operators are a central tool to detect entanglement or to distinguish among the different entanglement classes of multiparticle systems, which can be defined using stochastic local operations and classical communication (SLOCC). We show a one-to-one correspondence between general SLOCC witnesses and a class of entanglement witnesses in an extended Hilbert space. This relation can be used to derive SLOCC witnesses from criteria for full separability of quantum states; moreover, given SLOCC witness can be viewed as entanglement witnesses. As applications of this relation we discuss the calculation of overlaps between different SLOCC classes, and the SLOCC classification in $2\times 3\times 3$-dimensional systems.