# Multipartite States under Elementary Local Operations

## TL;DR

This paper introduces a new method based on elementary local operations and a multipartite Gauss-Jordan elimination to classify multipartite pure states under SLOCC, revealing new equivalences and potential physical implications.

## Contribution

It develops a multipartite Gauss-Jordan elimination approach for SLOCC classification, extending previous methods and enabling analysis of complex states like LME and hypergraph states.

## Key findings

- Identifies new SLOCC equivalences among multipartite states
- Provides an analytical classification method for previously unknown states
- Suggests physical implications for Dicke-like structured states

## Abstract

Multipartite pure states are equivalent under Stochastic Local Operations and Classical Communication (SLOCC) whenever they can be mapped into one another by Invertible Local Operations. It is shown that this is equivalent to the inter-convertibility through finite sequences of Elementary Local Operations. A multipartite version of the Gauss-Jordan elimination strategy is then obtained, enabling the analytical SLOCC-classification of previously unknown examples. It is argued that the problem of SLOCC-classification is equivalent to the problem of classifying the Multipartite Fully Reduced Forms that the coefficient matrix of a state can assume after being subjected to this Gauss-Jordan elimination procedure. The method is applied to examples of so-called LME and hypergraph states, showing new SLOCC-equivalences. Moreover, possible physical implications for states with a Dicke-like structure are sketched.