Classification of arbitrary-dimensional multipartite pure states under stochastic local operations and classical communication using the rank of coefficient matrix

TL;DR

This paper introduces a method for classifying multipartite pure states' entanglement under SLOCC using the rank of coefficient matrices, applicable to arbitrary dimensions, and identifies 22 entanglement classes in a specific four-partite system.

Contribution

It proposes a novel entanglement classification method based on coefficient matrix ranks for arbitrary-dimensional multipartite states, extending previous approaches.

Findings

Ranks of coefficient matrices are entanglement monotones.

Classified 22 SLOCC families in a 2x2x2x4 system.

Applicable to arbitrary-dimensional multipartite states.

Abstract

We study multipartite entanglement under stochastic local operations and classical communication (SLOCC) and propose the entanglement classification under SLOCC for arbitrary-dimensional multipartite ($n$-qudit) pure states via the rank of coefficient matrix, together with the permutation of qudits. The ranks of the coefficient matrices have been proved to be entanglement monotones. The entanglement classification of the $2 \otimes 2 \otimes 2 \otimes 4$ system is discussed in terms of the generalized method, and 22 different SLOCC families are found.