Classification of the Entangled states L\times N\times N

TL;DR

This paper introduces a comprehensive classification scheme for tripartite entangled states in systems of dimensions L×N×N, combining matrix canonical forms and symmetry analysis, with a specific example for 3×N×N systems.

Contribution

It provides a novel, systematic method for classifying tripartite entangled states under stochastic local operations and classical communication.

Findings

Developed a classification scheme based on matrix similarity transformations.

Applied the scheme to classify entanglement in 3×N×N systems.

Demonstrated the method with a concrete example.

Abstract

We presented a general classification scheme for the tripartite $L\times N\times N$ entangled system under stochastic local operation and classical communication. The whole classification procedure consists of two correlated parts: the simultaneous similarity transformation of a commuting matrix pair into a canonical form and the study of internal symmetry of parameters in the canonical form. Based on this scheme, a concrete example of entanglement classification for a $3\times N\times N$ system is given.