# SLOCC classification of n qubits invoking the proportional relationships for spectrums and for standard Jordan normal forms

## TL;DR

This paper introduces a classification method for n-qubit states based on proportional relationships in spectrums and Jordan normal forms, leading to a detailed partitioning of states under SLOCC, especially for four qubits.

## Contribution

It develops a novel classification framework for n-qubit states using spectral and Jordan form relationships, enhancing understanding of SLOCC equivalence classes.

## Key findings

- States of n ≥ 4 qubits are partitioned into 12 groups and 34 families under SLOCC.
- The classification method is specifically effective for four qubits.
- Proportional relationships in spectrums and Jordan forms are key to the classification.

## Abstract

We investigate the proportional relationships for spectrums and for SJNFs (Standard Jordan Normal Forms) of the matrices constructed from coefficient matrices of two SLOCC (stochastic local operations and classical communication) equivalent states of $n$ qubits. Invoking the proportional relationships for spectrums and for SJNFs, pure states of $n$ ($\geq 4$) qubits are partitioned into 12 groups and 34 families under SLOCC, respectively. Specially, it is true for four qubits.