# Entanglement classification via integer partitions

## TL;DR

This paper introduces a novel method for classifying entanglement in 4n-qubit systems by reducing the problem to integer partitions based on invariants of eigenvalues and Jordan normal forms, advancing quantum entanglement theory.

## Contribution

It establishes invariance properties of eigenvalue multiplicities and Jordan block sizes under SLOCC, enabling a classification approach via integer partitions for 4n-qubit states.

## Key findings

- Invariance of algebraic and geometric multiplicities under SLOCC
- Reduction of entanglement classification to integer partitions
- Application to 4n-qubit systems

## Abstract

In [M. Walter et al., Science 340, 1205, 7 June (2013)], they gave a sufficient condition for genuinely entangled pure states and discussed SLOCC classification via polytopes and the eigenvalues of the single-particle states. In this paper, for $4n$ qubits, we show the invariance of algebraic multiplicities (AMs) and geometric multiplicities (GMs) of eigenvalues and the invariance of sizes of Jordan blocks (JBs) of the coefficient matrices under SLOCC. We explore properties of spectra, eigenvectors, generalized eigenvectors, standard Jordan normal forms (SJNFs), and Jordan chains of the coefficient matrices. The properties and invariance permit a reduction of SLOCC classification of $4n$ qubits to integer partitions (in number theory) of the number $2^{2n}-k$ and the AMs.