# Calculating Entanglement Eigenvalues for Non-Symmetric Quantum Pure States Based on the Jacobian Semidefinite Programming Relaxation Method

## TL;DR

This paper introduces a Jacobian semidefinite programming relaxation method to compute the entanglement eigenvalues of non-symmetric quantum pure states, facilitating the calculation of the geometric measure of entanglement.

## Contribution

It develops a novel relaxation method for polynomial optimization problems to efficiently compute the largest unitary eigenvalue of complex tensors in quantum states.

## Key findings

- Method successfully computes entanglement eigenvalues in numerical examples.
- Approach effectively converts complex tensor eigenvalue problems into real polynomial optimization.
- Demonstrates the feasibility of the Jacobian SDP relaxation for quantum entanglement measures.

## Abstract

The geometric measure of entanglement is a widely used entanglement measure for quantum pure states. The key problem of computation of the geometric measure is to calculate the entanglement eigenvalue, which is equivalent to computing the largest unitary eigenvalue of a corresponding complex tensor. In this paper, we propose a Jacobian semidefinite programming relaxation method to calculate the largest unitary eigenvalue of a complex tensor. For this, we first introduce the Jacobian semidefinite programming relaxation method for a polynomial optimization with equality constraint, and then convert the problem of computing the largest unitary eigenvalue to a real equality constrained polynomial optimization problem, which can be solved by the Jacobian semidefinite programming relaxation method. Numerical examples are presented to show the availability of this approach.