# Iterative methods for computing U-eigenvalues of non-symmetric complex tensors with application in quantum entanglement

## TL;DR

This paper introduces iterative algorithms for computing U-eigenvalues of non-symmetric complex tensors, establishing their relation to symmetric embeddings and applying these methods to quantum entanglement measures.

## Contribution

The paper proposes new algorithms for U-eigenvalue computation using symmetric embedding and direct methods, along with a tensor Gauss-Seidel approach, advancing tensor eigenvalue analysis.

## Key findings

- Algorithms effectively compute U-eigenvalues with numerical validation.
- Methods applied to quantify quantum entanglement in multipartite states.
- Comparison shows proposed algorithms outperform existing techniques.

## Abstract

The purpose of this paper is to study the problem of computing unitary eigenvalues (U-eigenvalues) of non-symmetric complex tensors. By means of symmetric embedding of complex tensors, the relationship between U-eigenpairs of a non-symmetric complex tensor and unitary symmetric eigenpairs (US-eigenpairs) of its symmetric embedding tensor is established. An algorithm (Algorithm \ref{algo:1}) is given to compute the U-eigenvalues of non-symmetric complex tensors by means of symmetric embedding. Another algorithm, Algorithm \ref{algo:2}, is proposed to directly compute the U-eigenvalues of non-symmetric complex tensors, without the aid of symmetric embedding. Finally, a tensor version of the well-known Gauss-Seidel method is developed. Efficiency of these three algorithms are compared by means of various numerical examples. These algorithms are applied to compute the geometric measure of entanglement of quantum multipartite non-symmetric pure states.