Quantum entanglement and approximation by positive matrices

TL;DR

This paper presents an exact method for approximating Hermitian matrices with positive semi-definite matrices, applying it to determine quantum entanglement through an algorithm based on density matrix properties.

Contribution

It introduces a novel exact solution for approximating matrices and applies it to quantum entanglement detection, linking approximation to density matrix additivity.

Findings

Exact approximation of density matrices by positive semi-definite matrices is achievable.

The approximation method can determine whether a quantum state is entangled.

Additivity of the density matrix influences the approximation process.

Abstract

We give an exact solution to the nonlinear optimization problem of approximating a Hermitian matrix by positive semi-definite matrices. Our algorithm was then used to judge whether a quantum state is entangled or not. We show that the exact approximation of a density matrices by tensor product of positive semi-definite operators is determined by the additivity property of the density matrix.