Separation of unistochastic matrices from the double stochastic ones. Recovery of a 3 x 3 unitary matrix from experimental data

TL;DR

This paper develops a constructive method to distinguish unistochastic matrices from double stochastic matrices and recovers a 3x3 unitary matrix from experimental data, including criteria and a statistical test.

Contribution

It provides the first complete solution for the 3-dimensional case and introduces a $ ext{chi}^2$ test for reconstructing unitary matrices from noisy data.

Findings

Complete characterization of 3x3 unistochastic matrices

Necessary and sufficient criteria for separation from double stochastic matrices

A $ ext{chi}^2$ test for data reconstruction

Abstract

The aim of the paper is to provide a constructive method for recovering a unitary matrix from experimental data. Since there is a natural immersion of unitary matrices within the set of double stochastic ones, the problem to solve is to find necessary and sufficient criteria that separate the two sets. A complete solution is provided for the 3-dimensional case, accompanied by a $\chi^2$ test necessary for the reconstruction of a unitary matrix from error affected data.