On spurious fixed points in iterative maximum likelihood reconstruction for quantum tomography

TL;DR

This paper reveals that maximum likelihood iteration in quantum tomography can converge to spurious fixed points, challenging its reliability, and proposes a criterion to verify true solutions, also linking it to gradient descent methods.

Contribution

It identifies the existence of spurious fixed points in maximum likelihood quantum tomography and introduces a criterion to verify convergence to true solutions, connecting it to gradient descent.

Findings

Spurious fixed points can occur in maximum likelihood quantum tomography.

A criterion based on first order optimality conditions can verify true solutions.

The algorithm is equivalent to factorized gradient descent.

Abstract

Maximum likelihood iteration is one of the most commonly used reconstruction algorithms in quantum tomography. The main appeal of the method is that it is easy to implement and that it converges reliably to a physically meaningful density matrix in practice. Contradicting these practical observations, we will show that convergence to a true solution is not guaranteed in general by constructing examples for spurious fixed points. To deal with this newly found problem, we then provide a criterion based on first order optimality conditions to check if the result of the algorithm is indeed the desired solution. Furthermore, we generalize the algorithm and show that it is equivalent to factorized gradient descent.