Local solutions of Maximum Likelihood Estimation in Quantum State Tomography

TL;DR

This paper proves that for convex negative log-likelihood functions in quantum state tomography, all local minima are global, ensuring reliable maximum likelihood estimation when using parameterized density matrices.

Contribution

It demonstrates that in convex cases, local minima in parameterized quantum state estimation are globally optimal, clarifying a key source of potential errors.

Findings

All local minima are global in convex negative log-likelihood functions.

Optimization traps in quantum state tomography are less problematic than previously thought.

Practical sources of errors are discussed, beyond local minima issues.

Abstract

Maximum likelihood estimation is one of the most used methods in quantum state tomography, where the aim is to reconstruct the density matrix of a physical system from measurement results. One strategy to deal with positivity and unit trace constraints is to parameterize the matrix to be reconstructed in order to ensure that it is physical. In this case, the negative log-likelihood function in terms of the parameters, may have several local minima. In various papers in the field, a source of errors in this process has been associated to the possibility that most of these local minima are not global, so that optimization methods could be trapped in the wrong minimum, leading to a wrong density matrix. Here we show that, for convex negative log-likelihood functions, all local minima of the unconstrained parameterized problem are global, thus any minimizer leads to the maximum likelihood estimation for the density matrix. We also discuss some practical sources of errors.