Behavior of the maximum likelihood in quantum state tomography

TL;DR

This paper investigates how the positivity constraint in quantum state tomography affects the behavior of the maximum likelihood estimator and introduces a new theoretical framework to improve model selection methods.

Contribution

It introduces metric-projected local asymptotic normality and a new Wilks theorem replacement for quantum state space, enhancing model selection accuracy.

Findings

Quantum state space satisfies metric-projected local asymptotic normality.

A new Wilks theorem replacement is derived for quantum state tomography.

Results clarify the impact of positivity constraints on state estimation.

Abstract

Quantum state tomography on a d-dimensional system demands resources that grow rapidly with d. They may be reduced by using model selection to tailor the number of parameters in the model (i.e., the size of the density matrix). Most model selection methods typically rely on a test statistic and a null theory that describes its behavior when two models are equally good. Here, we consider the loglikelihood ratio. Because of the positivity constraint $\rho \geq 0$, quantum state space does not generally satisfy local asymptotic normality, meaning the classical null theory for the loglikelihood ratio (the Wilks theorem) should not be used. Thus, understanding and quantifying how positivity affects the null behavior of this test statistic is necessary for its use in model selection for state tomography. We define a new generalization of local asymptotic normality, metric-projected local asymptotic normality, show that quantum state space satisfies it, and derive a replacement for the Wilks theorem. In addition to enabling reliable model selection, our results shed more light on the qualitative effects of the positivity constraint on state tomography.