Model selection for quantum homodyne tomography

TL;DR

This paper develops model selection methods for quantum tomography, applying penalized estimators and oracle inequalities to improve quantum state estimation and measurement calibration.

Contribution

It introduces novel model selection procedures with oracle inequalities for quantum homodyne tomography and extends these ideas to photocounter calibration.

Findings

Polynomial rate of convergence for non-parametric quantum state estimation

Oracle inequalities established for various estimators

Improved calibration accuracy with better photocounters

Abstract

This paper deals with a non-parametric problem coming from physics, namely quantum tomography. That consists in determining the quantum state of a mode of light through a homodyne measurement. We apply several model selection procedures: penalized projection estimators, where we may use pattern functions or wavelets, and penalized maximum likelihood estimators. In all these cases, we get oracle inequalities. In the former we also have a polynomial rate of convergence for the non-parametric problem. We finish the paper with applications of similar ideas to the calibration of a photocounter, a measurement apparatus counting the number of photons in a beam. Here the mathematical problem reduces similarly to a non-parametric missing data problem. We again get oracle inequalities, and better speed if the photocounter is good.