Quantum non demolition measurements: Parameter estimation for mixtures of multinomials

Tristan Benoist,Clément Pellegrini,Fabrice Gamboa

doi:10.1214/18-ejs1396

Abstract

In Quantum Non Demolition measurements, the sequence of observations is distributed as a mixture of multinomial random variables. Parameters of the dynamics are naturally encoded into this family of distributions. We show the local asymptotic mixed normality of the underlying statistical model and the consistency of the maximum likelihood estimator. Furthermore, we prove the asymptotic optimality of this estimator as it saturates the usual Cram\'er Rao bound.

Highlights

Measuring directly a small quantum sized physical system is done by letting it interact with a macroscopic instrument
The goal is to infer the system state from the information obtained through this indirect measurement
From the laws of quantum mechanics, this procedure induces a back action on the system that may change its state

Summary

Introduction

Measuring directly a small quantum sized physical system is done by letting it interact with a macroscopic instrument. Statistical inference for QND measurement cannot fully rely on standard results on i.i.d. models. Our results rely on the fact that our model, thanks to the QND condition, is a mixture of i.i.d. statistical models. It has been show in [2, 3, 4, 5, 1] that the probability space describing these experiments can be divided into asymptotic events (belonging to the tail algebra) such that (Xn) conditioned to one of such asymptotic event is a sequence of i.i.d. random variables. Some numerical simulations illustrate our results on a QND toy model inspired by [8]

Mixture of multinomials

Local Asymptotic Mixed Normality

Maximum likelihood estimation

Applications to Quantum non-Demolition Measurement