Bayesian parameter inference from continuously monitored quantum systems

TL;DR

This paper explores how likelihood functions and Fisher information can be applied to quantum systems under continuous measurement, enabling parameter estimation through stochastic master equations and MCMC sampling.

Contribution

It introduces a method to define likelihood functions and Fisher information in quantum measurement theory, connecting classical estimation tools to quantum dynamics.

Findings

Likelihood functions can be derived from quantum stochastic master equations.

Fisher information quantifies estimation error in quantum parameter inference.

Markov Chain Monte Carlo methods efficiently sample likelihoods for large parameter spaces.

Abstract

We review the introduction of likelihood functions and Fisher information in classical estimation theory, and we show how they can be defined in a very similar manner within quantum measurement theory. We show that the stochastic master equations describing the dynamics of a quantum system subject to a definite set of measurements provides likelihood functions for unknown parameters in the system dynamics, and we show that the estimation error, given by the Fisher information, can be identified by stochastic master equation simulations. For large parameter spaces we describe and illustrate the efficient use of Markov Chain Monte Carlo sampling of the likelihood function.