Maximum likelihood versus likelihood-free quantum system identification in the atom maser

Abstract

We consider the system identification problem of estimating a dynamical parameter of a Markovian quantum open system (the atom maser), by performing continuous time measurements in the system's output (outgoing atoms). Two estimation methods are investigated and compared. On the one hand, the maximum likelihood estimator (MLE) takes into account the full measurement data and is asymptotically optimal in terms of its mean square error. On the other hand, the 'likelihood-free' method of approximate Bayesian computation (ABC) produces an approximation of the posterior distribution for a given set of summary statistics, by sampling trajectories at different parameter values and comparing them with the measurement data via chosen statistics.Our analysis is performed on the atom maser model, which exhibits interesting features such as bistability and dynamical phase transitions, and has connections with the classical theory of hidden Markov processes. Building on previous results which showed that atom counts are poor statistics for certain values of the Rabi angle, we apply MLE to the full measurement data and estimate its Fisher information. We then select several correlation statistics such as waiting times, distribution of successive identical detections, and use them as input of the ABC algorithm. The resulting posterior distribution follows closely the data likelihood, showing that the selected statistics contain 'most' statistical information about the Rabi angle.

Fisher informations and local asymptotic normality for continuous-time quantum Markov processes

“…The following theorem establishes the local asymptotic normality of the joint system and output state. and let Ψ u t be the joint system-output state at time t, as defined in (19).…”

“…, L k,θ , all of which depend smoothly on an unknown parameter θ ∈ R. We assume that the dynamics is irreducible for θ = θ 0 . Let θ = θ 0 + u/ √ t be the local parametrisation around θ 0 and let Ψu t be the joint system-output state at time t, as defined in (19).…”

Equivalence Classes and Local Asymptotic Normality in System Identification for Quantum Markov Chains

“…A detailed study comparing the quantum Fisher information with classical Fisher informations of various counting statistics for the atom maser model can be found in [9,10]. The continuous time version of the local asymptotic normality result is discussed in [8].…”

Information geometry and local asymptotic normality for multi-parameter estimation of quantum Markov dynamics

“…According to the identification of the Lie algebra as (12), the range of this map fills the whole Lie algebra. Furthermore, we can easily verify the compatibility condition (14), and G-covariance (15) follows from the fact that…”

“…Our goal here is not to propose or analyse specific measurement and estimation schemes (see e.g. [43,20,21,14] for related results), but rather investigate the statistical properties of the output state, which will provide the ultimate limits in estimation precision. We envisage that the structure of the output state uncovered here will be relevant not only for designing efficient measurement schemes (cf.…”