Fisher information and asymptotic normality in system identification for quantum Markov chains

Abstract

This paper deals with the problem of estimating the coupling constant θ of a mixing quantum Markov chain. For a repeated measurement on the chain's output we show that the outcomes' time average has an asymptotically normal (Gaussian) distribution, and we give the explicit expressions of its mean and variance. In particular we obtain a simple estimator of θ whose classical Fisher information can be optimized over different choices of measured observables. We then show that the quantum state of the output together with the system, is itself asymptotically Gaussian and compute its quantum Fisher information which sets an absolute bound to the estimation error. The classical and quantum Fisher informations are compared in a simple example. In the vicinity of θ = 0 we find that the quantum Fisher information has a quadratic rather than linear scaling in output size, and asymptotically the Fisher information is localised in the system, while the output is independent of the parameter.

System Identification for Passive Linear Quantum Systems

“…[8], [9], [10], [11], [12], [13], [14], [15], [16] for a shortlist of recent results. Further, detailed statistical analysis for some dynamical quantum identification problems have been demonstrated [17], [18], [19], [20].…”

Asymptotic inference in system identification for the atom maser

“…In particular, it is not clear what is the optimal measurement, what is the quantum Fisher information of the output and how it compares with the Fisher information of simple (counting) measurements. These questions were partly answered in Guţȃ [11] in the context of a discrete time quantum Markov chain, and here we extend the results to a continuous time set-up with an infinite-dimensional system. For a better grasp of the statistical model, we consider several thought and real experiments, and compute the Fisher information of the data collected in these experiments.…”

“…Here, we extend these results further to the atom maser, which is a continuous time Markov process with an infinite-dimensional system. We refer to Guţȃ [11] for more details on the physical and statistical interpretation of the results.…”

“…Our contribution is to go beyond the usual set-up of repeated measurements of identically prepared systems, or that of process tomography, and look at estimation in the quantum Markov set-up. The first step in this direction was made in Guţȃ [11], which deals with asymptotics of system identification in a discrete time setting with finitedimensional systems. Here, we extend these ideas to the atom maser, including the effect of the thermal bath.…”

“…quantum statistical models, and can be used to find asymptotically optimal measurement strategies for parameter estimation problems. In Guţȃ [11], these notions were extended to the non-i.i.d. framework of a quantum Markov chain with finite-dimensional systems.…”