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This paper deals with the problem of estimating the coupling constant$\theta$ of a mixing quantum Markov chain. For a repeated measurement on thechain's output we show that the outcomes' time average has an asymptoticallynormal (Gaussian) distribution, and we give the explicit expressions of itsmean and variance. In particular we obtain a simple estimator of $\theta$ whoseclassical Fisher information can be optimized over different choices ofmeasured observables. We then show that the quantum state of the outputtogether with the system, is itself asymptotically Gaussian and compute itsquantum Fisher information which sets an absolute bound to the estimationerror. The classical and quantum Fisher informations are compared in a simpleexample. In the vicinity of $\theta=0$ we find that the quantum Fisherinformation has a quadratic rather than linear scaling in output size, andasymptotically the Fisher information is localised in the system, while theoutput is independent of the parameter.

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