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EQUIVALENCE CLASSES AND LOCAL ASYMPTOTIC NORMALITY IN SYSTEM IDENTIFICATION FOR QUANTUM MARKOV CHAINS
"... Abstract. We consider the problems of identifying and estimating dynamical parameters of an ergodic quantum Markov chain, when only the stationary output is accessible for measurements. On the identifiability question, we show that the knowledge of the output state completely fixes the dynamics up t ..."
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Abstract. We consider the problems of identifying and estimating dynamical parameters of an ergodic quantum Markov chain, when only the stationary output is accessible for measurements. On the identifiability question, we show that the knowledge of the output state completely fixes the dynamics up to a ‘coordinate transformation’ consisting of a multiplication by a phase and a unitary conjugation of the Kraus operators. When the dynamics depends on an unknown parameter, we show that the latter can be estimated at the ‘standard ’ rate n−1/2, and give an explicit expression of the (asymptotic) quantum Fisher information of the output, which is proportional to the Markov variance of a certain ‘generator’. More generally, we show that the output is locally asymptotically normal, i.e. it can be approximated by a simple quantum Gaussian model consisting of a coherent state whose mean is related to the unknown parameter. As a consistency check we prove that a parameter related to the ‘coordinate transformation ’ unitaries, has zero quantum Fisher information. 1.
Fisher informations and local asymptotic normality for continuoustime quantum Markov processes
"... We consider the problem of estimating an arbitrary dynamical parameter of an quantum open system in the inputoutput formalism. For irreducible Markov processes, we show that in the limit of large times the systemoutput state can be approximated by a quantum Gaussian state whose mean is proportiona ..."
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We consider the problem of estimating an arbitrary dynamical parameter of an quantum open system in the inputoutput formalism. For irreducible Markov processes, we show that in the limit of large times the systemoutput state can be approximated by a quantum Gaussian state whose mean is proportional to the unknown parameter. This approximation holds locally in a neighbourhood of size t−1/2 in the parameter space, and provides an explicit expression of the asymptotic quantum Fisher information in terms of the Markov generator. Furthermore we show that additive statistics of the counting and homodyne measurements also satisfy local asymptotic normality and we compute the corresponding classical Fisher informations. The mathematical theorems are illustrated with the examples of a twolevel system and the atom maser. Our results contribute towards a better understanding of the statistical and probabilistic properties of the output process, with relevance for quantum control engineering, and the theory of nonequilibrium quantum open systems. 1