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

Abstract

This paper deals with the problem of estimating the coupling constant $\theta$ 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 $\theta$ 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 $\theta =0$ 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.

10 pages, 2 figures. final version

References in corpus (9)

• Scalable multi-particle entanglement of trapped ions
• Quantum teleportation between light and matter
• Experimental demonstration of quantum memory for light
• Efficient quantum state tomography
• Entanglement-free Heisenberg-limited phase estimation
• Quantum process tomography and Linblad estimation of a solid state qubit
• Process tomography of field damping and measurement of Fock state lifetimes by quantum non-demolition photon counting in a cavity
• Indirect Hamiltonian Identification through a small gateway
• Fast rate estimation of an unitary operation in SU(d)