Local Asymptotic Normality and Optimal Estimation of low-rank Quantum Systems
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In classical statistics, a statistical experiment consisting of n i.i.d observations from d-dimensional multinomial distributions can be well approximated by a d-1 dimensional Gaussian distribution. In a quantum version of the result it has been shown that a collection of n qudits of full rank can be well approximated by a quantum system containing a classical part, which is a d-1 dimensional Gaussian distribution, and a quantum part containing an ensemble of d(d-1)/2 shifted thermal states. In this paper, we obtain a generalization of this result when the qudits are not of full rank. We show that when the rank of the qudits is r, then the limiting experiment consists of an r-1 dimensional Gaussian distribution and an ensemble of both shifted pure and shifted thermal states. We also outline a two-stage procedure for the estimation of the low-rank qudit, where we obtain an estimator which is sharp minimax optimal. For the estimation of a linear functional of the quantum state, we construct an estimator, analyze the risk and use quantum LAN to show that our estimator is also optimal in the minimax sense.
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