Local asymptotic normality for finite dimensional quantum systems

Abstract

We extend our previous results on local asymptotic normality (LAN) for qubits, to quantum systems of arbitrary finite dimension $d$. LAN means that the quantum statistical model consisting of $n$ identically prepared $d$-dimensional systems with joint state ${\rho }^{\otimes n}$ converges as $n\to \infty$ to a statistical model consisting of classical and quantum Gaussian variables with fixed and known covariance matrix, and unknown means related to the parameters of the density matrix $\rho$. Remarkably, the limit model splits into a product of a classical Gaussian with mean equal to the diagonal parameters, and independent harmonic oscillators prepared in thermal equilibrium states displaced by an amount proportional to the off-diagonal elements. As in the qubits case, LAN is the main ingredient in devising a general two step adaptive procedure for the optimal estimation of completely unknown $d$-dimensional quantum states. This measurement strategy shall be described in a forthcoming paper.

64 pages

References in corpus (6)

• Experimental Quantum State Tomography of Optical Fields and Ultrafast Statistical Sampling
• Asymptotic Error Rates in Quantum Hypothesis Testing
• Efficient Quantum State Estimation by Continuous Weak Measurement and Dynamical Control
• Optimal estimation of qubit states with continuous time measurements
• Asymptotic performance of optimal state estimation in quantum two level system
• Quantum estimation and the quantum central limit theorem

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