Local Asymptotic Normality for Finite Dimensional Quantum Systems

Previous results on local asymptotic normality (LAN) for qubits [16,19] are extended 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(circle times n) converges as n -> infinity 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 [16], 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 [18].