Local asymptotic normality for finite dimensional quantum systems

Details Local asymptotic normality for finite dimensional quantum systems Local asymptotic normality for finite dimensional quantum systems

2008-04-24

arxiv.org/abs/0804.3876v1

Commun. Math. Phys. 289, 597-652 (2009)

Physics

Quantum Physics

64 pages

Scientific paper

10.1007/s00220-009-0787-3

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.

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