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Local Asymptotic Normality for Finite Dimensional Quantum Systems

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Abstract

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 \({\rho^{\otimes n}}\) converges as n → ∞ 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 ρ. 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].

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Authors and Affiliations

  1. Département de Mathématiques, Université Paris-Sud 11, Bât 425, 91405, Orsay Cedex, France

    Jonas Kahn

  2. School of Mathematical Sciences, University of Nottingham, University Park, NG7 2RD, Nottingham, UK

    Mădălin Guţă

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  1. Jonas Kahn
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  2. Mădălin Guţă
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Correspondence to Mădălin Guţă.

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Communicated by M. B. Ruskai

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Kahn, J., Guţă, M. Local Asymptotic Normality for Finite Dimensional Quantum Systems. Commun. Math. Phys. 289, 597–652 (2009). https://doi.org/10.1007/s00220-009-0787-3

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