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Local Asymptotic Normality in Quantum Statistics

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Abstract

The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family \(\varphi_{\theta_{0}+u/\sqrt{n}}^{n}\) consisting of joint states of n identically prepared quantum systems approaches in a statistical sense a family of Gaussian state ϕ u of an algebra of canonical commutation relations. The convergence holds for all “local parameters” \(u\in {\mathbb{R}}^{m}\) such that \(\theta = \theta_{0} + u/\sqrt{n}\) parametrizes a neighborhood of a fixed point \(\theta_{0} \in \Theta \subset {\mathbb{R}}^{m}\) .

In order to prove the result we define weak and strong convergence of quantum statistical experiments which extend to the asymptotic framework the notion of quantum sufficiency introduces by Petz. Along the way we introduce the concept of canonical state of a statistical experiment, and investigate the relation between the two notions of convergence. For the reader’s convenience and completeness we review the relevant results of the classical as well as the quantum theory.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, U.K.

    Mădălin Guţă

  2. Mathematical Institute of the Slovak Academy of Sciences, Stefanikova 49, 814 73, Bratislava, Slovakia

    Anna Jenčová

Authors
  1. Mădălin Guţă
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  2. Anna Jenčová
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Correspondence to Mădălin Guţă.

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Communicated by M.B. Ruskai

Dedicated to Slava Belavkin on the occasion of his 60th anniversary

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Guţă, M., Jenčová, A. Local Asymptotic Normality in Quantum Statistics. Commun. Math. Phys. 276, 341–379 (2007). https://doi.org/10.1007/s00220-007-0340-1

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