dorsal/arxiv
View SchemaLocal asymptotic normality in quantum statistics
| Authors | Madalin Guta, Anna Jencova |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0606213 |
| URL | https://arxiv.org/abs/quant-ph/0606213 |
| DOI | 10.1007/s00220-007-0340-1 |
| Journal | Communications in Mathematical Physics, 276, (2007), 341-379 |
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 phi_{u} of an algebra of canonical commutation relations. The convergence holds for all "local parameters" u\in R^{m} such that theta=theta_{0}+ u/sqrt{n} parametrizes a neighborhood of a fixed point theta_{0}\in Theta\subset 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 reader's convenience and completeness we review the relevant results of the classical as well as the quantum theory.
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"abstract": "The theory of local asymptotic normality for quantum statistical experiments\nis developed in the spirit of the classical result from mathematical statistics\ndue to Le Cam. Roughly speaking, local asymptotic normality means that the\nfamily varphi_{\\theta_{0}+ u/\\sqrt{n}}^{n} consisting of joint states of n\nidentically prepared quantum systems approaches in a statistical sense a family\nof Gaussian state phi_{u} of an algebra of canonical commutation relations. The\nconvergence holds for all \"local parameters\" u\\in R^{m} such that\ntheta=theta_{0}+ u/sqrt{n} parametrizes a neighborhood of a fixed point\ntheta_{0}\\in Theta\\subset R^{m}.\n In order to prove the result we define weak and strong convergence of quantum\nstatistical experiments which extend to the asymptotic framework the notion of\nquantum sufficiency introduces by Petz. Along the way we introduce the concept\nof canonical state of a statistical experiment, and investigate the relation\nbetween the two notions of convergence. For reader\u0027s convenience and\ncompleteness we review the relevant results of the classical as well as the\nquantum theory.",
"arxiv_id": "quant-ph/0606213",
"authors": [
"Madalin Guta",
"Anna Jencova"
],
"categories": [
"quant-ph"
],
"doi": "10.1007/s00220-007-0340-1",
"journal_ref": "Communications in Mathematical Physics, 276, (2007), 341-379",
"title": "Local asymptotic normality in quantum statistics",
"url": "https://arxiv.org/abs/quant-ph/0606213"
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