On Quantum Statistical Inference

This paper attempts to develop a theory of sufficiency in the setting of non-commutative algebras parallel to the ideas in classical mathematical statistics. Sufficiency of a coarse-graining means that all information is extracted about the mutual relation of a given family of states. In the paper sufficient coarse-grainings are characterized in several equivalent ways and the non-commutative analogue of the factorization theorem is obtained. Among the applications the equality case for the strong subadditivity of the von Neumann entropy, the Imoto-Koashi theorem and exponential families are treated. The setting of the paper allows the underlying Hilbert space to be infinite dimensional.MSC: 46L53, 81R15, 62B05.

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“…The concept of sufficient measurement appeared also in [3]. For a non-commuting family of states, there is no sufficient measurement.…”

Section: Sufficient Statistic and Coarse-grainingmentioning

confidence: 99%

Sufficiency in Quantum Statistical Inference

Jenčová

Petz

2006

Commun. Math. Phys.

129

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“…Later we will further extend the theory to cover approximate sufficiency through the notion of convergence of quantum statistical experiments. For a review of the complementary theory of quantum statistical inference we refer to [5].…”

Section: Quantum Statistical Experimentsmentioning

confidence: 99%

“…In recent years there has been a renewed interest in the field [21,22,36,5] and the advances in quantum engineering have led to the first practical implementations of theoretical methods [1,16,43]. An illustrating example is that of quantum homodyne tomography [53,9,30], a measurement technique developed in quantum optics, which allows the estimation with arbitrary precision [2,7] of the state of a monochromatic beam of light, by repeatedly measuring a sufficiently large number of identically prepared beams [44,42,57].…”

Section: Introductionmentioning

confidence: 99%

Local Asymptotic Normality in Quantum Statistics

Guţǎ

Jenčová

2007

Commun. Math. Phys.

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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 ϕ n θ 0 +u/ √ n consisting of joint states of n identically prepared quantum systems approaches in a statistical sense a family of Gaussian state φu of an algebra of canonical commutation relations. The convergence holds for all "local parameters" u ∈ R m such that θ = θ0 + u/ √ n parametrizes a neighborhood of a fixed point θ0 ∈ Θ ⊂ 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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“…The interested reader can get further acquaintance with these concepts through the textbooks [13] and [15] or the review article [3].…”

Section: J Kahǹmentioning

confidence: 99%

Model selection for quantum homodyne tomography

Kahn

2009

ESAIM: PS

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Abstract. This paper deals with a non-parametric problem coming from physics, namely quantum tomography. That consists in determining the quantum state of a mode of light through a homodyne measurement. We apply several model selection procedures: penalized projection estimators, where we may use pattern functions or wavelets, and penalized maximum likelihood estimators. In all these cases, we get oracle inequalities. In the former we also have a polynomial rate of convergence for the non-parametric problem. We finish the paper with applications of similar ideas to the calibration of a photocounter, a measurement apparatus counting the number of photons in a beam. Here the mathematical problem reduces similarly to a non-parametric missing data problem. We again get oracle inequalities, and better speed if the photocounter is good.Résumé. Nous nous intéressonsà un problème de statistique non-paramétrique issu de la physique, et plus précisémentà la tomographie quantique, c'est-à-dire la détermination de l'état quantique d'un mode de la lumière via une mesure homodyne. Nous appliquons plusieurs procédures de sélection de modèles : des estimateurs par projection pénalisés, où on peut utiliser soit des fonctions motif, soit des ondelettes, et l'estimateur du maximum de vraisemblance pénalisé. Dans chaque cas, nous obtenons une inégalité oracle. Nous prouvonségalement une vitesse de convergence polynomiale pour ce problème non-paramétrique, pour les estimateurs par projection. Nous appliquons ensuite des idéesà la calibration d'un photocompteur, l'appareil dénombrant le nombre de photons dans un rayon lumineux. Le problème mathématique se réduit dans ce casà un problème non-paramétriquè a données manquantes. Nous obtenonsà nouveau des inégalités oracle, qui nous assurent des vitesses de convergence d'autant meilleures que le photocompteur est bon.Mathematics Subject Classification. 62G05, 81V80, 62P35.

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On Quantum Statistical Inference

Cited by 167 publications

References 81 publications

Sufficiency in Quantum Statistical Inference

Sufficiency in Quantum Statistical Inference

Local Asymptotic Normality in Quantum Statistics

Model selection for quantum homodyne tomography

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