HAL: hal-00705755, version 2

arXiv: 1206.1711

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v1 (2012-06-08)

v2 (2012-06-19)

Rank penalized estimation of a quantum system

Pierre Alquier 1, 2, Cristina Butucea 2, 3, Mohamed Hebiri 3, Katia Meziani 4

(2012-06-18)

We introduce a new method to reconstruct the quantum matrix $\bar{\rho}$ of a system of $n$-qubits and estimate its rank $d$ from data obtained by quantum state tomography measurements repeated $m$ times. The procedure consists in minimizing the risk of a linear estimator $\hat{\bar{\rho}}$ of $\rho$ penalized by given rank (from 1 to $2^n$), where $\hat{\bar{\rho}}$ is previously obtained by the moment method. We obtain simultaneously an estimator of the rank and the resulting state matrix associated to this rank. We establish an upper bound for the error of penalized estimator, evaluated with the Frobenius norm, which is of order $dn(3/4)^n /m$ and consistency for the estimator of the rank. The proposed methodology is computationnaly efficient and is illustrated with synthetic and real data sets.

1:	Laboratoire de Probabilités et Modèles Aléatoires (LPMA)
	CNRS : UMR7599 – Université Pierre et Marie Curie [UPMC] - Paris VI – Université Paris VII - Paris Diderot
2:	Centre de Recherche en Économie et Statistique (CREST)
	INSEE – École Nationale de la Statistique et de l'Administration Économique
3:	Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA)
	Université Paris-Est Marne-la-Vallée (UPEMLV) – Université Paris-Est Créteil Val-de-Marne (UPEC) – CNRS : UMR8050 – Fédération de Recherche Bézout
4:	CEntre de REcherches en MAthématiques de la DEcision (CEREMADE)
	CNRS : UMR7534 – Université Paris IX - Paris Dauphine

Subject

:

Mathematics/Statistics Statistics/Statistics Theory

Keyword(s): Rank-penalized matrix estimation – quantum tomography – quantum state – rank estimation – adaptive estimation – oracle inequalities – low rank matrix approximation.

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From: Pierre Alquier <>

Submitted on: Tuesday, 19 June 2012 16:24:43

Updated on: Sunday, 3 March 2013 15:16:40