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Rank penalized estimation of a quantum system

dc.contributor.authorMeziani, Katia
HAL ID: 2110
dc.contributor.authorHebiri, Mohamed
dc.contributor.authorButucea, Cristina
dc.contributor.authorAlquier, Pierre
dc.date.accessioned2012-07-05T09:21:37Z
dc.date.available2012-07-05T09:21:37Z
dc.date.issued2013
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/9698
dc.language.isoenen
dc.subjectlow rank matrix approximationen
dc.subjectoracle inequalitiesen
dc.subjectadaptive estimationen
dc.subjectrank estimationen
dc.subjectquantum stateen
dc.subjectquantum tomographyen
dc.subjectRank-penalized matrix estimationen
dc.subject.ddc519en
dc.titleRank penalized estimation of a quantum systemen
dc.typeArticle accepté pour publication ou publié
dc.contributor.editoruniversityotherLaboratoire d'Analyse et de Mathématiques Appliquées (LAMA) http://umr-math.univ-mlv.fr/ Université Paris Est Marne-la-Vallée;France
dc.contributor.editoruniversityotherCentre de Recherche en Économie et Statistique (CREST) http://www.crest.fr/ INSEE – École Nationale de la Statistique et de l'Administration Économique;France
dc.contributor.editoruniversityotherLaboratoire de Probabilités et Modèles Aléatoires (LPMA) http://www.proba.jussieu.fr/ CNRS : UMR7599 – Université Paris VI - Pierre et Marie Curie – Université Paris VII - Paris Diderot;France
dc.description.abstractenWe 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.en
dc.relation.isversionofjnlnamePhysical Review. A, Atomic, Molecular and Optical Physics
dc.relation.isversionofjnlvol88
dc.relation.isversionofjnldate2013
dc.relation.isversionofjnlpagesn°032113
dc.relation.isversionofdoihttp://dx.doi.org/10.1103/PhysRevA.88.032113
dc.identifier.urlsitehttp://hal.archives-ouvertes.fr/hal-00705755en
dc.description.sponsorshipprivateouien
dc.relation.isversionofjnlpublisherAPS
dc.subject.ddclabelProbabilités et mathématiques appliquéesen


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