hal-00705755, version 1

Rank penalized estimation of a quantum system

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

(2012-06-08)

• 1:  Laboratoire de Probabilités et Modèles Aléatoires (LPMA)
• http://www.proba.jussieu.fr/
  CNRS : UMR7599 – Université Pierre et Marie Curie [UPMC] - Paris VI – Université Paris VII - Paris Diderot France
• 2:  Centre de Recherche en Économie et Statistique (CREST)
• http://www.crest.fr/
  INSEE – École Nationale de la Statistique et de l'Administration Économique France
• 3:  Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA)
• http://umr-math.univ-mlv.fr/
  Université Paris-Est Marne-la-Vallée (UPEMLV) France
• 4:  CEntre de REcherches en MAthématiques de la DEcision (CEREMADE)
• http://www.ceremade.dauphine.fr/index.html
  CNRS : UMR7534 – Université Paris IX - Paris Dauphine Place du Maréchal de Lattre de Tassigny 75775 - Paris Cedex 16 France

Bibliographic reference

• Type of document: Documents without publication reference (Preprint)
• Subject:

  Mathematics/Statistics

  Statistics/Statistics Theory

• Title: Rank penalized estimation of a quantum system
• Abstract: 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.
• Fulltext language: English
• Production date: 2012-06-08
• Keyword(s): Rank-penalized matrix estimation – quantum tomography – quantum state – rank estimation – adaptive estimation – oracle inequalities – low rank matrix approximation.
• Classification: 62G05, 62P35, 81P50, 81P15, 81P16, 81P40, 62J07, 15A03
• ANR Project:

  Project Id

  ANR JC07 205763 "StatQuant" / ANR-09-BLAN-0128 "PARCIMONIE"

Attached file list to this document: 

TEX

papier7.tex(53 KB)

BoxplotOpnorm_m50_n4.eps(10.3 KB)

BoxplotOpnorm_m100et50_n4_rk4.eps(8.7 KB)

BoxplotOpnorm_m100_n4.eps(10 KB)

BoxplotOpnorm_m100_n5.eps(10.2 KB)

n4eigenvalues.eps(9.4 KB)

n5eigenvalues.eps(9.6 KB)

n6eigenvalues.eps(9.9 KB)

SuccesOfMethodsVSrank_m50n4.eps(9.9 KB)

SuccesOfMethodsVSrank_m100n4.eps(9.9 KB)

PDF

papier7.pdf(237.5 KB)

PS

papier7.ps(1 MB)

• hal-00705755, version 1
• http://hal.archives-ouvertes.fr/hal-00705755
• oai:hal.archives-ouvertes.fr:hal-00705755
• From: Pierre Alquier <>
• Submitted on: Friday, 8 June 2012 11:10:57
• Updated on: Friday, 8 June 2012 11:43:30