1. Orthogonalization of Positive Operator Valued Measures
- Author
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de La Salle, Mikael, Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), École normale supérieure - Lyon (ENS Lyon)-Centre National de la Recherche Scientifique (CNRS), ANR-16-CE40-0022,AGIRA,Actions de Groupes, Isométries, Rigidité et Aléa(2016), ANR-19-CE40-0002,ANCG,Analyse non commutative sur les groupes et les groupes quantiques(2019), ANR-10-LABX-0070,MILYON,Community of mathematics and fundamental computer science in Lyon(2010), École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS), de la salle, mikael, Actions de Groupes, Isométries, Rigidité et Aléa - - AGIRA2016 - ANR-16-CE40-0022 - AAPG2016 - VALID, Analyse non commutative sur les groupes et les groupes quantiques - - ANCG2019 - ANR-19-CE40-0002 - AAPG2019 - VALID, and Community of mathematics and fundamental computer science in Lyon - - MILYON2010 - ANR-10-LABX-0070 - LABX - VALID
- Subjects
Quantum Physics ,Mathematics::Operator Algebras ,General Mathematics ,[MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA] ,[MATH.MATH-FA] Mathematics [math]/Functional Analysis [math.FA] ,Mathematics - Operator Algebras ,FOS: Physical sciences ,Computer Science::Computational Complexity ,[MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA] ,[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] ,FOS: Mathematics ,[MATH.MATH-OA] Mathematics [math]/Operator Algebras [math.OA] ,Operator Algebras (math.OA) ,Quantum Physics (quant-ph) ,[MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR] - Abstract
We show that a partition of the unity (or POVM) on a Hilbert space that is almost orthogonal is close to an orthogonal POVM in the same von Neumann algebra. This generalizes to infinite dimension previous results in matrix algebras by Kempe-Vidick and Ji-Natarajan-Vidick-Wright-Yuen. Quantitatively, our result are also finer, as we obtain a linear dependance, which is optimal. We also generalize to infinite dimension a duality result between POVMs and minimal majorants of finite subsets in the predual of a von Neumann algebra., Comment: 14 pages; v2: improvements in the presentation and small mistakes corrected. To appear in the open access journal Comptes Rendus - Math\'ematique
- Published
- 2022
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