Geometry of contextuality from Grothendieck’s coset space

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• 作者：Michel Planat
• 关键词：Quantum contextuality ; Dessins d’enfants ; Point/line geometries ; Coset space ; 11G32 ; 81P13 ; 81P45 ; 51A45 ; 14H57 ; 81Q35
• 刊名：Quantum Information Processing
• 出版年：2015
• 出版时间：July 2015
• 年：2015
• 卷：14
• 期：7
• 页码：2563-2575
• 全文大小：791 KB
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• 作者单位：Michel Planat (1)
  
  1. CNRS, Institut FEMTO-ST, 15 B Avenue des Montboucons, 25044, Besan?on, France
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Physics
  Mathematics
  Engineering, general
  Computer Science, general
  Characterization and Evaluation Materials
  
• 出版者：Springer Netherlands
• ISSN：1573-1332

文摘

The geometry of cosets in the subgroups \(H\) of the two-generator free group \(G=\langle a,b\rangle \) nicely fits, via Grothendieck’s dessins d’enfants, the geometry of commutation for quantum observables. In previous work, it was established that dessins stabilize point-line geometries whose incidence structure reflects the commutation of (generalized) Pauli operators. Now we find that the nonexistence of a dessin for which the commutator \((a,b)=a^{-1}b^{-1}ab\) precisely corresponds to the commutator of quantum observables \([\mathcal {A},\mathcal {B}] = \mathcal {A}\mathcal {B}-\mathcal {B}\mathcal {A}\) on all lines of the geometry is a signature of quantum contextuality. This occurs first at index \(|G\):\(H|=9\) in Mermin’s square and at index \(10\) in Mermin’s pentagram, as expected. Commuting sets of \(n\)-qubit observables with \(n>3\) are found to be contextual as well as most generalized polygons. A geometrical contextuality measure is introduced.