Projective ring line encompassing two-qubits View Full Text


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Article Info

DATE

2008-06

AUTHORS

M. Saniga, M. Planat, P. Pracna

ABSTRACT

We find that the projective line over the (noncommutative) ring of 2×2 matrices with coefficients in GF(2) fully accommodates the algebra of 15 operators (generalized Pauli matrices) characterizing two-qubit systems. The relevant subconfiguration consists of 15 points, each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified one-to-one with the points such that their commutation relations are exactly reproduced by the underlying geometry of the points with the ring geometric notions of neighbor and distant corresponding to the respective operational notions of commuting and noncommuting. This remarkable configuration can be viewed in two principally different ways accounting for the basic corresponding 9+6 and 10+5 factorizations of the algebra of observables: first, as a disjoint union of the projective line over GF(2) × GF(2) (the “Mermin” part) and two lines over GF(4) passing through the two selected points that are omitted; second, as the generalized quadrangle of order two with its ovoids and/or spreads corresponding to (maximum) sets of five mutually noncommuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open unexpected possibilities for an algebro-geometric modeling of finite-dimensional quantum systems and completely new prospects for their numerous applications. More... »

PAGES

905

References to SciGraph publications

  • 2005-03. On distant-isomorphisms of projective lines in AEQUATIONES MATHEMATICAE
  • 2001-09. Generalized Flatland in THE MATHEMATICAL INTELLIGENCER
  • 2007-04. Projective line over the finite quotient ring GF(2)[x]/〈x3 − x〉 and quantum entanglement: Theoretical background in THEORETICAL AND MATHEMATICAL PHYSICS
  • 2000-12. Projective representations i. projective lines over rings in ABHANDLUNGEN AUS DEM MATHEMATISCHEN SEMINAR DER UNIVERSITÄT HAMBURG
  • 2007-05. Projective line over the finite quotient ring GF(2)[x]/〈x3 ™ x〉 and quantum entanglement: The Mermin “magic” square/pentagram in THEORETICAL AND MATHEMATICAL PHYSICS
  • 1998. A Geometrical Picture Book in NONE
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s11232-008-0076-x

    DOI

    http://dx.doi.org/10.1007/s11232-008-0076-x

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1041700739


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