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Viewing sets of mutually unbiased bases as arcs in finite projective planes

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  • Saniga, Metod
  • Planat, Michel

Abstract

This note is a short conceptual elaboration of the conjecture of Saniga et al. [J. Opt. B: Quantum Semiclass 6 (2004) L19–L20] by regarding a set of mutually unbiased bases (MUBs) in a d-dimensional Hilbert space as an analogue of an arc in a (finite) projective plane of order d. Complete sets of MUBs thus correspond to (d+1)-arcs, i.e., ovals. In the Desarguesian case, the existence of two principally distinct kinds of ovals for d=2n and n⩾3, viz. conics and non-conics, implies the existence of two qualitatively different groups of the complete sets of MUBs for the Hilbert spaces of corresponding dimensions. A principally new class of complete sets of MUBs are those having their analogues in ovals in non-Desarguesian projective planes; the lowest dimension when this happens is d=9.

Suggested Citation

  • Saniga, Metod & Planat, Michel, 2005. "Viewing sets of mutually unbiased bases as arcs in finite projective planes," Chaos, Solitons & Fractals, Elsevier, vol. 26(5), pages 1267-1270.
    • Handle: RePEc:eee:chsofr:v:26:y:2005:i:5:p:1267-1270
    DOI: 10.1016/j.chaos.2005.03.008
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