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

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  = 2 n 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.

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