ar X iv : q ua nt - p h / 04 09 18 4 v 2 2 5 N ov 2 00 4 Sets of Mutually Unbiased Bases as Arcs in Finite Projective Planes ?
نویسنده
چکیده
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.
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ar X iv : q ua nt - p h / 04 09 18 4 v 1 2 7 Se p 20 04 Sets of Mutually Unbiased Bases as Arcs in Finite Projective Planes ?
This note is a short 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, d being a power of a prime, as an analogue of an arc in a (Desarguesian) projective plane of order d. Complete sets of MUBs thus correspond to (d+1)-arcs, i.e., ovals. The existence of two princ...
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