Sets of Mutually Unbiased Bases as Arcs in Finite Projective Planes?

Details Sets of Mutually Unbiased Bases as Arcs in Finite Projective Planes? Sets of Mutually Unbiased Bases as Arcs in Finite Projective Planes?

2004-09-27

arxiv.org/abs/quant-ph/0409184v2

Chaos, Solitons and Fractals 26 (2005) 1267 - 1270

Physics

Quantum Physics

3 pages, no figures

Scientific paper

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 principally distinct kinds of ovals for d even and greater than four, 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.

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