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

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 and n P 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. 2005 Elsevier Ltd. All rights reserved. It has for a long time been suspected but only recently fully recognized [1–4] that finite (projective and related) geometries may provide us with important clues for solving the problem of the maximum cardinality of MUBs for Hilbert spaces of finite dimensions d. It is well-known [5,6] that this number cannot be greater than d + 1 and that this limit is reached if d is a power of a prime. Yet, a still unanswered question is if there are non-prime-power values of d for which this bound is attained. On the other hand, the minimum number of MUBs was found to be three for all dimensions d P 2 [7]. Motivated by these facts, Saniga et al. [1] have conjectured that the question of the existence of the maximum, or complete, sets of MUBs in a d-dimensional Hilbert space if d differs from a prime power is intricately connected with the problem of whether there exist projective planes whose order d is not a power of a prime. This note aims at getting a deeper insight into this conjecture by introducing particular objects in a finite projective plane, the so-called ovals, which can be viewed as geometrical analogues of complete sets of MUBs. We shall start with a more general geometrical object of a projective plane, viz. a k-arc—a set of k points, no three of which are collinear [see, e.g. 8,9]. From the definition it immediately follows that k = 3 is the minimum cardinality of such an object. If one requires, in addition, that there is at least one tangent (a line meeting it in a single point only) at each of its points, then the maximum cardinality of a k-arc is found to be d + 1, where d is the order of the projective plane [8,9]; these (d + 1)-arcs are called ovals. It is striking to observe that such k-arcs in a projective plane of order d 0960-0779/$ see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.03.008 * Corresponding author. Tel.: +421 52 4467 866; fax: +421 52 4467 656. E-mail address: [email protected] (M. Saniga). 1268 M. Saniga, M. Planat / Chaos, Solitons and Fractals 26 (2005) 1267–1270 and MUBs of a d-dimensional Hilbert space have the same cardinality bounds. Can, then, individual MUBs (of a ddimensional Hilbert space) be simply viewed as points of some abstract projective plane (of order d) so that their basic combinatorial properties are qualitatively encoded in the geometry of k-arcs? A closer inspection of the algebraic geometrical properties of ovals suggests that this may indeed be the case. To this end in view, we shall first show that every proper (non-composite) conic in PG(2,d), a (Desarguesian) projective plane over the Galois field GF(d), is an oval. A conic is the curve of second order