Mutually unbiased bases and finite projective planes

Abstract

It is conjectured that the question of the existence of a set of d + 1 mutually unbiased bases in a ddimensional Hilbert space if d differs from a power of prime is intimatelly linked with the problem whether there exist projective planes whose order d is not a power of prime.Recently, there has been a considerable resurgence of interest in the concept of the so-called mutually unbiased bases [see, e.g., 1-7], especially in the context of quantum state determination, cryptography, quantum information theory and the King's problem. We recall that two different orthonormal bases A and B of a d-dimensional Hilbert space H d are called mutually unbiased if and only if | a|b | = 1/ √ d for all a∈A and all b∈B. An aggregate of mutually unbiased bases is a set of orthonormal bases which are pairwise mutually unbiased. It has been found that the maximum number of such bases cannot be greater than d + 1 [8,9]. It is also known that this limit is reached if d is a power of prime. Yet, a still unanswered question is if there are non-prime-power values of d for which this bound is attained. The purpose of this short note is to draw the reader's attention to the fact that the answer to this question may well be related with the (non-)existence of finite projective planes of certain orders.A finite projective plane is an incidence structure consisting of points and lines such that any two points lie on just one line, any two lines pass through just one point, and there exist four points, no three of them on a line [10]. From these properties it readily follows that for any finite projective plane there exists an integer d with the properties that any line contains exactly d + 1 points, any point is the meet of exactly d + 1 lines, and the number of points is the same as the number of lines, namely d 2 + d + 1. This integer d is called the order of the projective plane. The most striking issue here is that the order of known finite projective planes is a power of prime [10]. The question of which other integers occur as orders of finite projective planes remains one of the most challenging problems of contemporary mathematics. The only "no-go" theorem known so far in this respect is the Bruck-Ryser theorem [11] saying that there is no projective plane of order d if d − 1 or d − 2 is divisible by 4 and d is not the sum of two squares. Out of the first few non-prime-power numbers, this theorem rules out finite projective planes of order 6, 14, 21, 22, 30 and 33. Moreover, using massive computer calculations, it was proved by Lam [12] that there is no projective plane of order ten. It is surmised that the order of any projective plane is a power of a prime.¿From what has already been said it is quite tempting to hypothesize that the above described two problems are nothing but different aspects of one and the same problem. That is, we conjecture that non-existence of a projective plane of the given order d implies that there are less than d + 1 mutually unbiased bases (MUBs) in the corresponding H d , and vice versa. Or, slightly...

The Veldkamp Space of Two-Qubits

“…3, 4th row) of points/operators. This feature has also a very interesting aspect in connection with the conjecture relating the existence of mutually unbiased bases and finite projective planes raised in [10], because with each point x of W (2) there is associated a projective plane of order two (the Fano plane) whose points are the elements of x ⊥ and whose lines are the spans {u, v} ⊥⊥ , where u, v ∈ x ⊥ with u = v [4]. Identifying the Pauli operators of a two-qubit system with the points of the generalized quadrangle of order two led to the discovery of three distinguished subsets of the operators in terms of geometric hyperplanes of the quadrangle.…”

MUBs: From Finite Projective Geometry to Quantum Phase Enciphering

“…As already mentioned, we have recently conjectured [3] that the existence of the maximum set of MUBs in a given dimension d and that of a projective plane of the same dimension may well represent two aspects of one and the same problem. Perhaps the most serious backing of our surmise is found in a recent paper by Wootters [7].…”

SIC-POVMS and MUBS: Geometrical Relationships in Prime Dimension

“…There has been much interest in recent years in SIC-POVMs [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34] (symmetric informationally complete positive operator valued measures; citations in order of first appearance online or in print). SIC-POVMs have been constructed analytically in Hilbert space dimension d = 2-13, 15 and 19 (existence of analytic solutions for d = 11, 15 communicated to author privately [35]; for d = 15 also see ref.…”

“…MUBs [6,9,10,11,12,37,38,39,40,41,42,43,44,45,46,47] (mutually unbiased bases; only a representative selection of papers cited) have been the subject of intense investigation for a rather longer period of time. It is known that the number of MUBs in dimension d cannot exceed d+1, and that the maximum number of d+1 MUBs exist whenever d is a power of a prime number.…”