Evidence for and against Zauner's MUB conjecture in C^6

Abstract

The problem of finding provably maximal sets of mutually unbiased bases in $\CC^d$, for composite dimensions $d$ which are not prime powers, remains completely open. In the first interesting case,~$d=6$, Zauner predicted that there can exist no more than three MUBs. We explore possible algebraic solutions in~$d=6$ by looking at their~`shadows' in vector spaces over finite fields. The main result is that if a counter-example to Zauner's conjecture were to exist, then it would leave no such shadow upon reduction modulo several different primes, forcing its algebraic complexity level to be much higher than that of current well-known examples. In the case of prime powers~$q \equiv 5 \bmod 12$, however, we are able to show some curious evidence which --- at least formally --- points in the opposite direction. In $\CC^6$, not even a single vector has ever been found which is mutually unbiased to a set of three MUBs. Yet in these finite fields we find sets of three `generalised MUBs' together with an orthonormal set of four vectors of a putative fourth MUB, all of which lifts naturally to a number field.

9 $\times$ 4 = 6 $\times$ 6: Understanding the quantum solution to the Euler's problem of 36 officers

“…As there are no solutions to the classical problem of 36 officers of Euler, we cannot look for a quantum design (7) taking the bi-partite states |ψ ij in a product form as it was done for d = 3 in (6). Instead, one has to allow the officers to be entangled and use entangled states visualized by several chess figures occupying the same field of the chessboard.…”

“…Two permutation matrices that bring U Γ to P 1 U Γ P 2 = U Γ block , shown in Fig. D2, can be written as two vectors consisting of 36 integers, [6,2,36,24,13,29,22,10,32,27,1,17,31,26,3,19,23,9,18,5,12,33,28,16,11,34,25,15,20,7,14,30,8,4,35,21], 3,4,9,10,7,8,1,2,27,28,33,34,16,15,…”

“…Quantum combinatorial designs allow for various interpretations and offer several applications of quantum theory. For instance, mutually unbiased bases (MUBs) [3][4][5][6][7] and symmetric informationally complete positive operator valued measures (SIC POVMs) [2,[8][9][10][11] can provide schemes of quantum measurements with particularly appealing properties. A constellation of pure states of a simple system forms a quantum tdesign [15] if the average of any polynomial function of order t over the constellation agrees with the average over the entire space of pure quantum states with respect to the natural unitarily invariant measure.…”