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The problem of finding provably maximal sets of mutually unbiased bases in$\mathbb{C}^d$, for composite dimensions $d$ which are not prime powers,remains completely open. In the first interesting case, $d=6$, Zauner predictedthat there can exist no more than three MUBs. We explore possible algebraicsolutions in $d=6$ by looking at their `shadows' in vector spaces over finitefields. The main result is that if a counter-example to Zauner's conjecturewere to exist, then it would leave no such shadow upon reduction modulo severaldifferent primes, forcing its algebraic complexity level to be much higher thanthat 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 leastformally -- points in the opposite direction. In $\mathbb{C}^6$, not even asingle vector has ever been found which is mutually unbiased to a set of threeMUBs. 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, allof which lifts naturally to a number field.

Evidence for and against Zauner's MUB Conjecture in $\mathbb{C}^6$