# The number of mutually-unbiased bases in dimension $d$

This recent answer points to the concept of a mutually-unbiased pair of bases, which are orthonormal bases $\{e_1,\ldots, e_d\}$ and $\{f_1,\ldots, f_d\}$ of a $d$-dimensional Hilbert space $\mathcal H$ such that the overlap between basis vectors of the two bases, $|\langle e_i,f_j\rangle|$, is constant and independent of $i$ and $j$.

As Dr. Mitchison points out, a reasonable amount is known about the total number $m$ of pairwise-mutually-unbiased bases that one can have in a given complex vector space as a function of its dimension, including the case of a power-of-a-prime dimension $d=p^n$ having a maximal $m=d+1$ MUBs possible.

I would like to ask a bit more about the details of that result:

• Are any other infinite families of dimensions known at which the maximal number of MUBs is known? Are any finite cases known?
• If the latter but not the former, what's the largest dimension $d$ to have a known maximal $m$?
• What's the smallest dimension $d$ for which the maximal number of possible MUBs is an open question?

It would also be good to have solid (and, if possible, accessible) references for any quoted results.

• As far as linearity goes one choice should be enough.
– user97261
Commented Mar 9, 2018 at 21:56
• I don't know of any example of a non-prime-power dimension where the number of mutually unbiased bases is known. In simplest case of $d=6$ there is evidence that the number is $3$ but no proof as far as I know. Higher than that relatively little is known. Commented Mar 15, 2018 at 19:24
• You might want to look up the work of Stefan Weigert from York U. in the UK on this topic. (He has a GoogleScholar profile). He and his students have done what is probably the most extensive search for mutually unbiased based in composite dimensions. See as an example journals.aps.org/pra/abstract/10.1103/PhysRevA.78.042312 Commented Sep 3, 2018 at 15:07