M Theory Lesson 244

The list of $d+1$ operators, whose columns form a set of MUBs in any prime dimension $d$, is most easily described by the procedure outlined in the paper by Monique Combescure. For $d=3$, up to factors of $\sqrt{3}$, this operator set is The standard basis is read off the identity matrix, and M theorists will recognise the Fourier operator, which defines the second basis. In general, the third operator is defined by the 1-circulant

$M_3=(1,{\omega }^{-1},{\omega }^{-3},\cdots ,{\omega }^{-k(k+1)/2},1)$

and the remaining bases are specified by circulant powers of $M_3$. For $d=3$ there only remains $M_4=M_3^2$. The operator $M_3$ diagonalises $VU$, for the two Weyl generators $U$ and $V$. For $d=3$, $V=\left(231\right)$ and $U$ is the diagonal $(1,\omega ,{\omega }^2)$.

Combescure extends this result to all odd dimensions, in which case $j+1$ MUBs are constructed, where $j>2$ is the smallest divisor of $d$, and $M_3^{j-1}$ is the highest non trivial power of $M_3$. In even dimensions, there are only three operators which provide MUBs, and $M_3$ is defined differently. In particular, one requires the root $\sqrt{\omega }$, forcing factors of $i$ into the Pauli MUB algebra.