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I have been studying this paper by S. Bandyopadhyay et al. - "A new proof for the existence of mutually unbiased bases". Here is the link.

They explicitly construct MUB using generalized Pauli matrices. The statement of the theorem goes as follows:

"For any prime d, the set of the bases each consisting of the eigenvectors of $Z_d, X_d, X_dZ_d, X_dZ_d^2,...,X_d(Z_d)^{d-1}$ form a set of $d+1$ mutually unbiased bases"

Where $X_d$ and $Z_d$ are generalizations of Pauli matrices in $d$ dimensions and they satisfy $X_d|j> = |j+1>$ and $Z_d|j>=\omega^j|j>$ with $|d+1>=|1>$ (i.e. it is a cyclic system)

The proof doesn't seem to use the requirement of $d$ being prime. It would really appreciate any help in figuring this out.

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