On mutually unbiased unitary bases in prime-dimensional Hilbert spaces

Akin to the idea of complete sets of mutually unbiased bases for prime-dimensional Hilbert spaces, Hd, we study its analogue for a d-dimensional subspace of M(d,C), i.e. mutually unbiased unitary bases (MUUBs) comprising of unitary operators. We note an obvious isomorphism between the vector spaces and beyond that, we define a relevant monoid structure for Hd isomorphic to one for the subspace of M(d,C). This provides us not only with the maximal number of such MUUBs, but also a recipe for its construction.