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, \(\mathcal {H}_d\), we study its analogue for a d-dimensional subspace of \(M (d,\mathbb {C})\), 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 \(\mathcal {H}_d\) isomorphic to one for the subspace of \(M (d,\mathbb {C})\). This provides us not only with the maximal number of such MUUBs, but also a recipe for its construction.