  Analogous to the notion of mutually unbiased bases for Hilbert spaces, we
consider mutually unbiased unitary bases (MUUB) for the space of operators,
$M(d, \mathbb{C})$, acting on such Hilbert spaces. The notion of MUUB reflects
the equiprobable guesses of unitary in one bases of $M(d, \mathbb{C})$ when
estimating a unitary operator in another. Though, for prime dimension $d$, the
maximal number of MUUBs is known to be $d^{2}-1$, there is no known recipe for
constructing them, assuming they exist. However, one can always construct a
minimum of three MUUBs, and the maximal number is approached for very large
values of $d$. MUUBs can also exists for some $d$-dimensional subspace of $M(d,
\mathbb{C})$ with the maximal number being $d$.
