Mutually unbiased unitary bases of operators on d-dimensional hilbert space

Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUBs) for the space of operators, M(d,a), acting on such Hilbert spaces. The notion of MUUB reflects the equiprobable guesses of unitary operators in one basis of M(d,a) when estimating a unitary operator in another. Though, for prime dimension d, the maximal number of MUUBs is known to be d2-1, 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 exist for some d-dimensional subspace of M(d,a) with the maximal number being d.