We consider the notion of unitary transformations forming bases for subspaces of M(d,C) such that the square of the Hilbert-Schmidt inner product of matrices from the differing bases is a constant. Moving from the qubit case, we construct the maximal number of such bases for the four- and two-dimensional subspaces while proving the nonexistence of such a construction for the three-dimensional case. Extending this to higher dimensions, we commit to such a construct for the case of qutrits and provide evidence for the existence of such unitaries for prime dimensional quantum systems. Focusing on the qubit case, we show that the average fidelity for estimating any such transformation is equal to the case for estimating a completely unknown unitary from SU(2). This is then followed by a quick application for such unitaries in a quantum cryptographic setup

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, C), acting on such Hilbert spaces. The notion of MUUB reflects the equiprobable guesses of unitary operators in one basis of M (d, 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 exist for some d-dimensional subspace of M (d, C) with the maximal number being d.

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