  When an optimal measurement is made on a qubit and what we call an Unbiased
Mixture of the resulting ensembles is taken, then the post measurement density
matrix is shown to be related to the pre-measurement density matrix through a
simple and linear relation. It is shown that such a relation holds only when
the measurements are made in Mutually Unbiased Bases- MUB. For Spin-1/2 it is
also shown explicitly that non-orthogonal measurements fail to give such a
linear relation no matter how the ensembles are mixed. The result has been
proved to be true for arbitrary quantum mechanical systems of finite
dimensional Hilbert spaces. The result is true irrespective of whether the
initial state is pure or mixed.
