Optimal measurements and fixed-point maps

When an optimal measurement of (S_x,S_y,S_z) is made on a qubit and what we call a mutually unbiased mixture of the resulting ensembles is taken, then the post-measurement density matrix is shown to be related to the premeasurement density matrix through a simple linear relation. It is also shown that the form of this relation is the same for all quantum systems. It is shown that for a general quantum system such a relation holds only when the measurements are made in mutually unbiased bases. The post-measurement density matrix is shown to be a normalized incoherent superposition of the identity map and the pin map of Gorini and Sudarshan. A pin map is one which maps all density matrices to a fixed density matrix which in our case turns out to be the unit matrix. The result is shown to be true irrespective of whether the initial state is pure or mixed. For spin-1∕2 systems it is also shown explicitly that nonorthogonal measurements fail to give such a linear relation no matter how the ensembles are mixed.