Optimal single quantum measurement of multi-level quantum systems between pure state and mixed state

We derive analytic solution of the particular case of the optimal quantum measurement between pure state and mixed state of the multi-level system. In order to figure out the physical essence rather than the facial complex mathematical expression, we restrict the system to be 3-level and the mixed state to be a diagonal matrix to get a obvious result. Generally the optimal solution is determined by the 9 free elements of the transformation unitary matrix. To our surprise, the results actually are so simple that we just have to search 3-dimensional Hilbert space for the optimal solutions. However, the optimal solutions depend on the relative scale among the diagonal elements. Some guessing analytical solution of the optimal target function is given, but the general cases of transformation between mixed state and pure state without imposing these restrictions is still unavailable. © 2011 Springer-Verlag.