Mutually unbiased measurements with arbitrary purity

Mutually unbiased measurements are a generalization of mutually unbiased bases in which the measurement operators need not to be rank one projectors. In a d-dimensional space, the purity of measurement elements ranges from 1/d for the measurement operators corresponding to maximally mixed states to 1 for the rank one projectors. In this contribution, we provide a class of MUM that encompasses the full range of purity. Similar to the MUB in which the operators corresponding to different outcomes of the same measurement commute mutually, our class of MUM possesses this sense of compatibility within each measurement. The spectra of these MUMs provide a way to construct a class of d-dimensional orthogonal matrices which leave the vector of equal components invariant. Based on this property, and by using the MUM-based entanglement witnesses, we examine the minimal number of measurements needed to detect entanglement of bipartite states. For a general bipartite pure state, we need only two MUMs: The first one assigns a zero mean value for all pure states; however, a complementary measurement is needed to give a negative mean value for entangled states. Interestingly, the amount of this negative value is proportional to an entanglement monotone.