Minimal covariant observables identifying all pure states

It has been recently shown by Heinosaari, Mazzarella and Wolf (2013) [1] that an observable that identifies all pure states of a d-dimensional quantum system has minimally 4d - 4 outcomes or slightly less (the exact number depending on d). However, no simple construction of this type of minimal observable is known. We investigate covariant observables that identify all pure states and have minimal number of outcomes. It is shown that the existence of this kind of observables depends on the dimension of the Hilbert space.