Simultaneous measurement of non-commuting observables

It is shown that the full unknown state of a spin-1/2 system, S, described by its density matrix, can be determined with a simultaneous measurement with the help of another system A, called assistant, whose initial state is known. The idea is to let S and A interact with each other in a known way during a proper interaction time τ, and then to measure simultaneously two observables, one of S and one of A and then to determine their averages and their correlation. One thus determines the three unknown components of the polarization vector of S by means of repeated experiments using a unique stetting. In this way one can measure all the non-commutative observables of S at the same time, which may seem prohibited in quantum mechanics.