Simultaneous Measurement of Non-commuting Observables

It is shown that the full unknown state of a spin1 2 system, S, which, within Born’s statistical interpretation, is meant as the state of an ensemble of identically prepared systems and is described by its density matrix, can be determined with a simultaneous measurement with the help of an “assistant” system A 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 , to measure simultaneously two observables, one of S and one of A and their correlation. One thus determines the three unknown components of the polarization vector of S by means of repeated experiments using a unique setting. In this way one can measure simultaneously all the non-commutative observables of S, which might seem prohibited in quantum mechanics.