Simultaneous measurement of non-commuting observables

A dynamical model of a quantum measurement process is introduced, where the tested system S, a spin 12, is simultaneously coupled with two apparatuses A and A′A′. Alone, A would measure the component s^z whereas A′A′ alone would measure s^x. The apparatus A simulates an Ising magnetic dot involving N   spins weakly coupled to a bath of phonons at a temperature lower than the Curie point. Initially in its metastable paramagnetic state, A tends to reach either one of its two equilibrium ferromagnetic states, with magnetization +mF+mF or -mF-mF along z, triggered by its interaction with the z  -component s^z of S. Likewise, A′A′ is coupled to the x  -component s^x. The four probabilities of A+A′A+A′ depend on the polarizations 〈sz(0)〉〈sz(0)〉 and 〈sx(0)〉〈sx(0)〉 of S at the initial time. The counting rates for repeated experiments then determine both 〈s^z(0)〉 and 〈s^x(0)〉, although the process cannot be regarded as an ideal measurement. Three apparatuses simultaneously coupled to all three components of S provide full information on the initial density matrix of S through repeated runs. The lack of violation of Bell's inequalities by the indications of the apparatuses is discussed.