Simultaneous measurement of non-commuting observables

A dynamical model of a quantum measurement process is introduced, where the tested system S, a spin 1/2, is simultaneously coupled with two apparatuses A and A'. Alone, A would measure the component (s) over cap (z) whereas A' alone would measure (s) over cap (x). The apparatus A simulates an Ising magnetic dot involving N spins weakly coupled to a bath of phorrons 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 +m(F) or -m(F) along z, triggered by its interaction with the z-component (s) over cap (z) of S. Likewise, A' is coupled to the x-component (s) over cap (x). The four probabilities of A + A' depend on the polarizations < s(z)(0)> and < s(x)(0)> of S at the initial time. The counting rates for repeated experiments then determine both <(s) over cap (z)(0)> and <(s) over cap (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.