Simultaneous measurement of non-commuting observables

A dynamical model of a quantum measurement process is introduced, where the tested system S, a spin View the MathML source, is simultaneously coupled with two apparatuses A and A′. Alone, A would measure the component View the MathML source whereas A′ alone would measure View the MathML source. 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 or -mF along z, triggered by its interaction with the z-component View the MathML source of S. Likewise, A′ is coupled to the x-component View the MathML source. The four probabilities of A+A′ depend on the polarizations 〈sz(0)〉 and 〈sx(0)〉 of S at the initial time. The counting rates for repeated experiments then determine both View the MathML source and View the MathML source, 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 Bellʹs inequalities by the indications of the apparatuses is discussed.