Ideal measurements are described in quantum mechanics textbooks by two postulates: the collapse of the wave packet and BornÃ¢ÂÂs rule for the probabilities of outcomes. The quantum evolution of a system then has two components: a unitary (Hamiltonian) evolution in between measurements and non-unitary one when a measurement is performed. This situation was considered to be unsatisfactory by many people, including Einstein, Bohr, de Broglie, von Neumann and Wigner, but has remained unsolved to date.
The quantum measurement problem, that is, understanding why a unique outcome is obtained in each individual run of an experiment, is tackled by solving a Hamiltonian model within standard quantum statistical mechanics. The model describes the measurement of the z-component of a spin through interaction with a magnetic memory. The latter apparatus is modeled by a CurieÃ¢ÂÂWeiss magnet having N Ã¢ÂÂ« 1 spins weakly coupled to a phonon bath.
The Hamiltonian evolution exhibits several time scales. The reduction, a rapid decay of the off-diagonal blocks of the systemÃ¢ÂÂapparatus density matrix, arises from the many degrees of freedom of the pointer (the magnetization). The registration occurs due to a phase transition from the initial metastable state to one of the final stable states triggered by the tested system. It yields a stationary state in which the apparatus and the system are correlated. Under proper conditions the process satisfies all the features of ideal measurements, including collapse and BornÃ¢ÂÂs rule.
As usual, irreversibility is ensured by the macroscopic size of the apparatus, in particular by the large value of N. Nothing else than the usual quantum statistical mechanics and Schro ÃÂdinger equation is needed, and the results support a specified version of the statistical interpretation. The solution of the quantum measurement problem requires a combination of the reduction and the registration, the properties of which arise from the irreversible dynamics.