Understanding quantum measurement from the solution of dynamical models
Abstract
The quantum measurement problem, to wit, understanding why a unique outcome is obtained in each individual experiment, is currently tackled by solving models. After an introduction we review the many dynamical models proposed over the years for elucidating quantum measurements. The approaches range from standard quantum theory, relying for instance on quantum statistical mechanics or on decoherence, to quantumclassical methods, to consistent histories and to modifications of the theory. Next, a flexible and rather realistic quantum model is introduced, describing the measurement of the zcomponent of a spin through interaction with a magnetic memory simulated by a CurieWeiss magnet, including N≫1 spins weakly coupled to a phonon bath. Initially prepared in a metastable paramagnetic state, it may transit to its up or down ferromagnetic state, triggered by its coupling with the tested spin, so that its magnetization acts as a pointer. A detailed solution of the dynamical equations is worked out, exhibiting several time scales. Conditions on the parameters of the model are found, which ensure that the process satisfies all the features of ideal measurements. Various imperfections of the measurement are discussed, as well as attempts of incompatible measurements. The first steps consist in the solution of the Hamiltonian dynamics for the spinapparatus density matrix Dˆ(t). Its offdiagonal blocks in a basis selected by the spinpointer coupling, rapidly decay owing to the many degrees of freedom of the pointer. Recurrences are ruled out either by some randomness of that coupling, or by the interaction with the bath. On a longer time scale, the trend towards equilibrium of the magnet produces a final state Dˆ(t) that involves correlations between the system and the indications of the pointer, thus ensuring registration. Although Dˆ(t) has the form expected for ideal measurements, it only describes a large set of runs. Individual runs are approached by analyzing the final states associated with all possible subensembles of runs, within a specified version of the statistical interpretation. There the difficulty lies in a quantum ambiguity: There exist many incompatible decompositions of the density matrix Dˆ(t) into a sum of submatrices, so that one cannot infer from its sole determination the states that would describe small subsets of runs. This difficulty is overcome by dynamics due to suitable interactions within the apparatus, which produce a special combination of relaxation and decoherence associated with the broken invariance of the pointer. Any subset of runs thus reaches over a brief delay a stable state which satisfies the same hierarchic property as in classical probability theory; the reduction of the state for each individual run follows. Standard quantum statistical mechanics alone appears sufficient to explain the occurrence of a unique answer in each run and the emergence of classicality in a measurement process. Finally, pedagogical exercises are proposed and lessons for future works on models are suggested, while the statistical interpretation is promoted for teaching.
 Publication:

Physics Reports
 Pub Date:
 April 2013
 DOI:
 10.1016/j.physrep.2012.11.001
 arXiv:
 arXiv:1107.2138
 Bibcode:
 2013PhR...525....1A
 Keywords:

 Quantum Physics;
 Condensed Matter  Statistical Mechanics;
 High Energy Physics  Theory
 EPrint:
 Minor changes, matches published version. 201 pages Latex, 377 references, 15 figures