We present a graduate course on ideal measurements, analyzed as dynamical processes of interaction between the tested system S and an apparatus A, described by quantum statistical mechanics. The apparatus A = M + B involves a macroscopic measuring device M and a bath B. The requirements for ideality of the measurement allow us to specify the Hamiltonian of the isolated compound system S + M + B. The resulting dynamical equations may be solved for simple models. Conservation laws are shown to entail two independent relaxation mechanisms: truncation and registration. Approximations, justified by the large size of M and of B, are needed. The final density matrix (tf) of S + A has an equilibrium form. It describes globally the outcome of a large set of runs of the measurement. The measurement problem, i.e., extracting physical properties of individual runs from (tf), then arises due to the ambiguity of its splitting into parts associated with subsets of runs. To deal with this ambiguity, we postulate that each run ends up with a distinct pointer value Ai of the macroscopic M. This is compatible with the principles of quantum mechanics. Born’s rule then arises from the conservation law for the tested observable; it expresses the frequency of occurrence of the final indications Ai of M in terms of the initial state of S. Von Neumann’s reduction amounts to updating of information due to selection of Ai. We advocate the terms q-probabilities and q-correlations when analyzing measurements of non-commuting observables. These ideas may be adapted to different types of courses.
On présente un cours doctoral sur les mesures idéales, processus dynamiques couplant le système testé S et un appareil A analysés en mécanique statistique quantique. Cet appareil A = M + B comprend un dispositif de mesure macroscopique M et un bain B. Les conditions requises pour l’idéalité de la mesure impliquent une forme spécifique du Hamiltonien du système composite isolé S + M + B. Les equations dynamiques résultantes sont solubles pour des modèles simples. Les lois de conservation engendrent deux mécanismes de relaxation indépendants, la troncature et l’enregistrement. Des approximations, justifiées par la grande taille de M et de B, sont nécessaires. La matrice densité finale (tf) de S + A a une forme d’équilibre. Elle décrit globalement l’issue d’un large ensemble de processus similaires. Le problème de la mesure, extraire de (tf) des propriétés physiques de processus individuels, provient ici de l’impossibilité de le scinder sans ambiguïté en parties décrivant des sous-ensembles de processus. On lève cette ambiguïté en postulant que chaque mesure individuelle aboutit à une valeur distincte Ai du pointeur macroscopique. Ceci est compatible avec les principes de la mécanique quantique. La règle de Born résulte alors de la loi de conservation de l’observable mesurée ; elle exprime la fréquence de chaque indication finale Ai de M en termes de l’état initial de S. La réduction de von Neumann apparaît comme une mise à jour de l’information résultant de la sélection d’un résultat Ai. On préconise l’emploi des termes q-probabilités ou q-corrélations lors de l’analyse de mesures d’observables non-commutatives. Ces idées peuvent être adaptées à divers types de cours.
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Mot clés : mesures quantiques idéales, q-probabilités, dynamique système-appareil, problème de la mesure, règle de Born, réduction de von Neumann, interprétation minimaliste, contextualité
Armen E. Allahverdyan 1; Roger Balian 2; Theo M. Nieuwenhuizen 3
CC-BY 4.0
@article{CRPHYS_2024__25_G1_251_0,
author = {Armen E. Allahverdyan and Roger Balian and Theo M. Nieuwenhuizen},
title = {Teaching ideal quantum measurement, from dynamics to interpretation},
journal = {Comptes Rendus. Physique},
pages = {251--287},
publisher = {Acad\'emie des sciences, Paris},
volume = {25},
year = {2024},
doi = {10.5802/crphys.180},
language = {en},
}
TY - JOUR AU - Armen E. Allahverdyan AU - Roger Balian AU - Theo M. Nieuwenhuizen TI - Teaching ideal quantum measurement, from dynamics to interpretation JO - Comptes Rendus. Physique PY - 2024 SP - 251 EP - 287 VL - 25 PB - Académie des sciences, Paris DO - 10.5802/crphys.180 LA - en ID - CRPHYS_2024__25_G1_251_0 ER -
Armen E. Allahverdyan; Roger Balian; Theo M. Nieuwenhuizen. Teaching ideal quantum measurement, from dynamics to interpretation. Comptes Rendus. Physique, Volume 25 (2024), pp. 251-287. doi : 10.5802/crphys.180. https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.180/
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