Teaching ideal quantum measurement, from dynamics to interpretation
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...
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Académie des sciences
2024-05-01
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| Series: | Comptes Rendus. Physique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.180/ |
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| author | Allahverdyan, Armen E. Balian, Roger Nieuwenhuizen, Theo M. |
| author_facet | Allahverdyan, Armen E. Balian, Roger Nieuwenhuizen, Theo M. |
| author_sort | Allahverdyan, Armen E. |
| collection | DOAJ |
| description | 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 $\hat{\mathcal{D}}$(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 $\hat{\mathcal{D}}$(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. |
| format | Article |
| id | doaj-art-39d734564aa64385810f3e270d0bd4d7 |
| institution | Kabale University |
| issn | 1878-1535 |
| language | English |
| publishDate | 2024-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Physique |
| spelling | doaj-art-39d734564aa64385810f3e270d0bd4d72025-02-07T13:53:46ZengAcadémie des sciencesComptes Rendus. Physique1878-15352024-05-0125G125128710.5802/crphys.18010.5802/crphys.180Teaching ideal quantum measurement, from dynamics to interpretationAllahverdyan, Armen E.0Balian, Roger1Nieuwenhuizen, Theo M.2Yerevan Physics Institute, Alikhanian Brothers Street 2, Yerevan 375036, ArmeniaInstitut de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette cedex, FranceInstitute for Theoretical Physics, Science Park 904, 1098 XH Amsterdam, The NetherlandsWe 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 $\hat{\mathcal{D}}$(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 $\hat{\mathcal{D}}$(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.https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.180/ideal quantum measurements<i>q</i>-probabilitysystem-apparatus dynamicsmeasurement problemBorn rulevon Neumann reductionminimalist interpretationcontextuality |
| spellingShingle | Allahverdyan, Armen E. Balian, Roger Nieuwenhuizen, Theo M. Teaching ideal quantum measurement, from dynamics to interpretation Comptes Rendus. Physique ideal quantum measurements <i>q</i>-probability system-apparatus dynamics measurement problem Born rule von Neumann reduction minimalist interpretation contextuality |
| title | Teaching ideal quantum measurement, from dynamics to interpretation |
| title_full | Teaching ideal quantum measurement, from dynamics to interpretation |
| title_fullStr | Teaching ideal quantum measurement, from dynamics to interpretation |
| title_full_unstemmed | Teaching ideal quantum measurement, from dynamics to interpretation |
| title_short | Teaching ideal quantum measurement, from dynamics to interpretation |
| title_sort | teaching ideal quantum measurement from dynamics to interpretation |
| topic | ideal quantum measurements <i>q</i>-probability system-apparatus dynamics measurement problem Born rule von Neumann reduction minimalist interpretation contextuality |
| url | https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.180/ |
| work_keys_str_mv | AT allahverdyanarmene teachingidealquantummeasurementfromdynamicstointerpretation AT balianroger teachingidealquantummeasurementfromdynamicstointerpretation AT nieuwenhuizentheom teachingidealquantummeasurementfromdynamicstointerpretation |