Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics |
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| Authors: | Florio M. Ciaglia Fabio Di Cosmo Alberto Ibort Giuseppe Marmo |
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| Affiliation: | 1.Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany;2.ICMAT, Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Nicolás Cabrera, 13-15, Campus de Cantoblanco, UAM, 28049 Madrid, Spain; (F.D.C.); (A.I.);3.Departemento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain;4.Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II, 80126 Napoli, Italy; |
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| Abstract: | The evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently is discussed. It is shown that the notion of a classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. The impossibility of evolving a separable state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution is proven in accordance with Raggio’s theorem, which is extended to include a new family of separable states corresponding to the composition of a system with a totally disconnected space of outcomes and a quantum one. |
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| Keywords: | quantum mechanics, entanglement, Schwinger’ s selective measurements, composite systems, groupoids picture of quantum mechanics, groupoids, Birkhoff– von Neumann logic, foundations of quantum theories |
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