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AIP conference proceedings

The Representation of Mixtures in the ESR Model for QM

Claudio Garola, Sandro Sozzo, Andrei Yu. Khrennikov

Published Jan 1, 2010• 3 citations
Publisher

Abstract

The extended semantic realism (ESR) model proposes a new theoretical perspective which embodies the mathematical formalism of Hilbert space quantum mechanics (QM) into a broader noncontextual, hence local, framework, reinterpreting quantum probabilities as conditional (in a nonstandard sense). We have proven in a previous paper that each generalized observable introduced by the ESR model is represented by a family of positive operator valued measures (POVM) parametrized by the pure states of the physical system Ω that is considered. We show here that each mixture is represented in the ESR model by a family of density operators parametrized by the physical properties characterizing Ω. This representation implies predictions that may differ from those of QM and avoids some deep problems that arise in the interpretation of mixtures provided by QM. We also show that the state transformations induced by idealized nondestructive measurements can be obtained by means of a nontrivial generalization of the Lüders postulate, and discuss our results in the special case of discrete generalized observables.

DOI
10.1063/1.3431518

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The extended semantic realism (ESR) model proposes a new theoretical perspective which embodies the mathematical formalism of Hilbert space quantum mechanics (QM) into a broader noncontextual, hence local, framework, reinterpreting quantum probabilities as conditional (in a nonstandard sense). We have proven in a previous paper that each generalized observable introduced by the ESR model is represented by a family of positive operator valued measures (POVM) parametrized by the pure states of the physical system Ω that is considered. We show here that each mixture is represented in the ESR model by a family of density operators parametrized by the physical properties characterizing Ω. This representation implies predictions that may differ from those of QM and avoids some deep problems that arise in the interpretation of mixtures provided by QM. We also show that the state transformations induced by idealized nondestructive measurements can be obtained by means of a nontrivial generalization of the Lüders postulate, and discuss our results in the special case of discrete generalized observables.
The extended semantic realism (ESR) model proposes a new theoretical perspective which embodies the mathematical formalism of standard (Hilbert space) quantum mechanics (QM) into a noncontextual framework, reinterpreting quantum probabilities as conditional instead of absolute. We provide in this review an overall view on the present status of our research on this topic. We attain in a new, shortened way a mathematical representation of the generalized observables introduced by the ESR model and a generalization of the projection postulate of elementary QM. Basing on these results we prove that the Bell-Clauser-Horne-Shimony-Holt (BCHSH) inequality, a modified BCHSH inequality and quantum predictions hold together in the ESR model because they refer to different parts of the picture of the physical world supplied by the model. Then we show that a new mathematical representation of mixtures must be introduced in the ESR model which does not coincide with the standard representation in QM and avoids some deep problems that arise from the representation of mixtures provided by QM. Finally we get a nontrivial generalization of the Lüders postulate, which is justified in a special case by introducing a reasonable physical assumption on the evolution of the compound system made up of the measured system and the measuring apparatus.
The extended semantic realism ( ESR ) model proposes a new theoretical perspective that embodies the mathematical formalism of standard (Hilbert space) quantum mechanics (QM) in a noncontextual framework, reinterpreting quantum probabilities as conditional instead of absolute . We provide here a Hilbert space representation of the generalized observables introduced by the ESR model. By using this representation, we supply a straightforward generalization of the projection postulate and justify it in the case of discrete generalized observables by introducing a reasonable physical assumption on the evolution of the compound system made up of the (microscopic) measured system and the (macroscopic) measuring apparatus.
The extended semantic realism (ESR) model embodies the mathematical formalism of standard (Hilbert space) quantum mechanics (QM) in a noncontextual framework, reinterpreting quantum probabilities as conditional instead of absolute. We provide here a Hilbert space representation of the discrete generalized observables introduced by the ESR model that satisfy a simple physical condition, propose a generalization of the projection postulate and suggest a possible mathematical description of an ideal measurement process in terms of evolution of the compound system made up of the measured system and the measuring apparatus.
A crucial and problematical feature of quantum mechanics (QM) is nonobjectivity of properties. The ESR model restores objectivity reinterpreting quantum probabilities as conditional on detection and embodying the mathematical formalism of QM into a broader noncontextual (hence local) framework. We propose here an improved presentation of the ESR model containing a more complete mathematical representation of the basic entities of the model. We also extend the model to mixtures showing that the mathematical representations of proper mixtures does not coincide with the mathematical representation of mixtures provided by QM, while the representation of improper mixtures does. This feature of the ESR model entails that some interpretative problems raising in QM when dealing with mixtures are avoided. From an empirical point of view the predictions of the ESR model depend on some parameters which may be such that they are very close to the predictions of QM in most cases. But the nonstandard representation of proper mixtures allows us to propose the scheme of an experiment that could check whether the predictions of QM or the predictions of the ESR model are correct.
A crucial and problematical feature of quantum mechanics (QM) is nonobjectivity of properties. The ESR model restores objectivity reinterpreting quantum probabilities as conditional on detection and embodying the mathematical formalism of QM into a broader noncontextual (hence local) framework. We propose here an improved presentation of the ESR model containing a more complete mathematical representation of the basic entities of the model. We also extend the model to mixtures showing that the mathematical representations of proper mixtures does not coincide with the mathematical representation of mixtures provided by QM, while the representation of improper mixtures does. This feature of the ESR model entails that some interpretative problems raising in QM when dealing with mixtures are avoided. From an empirical point of view the predictions of the ESR model depend on some parameters which may be such that they are very close to the predictions of QM in most cases. But the nonstandard representation of proper mixtures allows us to propose the scheme of an experiment that could check whether the predictions of QM or the predictions of the ESR model are correct.
The extended semantic realism (ESR) model recently worked out by one of the authors embodies the mathematical formalism of standard (Hilbert space) quantum mechanics in a noncontextual framework, reinterpreting quantum probabilities as conditional instead of absolute. We provide here a Hilbert space representation of the generalized observables introduced by the ESR model that satisfy a simple physical condition, propose a generalization of the projection postulate, and suggest a possible mathematical description of the measurement process in terms of evolution of the compound system made up of the measured system and the measuring apparatus.