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Generalized observables, Bell's inequalities and mixtures in the ESR model for QM

Claudio Garola

Published Jan 1, 2010• 12 citations
OpenAlex

Abstract

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.

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•Jan 1, 2008•11 citations
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, recently worked out by the author, embodies the mathematical formalism of standard (Hilbert space) quantum mechanics (QM) in a noncontextual framework and reinterprets quantum probabilities as conditional instead of absolute. The predictions of the ESR model do not coincide with the predictions of QM, which makes the model falsifiable. In particular, the ESR model predicts that, whenever idealized measurements are performed, modified Bell–Clauser–Horne–Shimony–Holt (BCHSH) inequalities hold if one takes into account all individual systems that are prepared, standard quantum inequalities hold if one considers only the individual systems that are detected, and standard BCHSH inequalities hold at a microscopic (purely theoretical) level. These results constitute a first example of the unified perspective that can be attained by adopting the ESR model.
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 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.
Most physicists uphold that the tests of the Bell inequalities (BI) performed up to now confirm the predictions of standard quantum mechanics (SQM) and refute local realism. But some scholars criticize this conviction, defending local realism in various ways. We present here a new viewpoint based on an improved version of the extended semantic realism (ESR) model that has been recently worked out by one of the authors. The ESR model embodies the mathematical formalism of SQM into a more general framework in which not only local realism but also objectivity of physical properties holds, and the probabilities of SQM are reinterpreted as conditional instead of absolute. Hence the ESR model provides some predictions that are formally identical to those of SQM but have a different physical interpretation, and further predictions that differ also formally from those of SQM. In particular, we show here that the BI introduced by Clauser, Horne, Shimony and Holt (BCHSH inequalities) must be replaced by modified BCHSH inequalities. These depend on detection probabilities which may be such that the new inequalities are never violated by the conditional expectation values predicted by the model. The condition that no violation occurs implies the existence of upper bounds on detection probabilities, which makes the ESR model falsifiable. These results admit an intuitive explanation in terms of unfair sampling but basically differ from the seemingly similar results obtained by other approaches in which the efficiency problem is discussed in order to vindicate some kind of local realism.
The extended semantic realism (ESR) model reinterprets quantum probabilities as conditional rather than absolute probabilities, which provides a general framework that assumes local realism and incorporates the mathematical apparatus of standard quantum mechanics (SQM). We discuss expectation values, sequential measurements and correlation functions within this model. Then, we consider the standard Bell‐Clauser, Horne, Shimony and Holt (BCHSH) inequalities and show that they must be replaced by modified BCHSH inequalities. These contain detection probabilities that are not predicted by SQM and do not depend on features of the measuring apparatuses or environment. In order to avoid contradiction, the detection probabilities must satisfy some no‐violation conditions which can be experimentally checked, at least in principle, and make the ESR model falsifiable.
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.