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Possible experience: From Boole to Bell

K. Hess, Kristel F. L. Michielsen, Hans A. De Raedt

Published Jan 1, 2009• 39 citations
Publisher

Abstract

Mainstream interpretations of quantum theory maintain that violations of the Bell inequalities deny at least either realism or Einstein locality. Here we investigate the premises of the Bell-type inequalities by returning to earlier inequalities presented by Boole and the findings of Vorob'ev as related to these inequalities. These findings together with a space-time generalization of Boole's elements of logic lead us to a completely transparent Einstein local counterexample from everyday life that violates certain variations of the Bell inequalities. We show that the counterexample suggests an interpretation of the Born rule as a pre-measure of probability that can be transformed into a Kolmogorov probability measure by certain Einstein local space-time characterizations of the involved random variables. Copyright (C) EPLA, 2009

DOI
10.1209/0295-5075/87/60007/pdf

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Possible experience: From Boole to Bell

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Mainstream interpretations of quantum theory maintain that violations of the Bell inequalities deny at least either realism or Einstein locality. Here we investigate the premises of the Bell-type inequalities by returning to earlier inequalities presented by Boole and the findings of Vorob'ev as related to these inequalities. These findings together with a space-time generalization of Boole's elements of logic lead us to a completely transparent Einstein local counterexample from everyday life that violates certain variations of the Bell inequalities. We show that the counterexample suggests an interpretation of the Born rule as a pre-measure of probability that can be transformed into a Kolmogorov probability measure by certain Einstein local space-time characterizations of the involved random variables.

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Extended Boole-Bell Inequalities Applicable to Quantum Theory

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We address the basic meaning of apparent contradictions of quantum theory and probability frameworks as expressed by Bell's inequalities. We show that these contradictions have their origin in the incomplete considerations of 1:he premises of the derivation of the inequalities. A careful consideration of past work, including that of Boole and Vorob'ev, has lead us to the formulation of extended Boole-Bell inequalities that are binding for both classical and quantum models. The Einstein-Podolsky-Rosen-Bohm gedanken experiment and a macroscopic quantum coherence experiment proposed by Leggett and (Garg are both shown to obey the extended Boole-Bell inequalities. These examples as well as additional discussions also provide reasons for apparent violations of these inequalities.

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Extended Boole-Bell inequalities applicable to quantum theory

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Logical Views of Bell Inequalities

Zeno Toffano

Bell's Inequalites structure can be considered in a logical-probabilistic framework, as outlined by Pitowsky [1], where the origin of the inequality was attributed to George Boole’s “conditions of possible experience”. The inequality has then been often renamed the Bell-Boole inequality. An interesting fact is that G. Boole’s method used to establish his probability inequalities stems directly from his mathematical treatment of logic using idempotent symbols (which he called “elective symbols”). The “Boolean Algebra” we consider today bears some major differences with Boole’s original method as thoroughly discussed by Hailperin [2].We proposed recently the “Eigenlogic” approach [3] where we translate Boole’s original logical method to linear algebra using idempotent and also unitary operators and outline the bridges to quantum theories in particular to quantum information [4]. Eigenlogic can thus be a framework for transposing logic in a quantum context, the probabilistic aspects are then simply derived by measurements on logical operators using the Born rule.Bell Inequalities (BI) can be expressed in an operator form where the measurement observables of an experiment are combined to form Bell operators, the outcomes are then derived by using the quantum mechanical mean value (again the Born rule). The popular form adapted for this method is the CHSH form of the BI for spin 1/2 particles or polarized photons with outcomes ±1. Using the isomorphism between idempotent and self-inverse unitary operators (Householder transform) the CHSH Bell operator transforms in Eigenlogic in a combination of idempotent operators from which by using the Born rule one gets the Fine inequality on probabilities.As stated by Khrennikov [5] the BI violation occurs only when incompatible (non-commuting) observables are used in the BI associated experiments. Particularly in the CHSH-BI experiments on an entangled state, mutually unbiased bases (MUB), representing maximally dissimilar contexts, lead to the maximum quantum violation represented by the number 2√2>2. This is the case for the couple of MUB (z,x) associated to Pauli spin operators σ_x and σ_z. In Eigenlogic these (z,x) contexts represent dual logical systems which are at interplay and have an original syntax-semantic interpretation [4]. It has to be outlined that in many quantum algorithms the MUB (z,x), linked by the Hadamard gate which represents the Fourier Transformation between the two bases, provides a quantum algorithmic advantage.The situations of the BI in the so called post-quantum regime are also analyzed by generalizing the concept of PR-Box which relates outputs (a,b) to inputs (x,y) in a two-party correlation by means of the logical equation: a⊕b=x∧y. Using Eigenlogic, families of boxes with all possible combinations of logical functions are analyzed through the equations: f(a,b)=g(x,y), where f and g correspond to the 16 different Boolean functions, giving 256 possible logical equations. The logical equations are transformed in logical operators and used to define the associated Bell operator. 16 cases of the PR-Box type (considering all the possible complementations of the variables a,b,x,y) violate the CHSH-BI maximally with the number 4>2√2 (Tsirelson’s bound). Another family of 32 logical cases gives a CHSH-BI violation of 3.33 also beyond the quantum limit; this is the case for the logical equation: a∨b=x∧y.[1] Pitowsky, I., George Boole’s ‘conditions of possible experience’ and the quantum puzzle. Brit. J. Phil. Sci., 45:95–125 (1994).[2] Hailperin, T. Boole’s Algebra isn’t Boolean algebra. A description using modern algebra, of what Boole really did create. Math. Mag. 54(4): 172–184 (1981).[3] Toffano, Z. Eigenlogic in the Spirit of George Boole. Log. Univers. 14, 175–207 (2020). https://doi.org/10.1007/s11787-020-00252-3[4] Toffano, Z.; Dubois, F. Adapting Logic to Physics: The Quantum-Like Eigenlogic Program. Entropy 2020, 22, 139. https://doi.org/10.3390/e22020139[5] Khrennikov, A. Contextuality, Complementarity, Signaling, and Bell Tests. Entropy 2022, 24, 1380. https://doi.org/10.3390/e24101380

•Jun 13, 2023•0 citations

Bell’s theorem: Critique of proofs with and without inequalities

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Most of the standard proofs of the Bell theorem are based on the Kolmogorov axioms of probability theory. We show that these proofs contain mathematical steps that cannot be reconciled with the Kolmogorov axioms. Specifically we demonstrate that these proofs ignore the conclusion of a theorem of Vorob'ev on the consistency of joint distributions. As a consequence Bell's theorem stated in its full generality remains unproven, in particular, for extended parameter spaces that are still objective local and that include instrument parameters that are correlated by both time and instrument settings. Although the Bell theorem correctly rules out certain small classes of hidden variables, for these extended parameter spaces the standard proofs come to a halt. The Greenberger-Horne-Zeilinger (GHZ) approach is based on similar fallacious arguments. For this case we are able to present an objective local computer experiment that simulates the experimental test of GHZ performed by Pan, Bouwmeester, Daniell, Weinfurter and Zeilinger and that directly contradicts their claim that Einstein-local elements of reality can neither explain the results of quantum mechanical theory nor their experimental results.

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The interpretation of the meaning of Quantum Mechanics has faced controversy since its inception. Bell's inequalities are a touchstone in this controversy. Their observed violation demonstrates that at least one of the hypotheses involved in their derivation and test is false in Nature. In principle, one has to choose between accepting that Locality is false, what implies a possible contradiction with the Theory of Relativity, or accepting that Realism is false, what means to give up the existence of a physical world independent of the observer. The right answer has consequences both foundational and practical, and theoretical discussions have searched it for decades. We report the results of a Bell's experiment designed and performed to add observational information to the discussion. Three proposals to reveal the false hypothesis are carried out, namely: search of attractors in time series of observations, variation of randomness of binary series of outcomes between space-like and not-space-like separated conditions of observation, and test of a bound of Kolmogorov's complexity. The results are consistent with the absence of the specific form of Realism usually involved in the derivation of Bell's inequalities, while remaining compatible with Locality within the sensitivity of the tests. Independently of the foundational problem and of any interpretation, some bare observations have immediate practical impact on the best use of device-independent quantum Random Number Generators and Quantum Key Distribution.

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Mainstream interpretations of quantum theory maintain that violations of the Bell inequalities deny at least either realism or Einstein locality. Here we investigate the premises of the Bell-type inequalities by returning to earlier inequalities presented by Boole and the findings of Vorob'ev as related to these inequalities. These findings together with a space-time generalization of Boole's elements of logic lead us to a completely transparent Einstein local counterexample from everyday life that violates certain variations of the Bell inequalities. We show that the counterexample suggests an interpretation of the Born rule as a pre-measure of probability that can be transformed into a Kolmogorov probability measure by certain Einstein local space-time characterizations of the involved random variables. Copyright (C) EPLA, 2009
Mainstream interpretations of quantum theory maintain that violations of the Bell inequalities deny at least either realism or Einstein locality. Here we investigate the premises of the Bell-type inequalities by returning to earlier inequalities presented by Boole and the findings of Vorob'ev as related to these inequalities. These findings together with a space-time generalization of Boole's elements of logic lead us to a completely transparent Einstein local counterexample from everyday life that violates certain variations of the Bell inequalities. We show that the counterexample suggests an interpretation of the Born rule as a pre-measure of probability that can be transformed into a Kolmogorov probability measure by certain Einstein local space-time characterizations of the involved random variables.
We address the basic meaning of apparent contradictions of quantum theory and probability frameworks as expressed by Bell's inequalities. We show that these contradictions have their origin in the incomplete considerations of 1:he premises of the derivation of the inequalities. A careful consideration of past work, including that of Boole and Vorob'ev, has lead us to the formulation of extended Boole-Bell inequalities that are binding for both classical and quantum models. The Einstein-Podolsky-Rosen-Bohm gedanken experiment and a macroscopic quantum coherence experiment proposed by Leggett and (Garg are both shown to obey the extended Boole-Bell inequalities. These examples as well as additional discussions also provide reasons for apparent violations of these inequalities.
We address the basic meaning of apparent contradictions of quantum theory and probability frameworks as expressed by Bell's inequalities. We show that these contradictions have their origin in the incomplete considerations of the premisses of the derivation of the inequalities. A careful consideration of past work, including that of Boole and Vorob'ev, has lead us to the formulation of extended Boole-Bell inequalities that are binding for both classical and quantum models. The Einstein-Podolsky-Rosen-Bohm gedanken experiment and a macroscopic quantum coherence experiment proposed by Leggett and Garg are both shown to obey the extended Boole-Bell inequalities. These examples as well as additional discussions also provide reasons for apparent violations of these inequalities.
Bell's Inequalites structure can be considered in a logical-probabilistic framework, as outlined by Pitowsky [1], where the origin of the inequality was attributed to George Boole’s “conditions of possible experience”. The inequality has then been often renamed the Bell-Boole inequality. An interesting fact is that G. Boole’s method used to establish his probability inequalities stems directly from his mathematical treatment of logic using idempotent symbols (which he called “elective symbols”). The “Boolean Algebra” we consider today bears some major differences with Boole’s original method as thoroughly discussed by Hailperin [2].We proposed recently the “Eigenlogic” approach [3] where we translate Boole’s original logical method to linear algebra using idempotent and also unitary operators and outline the bridges to quantum theories in particular to quantum information [4]. Eigenlogic can thus be a framework for transposing logic in a quantum context, the probabilistic aspects are then simply derived by measurements on logical operators using the Born rule.Bell Inequalities (BI) can be expressed in an operator form where the measurement observables of an experiment are combined to form Bell operators, the outcomes are then derived by using the quantum mechanical mean value (again the Born rule). The popular form adapted for this method is the CHSH form of the BI for spin 1/2 particles or polarized photons with outcomes ±1. Using the isomorphism between idempotent and self-inverse unitary operators (Householder transform) the CHSH Bell operator transforms in Eigenlogic in a combination of idempotent operators from which by using the Born rule one gets the Fine inequality on probabilities.As stated by Khrennikov [5] the BI violation occurs only when incompatible (non-commuting) observables are used in the BI associated experiments. Particularly in the CHSH-BI experiments on an entangled state, mutually unbiased bases (MUB), representing maximally dissimilar contexts, lead to the maximum quantum violation represented by the number 2√2>2. This is the case for the couple of MUB (z,x) associated to Pauli spin operators σ_x and σ_z. In Eigenlogic these (z,x) contexts represent dual logical systems which are at interplay and have an original syntax-semantic interpretation [4]. It has to be outlined that in many quantum algorithms the MUB (z,x), linked by the Hadamard gate which represents the Fourier Transformation between the two bases, provides a quantum algorithmic advantage.The situations of the BI in the so called post-quantum regime are also analyzed by generalizing the concept of PR-Box which relates outputs (a,b) to inputs (x,y) in a two-party correlation by means of the logical equation: a⊕b=x∧y. Using Eigenlogic, families of boxes with all possible combinations of logical functions are analyzed through the equations: f(a,b)=g(x,y), where f and g correspond to the 16 different Boolean functions, giving 256 possible logical equations. The logical equations are transformed in logical operators and used to define the associated Bell operator. 16 cases of the PR-Box type (considering all the possible complementations of the variables a,b,x,y) violate the CHSH-BI maximally with the number 4>2√2 (Tsirelson’s bound). Another family of 32 logical cases gives a CHSH-BI violation of 3.33 also beyond the quantum limit; this is the case for the logical equation: a∨b=x∧y.[1] Pitowsky, I., George Boole’s ‘conditions of possible experience’ and the quantum puzzle. Brit. J. Phil. Sci., 45:95–125 (1994).[2] Hailperin, T. Boole’s Algebra isn’t Boolean algebra. A description using modern algebra, of what Boole really did create. Math. Mag. 54(4): 172–184 (1981).[3] Toffano, Z. Eigenlogic in the Spirit of George Boole. Log. Univers. 14, 175–207 (2020). https://doi.org/10.1007/s11787-020-00252-3[4] Toffano, Z.; Dubois, F. Adapting Logic to Physics: The Quantum-Like Eigenlogic Program. Entropy 2020, 22, 139. https://doi.org/10.3390/e22020139[5] Khrennikov, A. Contextuality, Complementarity, Signaling, and Bell Tests. Entropy 2022, 24, 1380. https://doi.org/10.3390/e24101380
Most of the standard proofs of the Bell theorem are based on the Kolmogorov axioms of probability theory. We show that these proofs contain mathematical steps that cannot be reconciled with the Kolmogorov axioms. Specifically we demonstrate that these proofs ignore the conclusion of a theorem of Vorob'ev on the consistency of joint distributions. As a consequence Bell's theorem stated in its full generality remains unproven, in particular, for extended parameter spaces that are still objective local and that include instrument parameters that are correlated by both time and instrument settings. Although the Bell theorem correctly rules out certain small classes of hidden variables, for these extended parameter spaces the standard proofs come to a halt. The Greenberger-Horne-Zeilinger (GHZ) approach is based on similar fallacious arguments. For this case we are able to present an objective local computer experiment that simulates the experimental test of GHZ performed by Pan, Bouwmeester, Daniell, Weinfurter and Zeilinger and that directly contradicts their claim that Einstein-local elements of reality can neither explain the results of quantum mechanical theory nor their experimental results.
The interpretation of the meaning of Quantum Mechanics has faced controversy since its inception. Bell's inequalities are a touchstone in this controversy. Their observed violation demonstrates that at least one of the hypotheses involved in their derivation and test is false in Nature. In principle, one has to choose between accepting that Locality is false, what implies a possible contradiction with the Theory of Relativity, or accepting that Realism is false, what means to give up the existence of a physical world independent of the observer. The right answer has consequences both foundational and practical, and theoretical discussions have searched it for decades. We report the results of a Bell's experiment designed and performed to add observational information to the discussion. Three proposals to reveal the false hypothesis are carried out, namely: search of attractors in time series of observations, variation of randomness of binary series of outcomes between space-like and not-space-like separated conditions of observation, and test of a bound of Kolmogorov's complexity. The results are consistent with the absence of the specific form of Realism usually involved in the derivation of Bell's inequalities, while remaining compatible with Locality within the sensitivity of the tests. Independently of the foundational problem and of any interpretation, some bare observations have immediate practical impact on the best use of device-independent quantum Random Number Generators and Quantum Key Distribution.