Bell's Theorem, Uncertainty, and Conditional Events

Ian T. Durham

Published Jul 1, 2005• 0 citations

Abstract

This article explores a relationship between the generalized form of Heisenberg’s uncertainty relations and Bell-type inequalities in the context of their associated algebras. I begin by exploring the algebraic and logical background for each, drawing parallels and a noticeable symmetry. In addition I describe a thought experiment linking the conceptual foundation of one to a mathematical representation of the other. Finally, I explore the requirements for a more inscrutable relationship between the two pointing out the tantalizing questions this suggestion raises as well as potential answers. The purpose of this article is to show that there is more to this relationship than meets the eye and suggests that a very general Bell-like theorem can be interpreted as a limiting case of the broader generalized uncertainty principle.

Related papers

Uncertainty Relations and Possible Experience

Gregg S. Jaeger

The uncertainty principle can be understood as a condition of joint indeterminacy of classes of properties in quantum theory. The mathematical expressions most closely associated with this principle have been the uncertainty relations, various inequalities exemplified by the well known expression regarding position and momentum introduced by Heisenberg. Here, recent work involving a new sort of “logical” indeterminacy principle and associated relations introduced by Pitowsky, expressable directly in terms of probabilities of outcomes of measurements of sharp quantum observables, is reviewed and its quantum nature is discussed. These novel relations are derivable from Boolean “conditions of possible experience” of the quantum realm and have been considered both as fundamentally logical and as fundamentally geometrical. This work focuses on the relationship of indeterminacy to the propositions regarding the values of discrete, sharp observables of quantum systems. Here, reasons for favoring each of these two positions are considered. Finally, with an eye toward future research related to indeterminacy relations, further novel approaches grounded in category theory and intended to capture and reconceptualize the complementarity characteristics of quantum propositions are discussed in relation to the former.

MathematicsJun 3, 20165 citations

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, 20230 citations

Uncertainty in Quantum Measurements

David Deutsch

The object of this Letter is to show that except in the case of canonically conjugate observables, the generalized Heisenberg inequality does not properly express the quantum uncertainty principle. It is, in general, too weak. An inequality is obtained which does express the principle.

Physical Review LettersFeb 28, 1983908 citations

Algebraic computations in quantum logics

Jeannine J. M. Gabriëls

Mathematical description of quantum phenomena requires an event structure more general than a Boolean algebra. For this purpose, Birkhoff and von Neumann proposed the notion of an orthomodular lattice. Its typical feature is the existence of so–called non–commuting events, which are not simultaneously observable (like position and momentum, according to Heisenberg’s uncertainty principle). There is an old open problem whether the word problem for orthomodular lattices is solvable. Is it possible to decide whether two formulas are equivalent? In Boolean algebras, an easy positive answer is given by a transformation of the formula to a unique normal form. This requires the commutativity, associativity, and distributivity of the Boolean operations (disjunction and conjunction). In orthomodular lattices, the corresponding lattice operations (join and meet) violate distributivity. This disables the use of normal forms. We looked for alternative approaches. E.g., the join is only one of six orthomodular lattice operations generalizing the disjunction. In the thesis, we study the question whether some of the 96 binary operations in orthomodular lattices admit normal forms similar to the classical logic. The first question was which operations satisfy the associative identity, eventually under the assumption that some variables commute. Then we studied monotonicity because it is related to distributivity over the meet and join. The conclusion is that there is no pair of operations in orthomodular lattices admitting “Boolean–like” normal forms. In the last chapter we study “Moufang–like” identities, which were inspired by the algebras of quaternions and octonions. These identities generalize associativity and may enable further progress. As a by–product, we proved interesting, yet unknown, properties of some orthomodular lattice operations (e.g., the Sasaki projection). These new tools simplify algebraic computations and give a chance to develop algorithms more general than the current specialized software.

Jan 1, 20150 citations

Whose Uncertainty Have We Actually Proved? -- Fourier Limits, Coarse-Graining, and the Boundary Between Description and Reality

JOHN DELATORRE

The more technical companion paper is at: https://doi.org/10.5281/zenodo.22118747 The uncertainty principle is commonly presented as one of the deepest declarations quantummechanics makes about physical reality: position and momentum cannot simultaneously possessarbitrarily sharp values. Yet the mathematical inequality underlying the familiar position–momentum relation belongs to a much broader class of Fourier uncertainty theorems that arenot intrinsically quantum. A localized function necessarily possesses a broad Fourier spectrumwhether that function represents an electron, an optical pulse, an acoustic signal, or an entirelyabstract mathematical object. This distinction raises a foundational question: at what point does mathematical uncertainty become physical uncertainty?We examine this question through a coarse-graining construction Cε that maps a finer arithmetic ensemble to a complex field G. A Heisenberg–Weyl-type uncertainty relation for G establishes an exact localization–bandwidth tradeoff, but it does not by itself identify either conjugatevariable with a physical observable or assign a probabilistic interpretation to |G|2. The resulttherefore occupies a precise epistemic position: the mathematical theorem is complete while itsphysical interpretation remains blocked by realization — and, as we make precise below, blockedby at least two logically distinct realization problems rather than one: a coordinate realizationR connecting arithmetic structure to a physical field, and a separate spectral realization Rp connecting the Fourier-conjugate variable to a physical observable such as momentum. Resolvingthe former does not resolve the latter.The situation suggests a broader lesson. An uncertainty relation may characterize limitationsof a representation without establishing indeterminacy of the reality represented. If a physicalrealization of the coarse-graining construction is eventually established, its uncertainty wouldnaturally admit an interpretation as uncertainty of a compressed description relative to anunderlying finer structure. This possibility does not establish hidden variables, determinism, orthe physical validity of the underlying arithmetic ontology. It instead exposes a question that islogically prior to such conclusions: whose uncertainty is an uncertainty relation describing—theworld’s, or ours?

Aug 31, 20260 citations

Relation of the Bell inequalities with quantum logic, hidden variables and information theory

Emilio Santos

I review the relation of the Bell inequalities - characteristic of (classical) probabilities defined on Boolean logics - with noncontextual and local hidden variables theories of quantum mechanics and with quantum information.

arXiv:quant-ph/0207062Jul 10, 20021 citation