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arXiv:1709.01207 · Journal of Mathematical Physics

Quantum supervaluationism

Arkady Bolotin

Published Dec 1, 2017• 7 citations
arXivHTMLPDF

Abstract

As it is known, neither classical logical conjunction “and” nor classical logical alternative “either…or” can replace “+” representing a linear superposition of two quantum states. Therefore, to provide a logical account of the quantum superposition, one must either reconsider the standard interpretation of quantum mechanics (making it fit for classical bivalent logic) or replace the standard logic with a deviant logic suitable for describing the superposition. In this paper, a supervaluation approach to the description of the quantum superposition is considered. In accordance with this approach, the indefinite propositions, which correspond to the superposition states, lack truth-values of any kind, even granting that their compounds (such as logical alternative “either…or”) can have truth-values. As an illustration, the supervaluationist account of the superposition of spin states is presented.

DOI
10.1063/1.5008374

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As it is known, neither classical logical conjunction “and” nor classical logical alternative “either…or” can replace “+” representing a linear superposition of two quantum states. Therefore, to provide a logical account of the quantum superposition, one must either reconsider the standard interpretation of quantum mechanics (making it fit for classical bivalent logic) or replace the standard logic with a deviant logic suitable for describing the superposition. In this paper, a supervaluation approach to the description of the quantum superposition is considered. In accordance with this approach, the indefinite propositions, which correspond to the superposition states, lack truth-values of any kind, even granting that their compounds (such as logical alternative “either…or”) can have truth-values. As an illustration, the supervaluationist account of the superposition of spin states is presented.
In the paper, the EPR paradox is explored by the approach of quantum supervaluationism that leads to a "gappy" semantics with the propositions giving rise to truth-value gaps. Within this approach, the statement, which asserts that in the singlet state the system of two (i.e., A and B) spin-1/2 particles possesses the a priori property "spin A is up and spin B is down along the same axis" or "spin A is down and spin B is up along the same axis", does not have the truth-value at all. Consequently, after the verification of, say, the proposition "spin A is up along the z-axis", the statistical population describing the valuation of the logical connective "spin B is down along the z-axis and spin B is up (down) along the x-axis" would have no elements.
The topos approach to the formulation of physical theories includes a new form of quantum logic. We present this topos quantum logic, including some new results, and compare it to standard quantum logic, all with an eye to conceptual issues. In particular, we show that topos quantum logic is distributive, multi-valued, contextual and intuitionistic. It incorporates superposition without being based on linear structures, has a built-in form of coarse-graining which automatically avoids interpretational problems usually associated with the conjunction of propositions about incompatible physical quantities, and provides a material implication that is lacking from standard quantum logic. Importantly, topos quantum logic comes with a clear geometrical underpinning. The representation of pure states and truth-value assignments are discussed. It is briefly shown how mixed states fit into this approach.
The topos approach to the formulation of physical theories includes a new form of quantum logic. We present this topos quantum logic, including some new results, and compare it to standard quantum logic, all with an eye to conceptual issues. In particular, we show that topos quantum logic is distributive, multi-valued, contextual and intuitionistic. It incorporates superposition without being based on linear structures, has a built-in form of coarse-graining which automatically avoids interpretational problems usually associated with the conjunction of propositions about incompatible physical quantities, and provides a material implication that is lacking from standard quantum logic. Importantly, topos quantum logic comes with a clear geometrical underpinning. The representation of pure states and truth-value assignments are discussed. It is briefly shown how mixed states fit into this approach.
Abstract This paper gives a formulation of quantum logic in the abstract algebraic setting laid out by Dunn and Hardegree (2001). On this basis, it provides a comparative analysis of viable quantum logical bivalent semantics and their classical counterparts, thereby showing that the truth-functional status of classical and quantum connectives is not as different as usually thought. Then it points out that bivalent semantics for quantum logic – compatible with realism about quantum mechanics – can be maintained, albeit at the price of truth-functionality. Finally, the paper critically addresses Geoffrey Hellman’s argument (1980) that this lack of truth-functionality entails a change of meaning between classical and quantum connectives.
The term proposition usually denotes in quantum mechanics (QM) an element of (standard) quantum logic (QL). Within the orthodox interpretation of QM the propositions of QL cannot be associated with sentences of a language stating properties of individual samples of a physical system, since properties are nonobjective in QM. This makes the interpretation of propositions problematical. The difficulty can be removed by adopting the objective interpretation of QM proposed by one of the authors (semantic realism, or SR, interpretation). In this case, a unified perspective can be adopted for QM and classical mechanics (CM), and a simple first order predicate calculus L(x) with Tarskian semantics can be constructed such that one can associate a physical proposition (i.e., a set of physical states) with every sentence of L(x). The set PfP^{f}Pf of all physical propositions is partially ordered and contains a subset PTfP^{f}_{T}PTf​ of testable physical propositions whose order structure depends on the criteria of testability established by the physical theory. In particular, PTfP^{f}_{T}PTf​ turns out to be a Boolean lattice in CM, while it can be identified with QL in QM. Hence the propositions of QL can be associated with sentences of L(x), or also with the sentences of a suitable quantum language LTQ(x)L_{TQ}(x)LTQ​(x), and the structure of QL characterizes the notion of testability in QM. One can then show that the notion of quantum truth does not conflict with the classical notion of truth within this perspective. Furthermore, the interpretation of QL propounded here proves to be equivalent to a previous pragmatic interpretation worked out by one of the authors, and can be embodied within a more general perspective which considers states as first order predicates of a broader language with a Kripkean semantics.
In this paper we consider the phenomenon of superpositions in quantum mechanics and suggest a way to deal with the idea in a logical setting from a syntactical point of view, that is, as subsumed in the language of the formalism, and not semantically. We restrict the discussion to the propositional level only. Then, after presenting the motivations and a possible world semantics, the formalism is outlined and we also consider within this schema the claim that superpositions may involve contradictions, as in the case of the Schrödinger's cat, which (it is usually said) is both alive and dead. We argue that this claim is a misreading of the quantum case. Finally, we sketch a new form of quantum logic that involves three kinds of negations and present the relationships among them. The paper is a first approach to the subject, introducing some main guidelines to be developed by a `syntactical' logical approach to quantum superpositions.