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A Topological Study of Contextuality and Modality in Quantum Mechanics

Graciela Domenech, Freytes Solari, Hector Carlos, Christian DE RONDE

Published Dec 29, 2006• 16 citations
OpenAlex

Abstract

Kochen-Specker theorem rules out the non-contextual assignment of values to physical magnitudes. Here we enrich the usual orthomodular structure of quantum mechanical propositions with modal operators. This enlargement allows to refer consistently to actual and possible properties of the system. By means of a topological argument, more precisely in terms of the existence of sections of sheaves, we give an extended version of Kochen-Specker theorem over this new structure. This allows us to prove that contextuality remains a central feature even in the enriched propositional system.

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Kochen-Specker theorem rules out the non-contextual assignment of values to physical magnitudes. Here we enrich the usual orthomodular structure of quantum mechanical propositions with modal operators. This enlargement allows to refer consistently to actual and possible properties of the system. By means of a topological argument, more precisely in terms of the existence of sections of sheaves, we give an extended version of Kochen-Specker theorem over this new structure. This allows us to prove that contextuality remains a central feature even in the enriched propositional system.
Modal quantum theory (MQT) is a simplified cousin of ordinary Hilbert space quantum theory. We show that two important theorems of actual quantum theory, the Kochen-Specker theorem excluding non-contextual hidden variables and the Conway-Kochen "free will theorem" about entangled systems, have direct analogues in MQT. The structure of possible measurement results for an entangled system in MQT cannot be represented by probability assignments satisfying the no-signaling principle, such as those given by ordinary quantum theory.
The semantics underlying the propositional structure of Hilbert space quantum mechanics involves an inherent ambiguity concerning the impossibility of assigning definite truth values to all propositions pertaining to a quantum system without generating a Kochen-Specker contradiction. Although the preceding Kochen-Specker result forbids a global, absolute assignment of truth values to quantum mechanical propositions, it does not exclude ones that are contextual. In this respect, the Bub-Clifton “uniqueness theorem” is utilized for arguing that truth-value definiteness is consistently restored with respect to a determinate sublattice of propositions defined by the state of the quantum system concerned and a suitable “preferred” observable. It is suggested that the most natural choice of the latter, especially for confronting the problem of truth valuation in quantum mechanics, results in the determinateness of the observable to be measured. An account of truth of contextual correspondence is thereby provided that is appropriate to the quantum domain of discourse. The resulting account of truth is compatible with a realist conception of truth. Such an account essentially denies that there can be a universal context of reference or an Archimedean standpoint from which to evaluate logically the totality of facts of nature.
Modal quantum theory (MQT) is a simplified cousin of ordinary Hilbert space quantum theory. We show that two important theorems of actual quantum theory, the Kochen-Specker theorem excluding non-contextual hidden variables and the Conway-Kochen "free will theorem" about entangled systems, have direct analogues in MQT. The proofs of these analogue theorems are similar to, but much simpler than, the originals. We also show that the structure of possible measurement results for an entangled system in MQT cannot be represented by probability assignments satisfying the no-signaling principle, such as those given by ordinary quantum theory.
Contextuality is a key distinguishing feature between classical and quantum physics. It expresses a fundamental obstruction to describing quantum theory using classical concepts. In turn, understood as a resource for quantum computation, it is expected to hold the key to quantum advantage. Yet, despite its long recognised importance in quantum foundations and, more recently, in quantum computation, the structural essence of contextuality has remained somewhat elusive - different frameworks address different aspects of the phenomenon, yet their precise relationship often remains unclear. This issue already looms large at the level of the Bell-Kochen-Specker theorem: while traditional proofs proceed by showing the nonexistence of valuations, the notion of state-independent contextuality in the marginal approach allows to prove contextuality from seemingly weaker assumptions. In the light of this, and at the absence of a unified mathematical framework for Kochen-Specker contextuality, the original algebraic approach has been widely abandoned, in favour of the study of contextual correlations. Here, we reinstate the algebraic perspective on contextuality. Concretely, by building on the novel concept of context connections, we reformulate the algebraic relations between observables originally postulated by Kochen and Specker, and we explicitly demonstrate their consistency with the notion of state-independent contextuality. In the present paper, we focus on the new conceptual ideas and discuss them in the concrete setting of spin-1 observables, specifically those in the example of [S. Yu and C.H. Oh, Phys. Rev. Lett., 108, 030402 (2012)]; in a companion paper, we generalise these ideas, obtain a complete characterisation of Kochen-Specker contextuality and provide a detailed comparison with the related notions of contextuality in the marginal and graph-theoretic approach.