Properties and operational propositions in quantum mechanics

Enrico G. Beltrametti, Gianni Cassinelli

Charles H. Randall, David J. Foulis

Joachim Schröter

Abstract It is shown that the interpretation of quantum mechanics as a theory describing systems for which their propositions are always valid or not leads to a contradiction within the theory. The proof does not depend on any specific property of measurements, but only the usual description of ensembles of quantum objects by statistical operators is used.

Hatice Kırmacı

Quantum Logic Distributivity And The Algebraic Structure Of Propositions As with many subatomic particles, the question of under what conditions the electron behaves as a particle or as a wave continues to be the subject of research. It was necessary to employ appropriate logic to organise the propositions describing the behaviour of such an object at the subatomic scale into a precise from and to establish coherent relationships between them. Quantum logic, which can be regarded as an interpretation of quantum mechanics, arose from this need. In classical mechanics, the state and properties of a system are described using the propositions of classical logic, which is based on Boolean algebra. In quantum mechanics, however, the fact that the operators representing the measured quantities do not commute has shown that quantum phenomena cannot be described using the propositions of classical logic. It is therefore necessary to have a formal framework suited to the propositions describing quantum phenomena. In our study, we highlight the various differences between classical logic and quantum logic, examine the distributive property and discuss, from the perspective of Birkhoff and von Neumann, the algebraic structure to which quantum logic might correspond. At the conclusion of our research, we find that, in order to assert that all experimental propositions logically equivalent to a given experimental proposition constitute a physical property, the operators corresponding to the measured quantities must commute with one another. Boolean algebra has an algebraic structure that allows classical logic to be formalised. In such algebra, the distributive property holds; thus, distributivity does not pose a problem for classical logic. Consequently, the distributive property does not always hold in quantum logic and quantum systems can be modelled using orthomodular lattices associated with closed subspaces of Hilbert space. We believe that this model must evolve in parallel with the discovery of new quantum phenomena and that it provides a framework conductive to this evolution. Keywords: Quantum logic; distributivity; partially order sets (posets); lattice; orthomodular lattice; Hilbert space; Boolean algebra.

Jeffrey Bub

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

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.