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A LOGIC-ALGEBRAIC FRAMEWORK FOR CONTEXTUALITY AND MODALITY IN QUANTUM SYSTEMS

Héctor Freytes

Published Mar 5, 2014• 0 citations
Publisher

Abstract

In this work we develop a modal structure for the simultaneous treatment of \nactual and possible properties of quantum systems. A logical system based on \northomodular lattices enriched with a modal operator is given, obtaining algebraic \ncompleteness and completeness with respect to a Kripke-style semantic. \nWe show that, in spite of the fact that, the language is enriched with the addition \nof a modal operator, contextuality remains a central feature of quantum \nsystems.

DOI
10.1142/9789814596299_0014

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In this work we develop a modal structure for the simultaneous treatment of \nactual and possible properties of quantum systems. A logical system based on \northomodular lattices enriched with a modal operator is given, obtaining algebraic \ncompleteness and completeness with respect to a Kripke-style semantic. \nWe show that, in spite of the fact that, the language is enriched with the addition \nof a modal operator, contextuality remains a central feature of quantum \nsystems.
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.
According to quantum mechanics, the pure states of a microsystem are represented by vectors in a Hilbert Space. Sentences of the form, "x є L" (where x is the state vector for a system, L a subspace of the appropriate Hilbert space), may be called Q-propositions: such sentences serve to summarise our information about the results of possible experiments on the system. Quantum logic investigates the relations which hold among the Q-propositions about a given physical sys tem. These logical relations correspond to algebraic relations among the subspaces of Hilbert space. The algebra of this set of subspaces is non-Boolean, and may be regarded either as an orthomodular lattice or as a partial Boolean algebra. With each type of structure we can associate a logic. A general approach to the semantics for such a logic is provided in terms of interpretations of a formal language within an algebraic structure; an interpretation maps sentences of the language homomorphically onto elements of the structure. When the structure in question is a Boolean algera, the resulting logic is classical; here we develop a semantics for the logic associated with partial Boolean algebras. Two systems of proof, based on the natural deduction systems of Gentzen, are shown for this logic. With respect to the given sematics, these calculi are sound and weakly complete. Strong completeness is conjectured. Quantum logic deals with the logical relations between sentences, and so is properly called a logic. However, it is the logic appropriate to a limited class of sentences: proposals that it should replace classical logic wherever the latter is used should be viewed with suspicion.
We introduce two modal natural deduction systems that are suitable to represent and reason about transformations of quantum registers in an abstract, qualitative, way. Quantum registers represent quantum systems, and can be viewed as the structure of quantum data for quantum operations. Our systems provide a modal framework for reasoning about operations on quantum registers (unitary transformations and measurements) in terms of possible worlds (as abstractions of quantum registers) and accessibility relations between these worlds. We give a Kripke--style semantics that formally describes quantum register transformations, and prove the soundness and completeness of our systems with respect to this semantics.
Quantum logic has developed in various directions since its inception.However, many existing logical approaches either focus only on pure states, or use Hilbert-space models in which the abstract relationship between pure states, mixed states, and probabilities is not made explicit.In this work, we introduce a model designed to handle quantum mixed states abstractly.Furthermore, we construct a proof system for the proposed logic and establish its completeness with respect to the intended semantics.