Operational Quantum Logic in Dynamic Perspective

Bob Coecke

Alexandru Baltag, Sonja Smets

Sonja Smets

Sonja Smets

Research within the operational approach to the logical foundations of physics has recently pointed out a new perspective in which quantum logic can be viewed as an intuitionistic logic with an additional operator to capture its essential, i.e., non-distributive, properties. In this paper we will offer an introduction to this approach. We will focus further on why quantum logic has an inherent dynamic nature which is captured in the meaning of “orthomodularity” and on how it motivates physically the introduction of dynamic implication operators, each for which a deduction theorem holds with respect to a dynamic conjunction. As such we can offer a positive answer to the many who pondered about whether quantum logic should really be called a logic. Doubts to answer the question positively were in first instance due to the former lack of an implication connective which satisfies the deduction theorem within quantum logic.

Bob Coecke, David Moore, Sonja Smets

This paper surveys some recent developments towards a dynamic quantum logic and outlines its explicite construction -- some analogies and contrasts with other logics of dynamics are indicated. Abstract: The development of ``(static) operational quantum logic" points out that classical boolean structures are too rigid to describe the actual and potential properties of quantum systems. On the other hand, an intuitionistic perspective on operational quantum logic, guides us in the direction of incorporating dynamics logically by reconsidering the primitive propositions required to describe the behavior of a quantum system, in particular in view of the emergent disjunctivity due to the non-determinism of quantum measurements. A further elaboration on "intuitionistic quantum logic" emerges into a "dynamic operational quantum logic", which allows us to express dynamic reasoning in the sense that we can capture how actual properties propagate, including their temporal causal structure, and provides a unified logical description of systems which evolve or which are submitted to measurements. This setting reveals that even static operational quantum logic bears a hidden dynamic ingredient in terms of what is called "the orthomodularity" of the lattice-structure. Focusing on the quantale semantics for dynamic operational quantum logic, we delineate some points of difference with the existing quantale semantics for (non)-commutative linear logic.

Bob Coecke, David Moore, Alexander Wilce