Coalgebras with Symmetries and Modelling Quantum Systems

Abstract

This paper describes a generalization of the usual category in which coalgebras are considered, and its application to modelling quantum systems and their physical symmetries. Following the programme of work initiated in [1], [2], we aim to model systems described by the laws of quantum physics using coalgebraic techniques. A broader notion of the morphisms of coalgebras is given, in which diagrams are allowed to commute only up to appropriate natural isomorphism. This relaxed setting is then shown to have analogues of coalgebraic notions such as bisimulations, with properties that parallel the usual coalgebraic ones closely. This new setting is then exploited to give coalgebraic models of quantum systems in which the conceptually important physical symmetries are given as automorphisms of a suitable coalgebra.

Finally, we investigate coalgebraic logic in this setting, showing that there is a natural notion of “symmetry modality” that can be exploited. The notions of Schrödinger and Heisenberg evolution are discussed, and it is argued that Heisenberg evolution is more natural in the coalgebraic setting. It is then shown that these additional modalities can be used to give an adequate and expressive coalgebraic logic for quantum system in which state evolution and measurement outcomes can be described by suitable modal operators. An appropriate model of this logic then gives predictions consistent with the laws of quantum mechanics.