Coalgebras, Chu Spaces, and Representations of Physical Systems

TL;DR

This paper explores coalgebraic and Chu space frameworks for representing quantum systems, addressing limitations of standard coalgebras by introducing fibrational structures to incorporate contravariance and symmetries.

Contribution

It introduces a fibrational structure on coalgebras and Chu spaces to better model quantum systems, including symmetries and dynamics, extending previous representations.

Findings

Coalgebraic semantics for quantum systems via final coalgebras.

A fibrational structure captures contravariance and symmetries.

A truncation functor links coalgebras and Chu spaces.

Abstract

We revisit our earlier work on the representation of quantum systems as Chu spaces, and investigate the use of coalgebra as an alternative framework. On the one hand, coalgebras allow the dynamics of repeated measurement to be captured, and provide mathematical tools such as final coalgebras, bisimulation and coalgebraic logic. However, the standard coalgebraic framework does not accommodate contravariance, and is too rigid to allow physical symmetries to be represented. We introduce a fibrational structure on coalgebras in which contravariance is represented by indexing. We use this structure to give a universal semantics for quantum systems based on a final coalgebra construction. We characterize equality in this semantics as projective equivalence. We also define an analogous indexed structure for Chu spaces, and use this to obtain a novel categorical description of the category of Chu spaces. We use the indexed structures of Chu spaces and coalgebras over a common base to define a truncation functor from coalgebras to Chu spaces. This truncation functor is used to lift the full and faithful representation of the groupoid of physical symmetries on Hilbert spaces into Chu spaces, obtained in our previous work, to the coalgebraic semantics.