Schr\"{o}dinger's problem with cats: measurements and states in the Groupoid Picture of Quantum Mechanics

TL;DR

This paper explores the evolution of classical-quantum system states using the groupoid formalism, demonstrating the classical system's structure and the impossibility of certain entanglement evolutions.

Contribution

It introduces a groupoid-based framework for classical and quantum systems, proving classical systems correspond to totally disconnected groupoids and showing entanglement evolution constraints.

Findings

Classical systems correspond to totally disconnected groupoids with Abelian von Neumann algebras.

The groupoid formalism confirms the impossibility of evolving classical-quantum product states into entangled states via internal unitaries.

The approach links classical systems to Birkhoff-von Neumann notions within the groupoid framework.

Abstract

Schr\"{o}dinger's famous Gedankenexperiment involving a cat is used as a motivation to discuss the evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently. It is shown that the notion of classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. In accordance with Raggio's theorem, the impossibility of evolving the product state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution which is internal to the composite system itself is proved in the groupoid formalism.