Continuum canonical purifications

TL;DR

This paper develops a universal method for constructing canonical purifications of algebraic states applicable across all quantum theories, extending previous work and exploring their properties and conditions for purity.

Contribution

It introduces a general construction of canonical purifications for algebraic states, applicable to all quantum theories, and analyzes their properties including purity and modular conjugations.

Findings

Canonical purifications can be constructed for all algebraic states.

Conditions for purity of canonical purifications are identified.

Connections to GNS and natural-cone purifications are established.

Abstract

We construct and characterize canonical purifications for general algebraic states, extending prior constructions by Woronowicz and by Dutta/Faulkner to general quantum theories. Given a state on a $*$-algebra, the canonical purification is a state on a "doubled" algebra that admits an interpretation in terms of CRT reflection. This interpretation holds for all quantum theories, even in the absence of gravity. We then identify conditions under which canonical purifications are "pure" in the technical sense, compute their modular conjugations, and relate them to GNS and natural-cone purifications in certain settings. In an appendix, we develop a general theory of von Neumann algebras generated by unbounded $*$-algebras. In a forthcoming paper with Caminiti and Capeccia, we provide an application of this general theory to the problem of excitability in quantum field theory.