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Continuum canonical purifications

Jonathan Sorce

TL;DR

The work develops a general theory of canonical purifications for states on *-algebras, extending prior constructions to general quantum-field-theoretic settings. It defines the canonical purification on $\mathcal{A}_0 \otimes \mathcal{A}_0^{\text{op}}$ using time-reversal (CRT reflection) and connects it to modular data, GNS purifications, and the natural cone, with detailed results on positivity, GNS isomorphisms, modular conjugation, and purity (factorial vs non-factorial). The paper further generalizes the construction to non-separating states, establishing positivity and a robust GNS-structure via separating extensions, and clarifies the relation between canonical and natural purifications. Finally, it analyzes how canonical purifications relate to natural-cone purifications under unitary/center adjustments, providing a framework that unifies several purification notions and sets the stage for applications in quantum field theory phenomena such as excitability.

Abstract

We construct and characterize canonical purifications for general algebraic states, extending prior constructions by Woronowicz and by Dutta/Faulkner to general quantum field theories. Given a quantum state on a *-algebra, the canonical purification is a state on a "doubled" algebra that admits an interpretation in terms of CRT reflection. We study the conditions under which these enlarged states are "pure" in the technical sense, compute their modular conjugations, and relate them to GNS and natural-cone purifications in certain settings. 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.

Continuum canonical purifications

TL;DR

The work develops a general theory of canonical purifications for states on *-algebras, extending prior constructions to general quantum-field-theoretic settings. It defines the canonical purification on using time-reversal (CRT reflection) and connects it to modular data, GNS purifications, and the natural cone, with detailed results on positivity, GNS isomorphisms, modular conjugation, and purity (factorial vs non-factorial). The paper further generalizes the construction to non-separating states, establishing positivity and a robust GNS-structure via separating extensions, and clarifies the relation between canonical and natural purifications. Finally, it analyzes how canonical purifications relate to natural-cone purifications under unitary/center adjustments, providing a framework that unifies several purification notions and sets the stage for applications in quantum field theory phenomena such as excitability.

Abstract

We construct and characterize canonical purifications for general algebraic states, extending prior constructions by Woronowicz and by Dutta/Faulkner to general quantum field theories. Given a quantum state on a *-algebra, the canonical purification is a state on a "doubled" algebra that admits an interpretation in terms of CRT reflection. We study the conditions under which these enlarged states are "pure" in the technical sense, compute their modular conjugations, and relate them to GNS and natural-cone purifications in certain settings. 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.

Paper Structure

This paper contains 34 sections, 4 theorems, 219 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Finite dimensions and beyond
  3. Finite-dimensional formulas
  4. Time-reversal and GNS
  5. The need for generalization
  6. Canonical purifications of separating states
  7. Basic algebraic properties
  8. Positivity
  9. GNS isomorphism
  10. Computing the modular conjugation
  11. Purity in the factorial case
  12. Canonical purifications of non-separating states
  13. The general strategy
  14. Positivity
  15. Structure of the GNS space
  16. ...and 19 more sections

Key Result

Lemma 3.1

On $\mathcal{H}_{\widehat{\omega}}$, we will call $(\mathcal{A}_0 \otimes 1)_{\widehat{\omega}}, (1 \otimes \mathcal{A}_0^{\text{op}})_{\widehat{\omega}},$ and $(\mathcal{A}_0 \otimes \mathcal{A}_0^{\text{op}})_{\widehat{\omega}}$ the minimal von Neumann algebras generated by the $*$-algebras in par

Figures (2)

  • Figure 1: The canonical purification is an entangled algebraic state, $\widehat{\omega},$ defined on two copies of the same degrees of freedom, with one copy subject to time reversal. In the language of the text, the left causal diamond contains the degrees of freedom $\mathcal{A}_0$, while the right causal diamond contains the degrees of freedom $\mathcal{A}_0^{\text{op}}$.
  • Figure 2: The GNS purification introduces entangled degrees of freedom with no intrinsic structure; this is represented by the left side of the above figure. In the case that $\omega$ is faithful, the canonical purification arises by using the modular conjugation to identify the new degrees of freedom with a second copy of the original ones.

Theorems & Definitions (8)

  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 4.1
  • proof
  • Corollary 4.2
  • proof