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Lectures on entanglement in quantum field theory

Horacio Casini, Marina Huerta

TL;DR

This work develops a cohesive information-theoretic framework for quantum field theory by organizing QFT around algebras of observables, states, and entropic measures. It connects entanglement entropy, mutual information, and relative entropy to fundamental QFT structures (causal algebras, replica techniques, modular theory) and uses them to derive irreversible RG behavior, energy-entropy bounds, and symmetry-encoded entanglement phenomena. Central results include the replica-trick foundation for EE, the emergence of universal logarithmic terms tied to conformal anomalies, and entropy-based formulations of the second law, ANEC, and QNEC. The treatment further integrates generalized symmetries and topological aspects into entropic order parameters, providing a robust, geometry-driven lens to study phases and RG flows in QFT.

Abstract

These notes grew from a series of lectures given by the authors during the last decade. They will be published in the proceedings of TASI 2021. After a brief introduction to quantum information theory tools, they are organized in four chapters covering the following subjects: Entanglement in quantum field theory, Irreversibility theorems, Energy-entropy bounds, Entanglement and symmetries.

Lectures on entanglement in quantum field theory

TL;DR

This work develops a cohesive information-theoretic framework for quantum field theory by organizing QFT around algebras of observables, states, and entropic measures. It connects entanglement entropy, mutual information, and relative entropy to fundamental QFT structures (causal algebras, replica techniques, modular theory) and uses them to derive irreversible RG behavior, energy-entropy bounds, and symmetry-encoded entanglement phenomena. Central results include the replica-trick foundation for EE, the emergence of universal logarithmic terms tied to conformal anomalies, and entropy-based formulations of the second law, ANEC, and QNEC. The treatment further integrates generalized symmetries and topological aspects into entropic order parameters, providing a robust, geometry-driven lens to study phases and RG flows in QFT.

Abstract

These notes grew from a series of lectures given by the authors during the last decade. They will be published in the proceedings of TASI 2021. After a brief introduction to quantum information theory tools, they are organized in four chapters covering the following subjects: Entanglement in quantum field theory, Irreversibility theorems, Energy-entropy bounds, Entanglement and symmetries.
Paper Structure (70 sections, 294 equations, 33 figures)

This paper contains 70 sections, 294 equations, 33 figures.

Table of Contents

  1. Introduction
  2. Some general bibliography
  3. Algebras, states, and some basic tools of quantum information
  4. Algebras
  5. States and density matrices
  6. Entanglement
  7. Entropy
  8. Modular flow and modular Hamiltonian
  9. Relative entropy
  10. Mutual information and strong subadditivity
  11. Evolution and the second law
  12. Purification and Tomita-Takesaki theory
  13. Exercises
  14. Notes and references
  15. Entanglement entropy in QFT
  16. ...and 55 more sections

Figures (33)

  • Figure 1: The causal complement $V'$ of the region $V$ (shaded), and the causal completion $V"=W$.
  • Figure 2: Algebras of small regions near $t=0$ generating the algebra containing $W$.
  • Figure 3: A square on the lattice. Area law and logarithmic term.
  • Figure 4: Left: a shape with $6$ right angle corners. Right: a square rotated with respect to the lattice.
  • Figure 5: Mutual information of two circles of radius $R$ separated by $R$ in a square lattice.
  • ...and 28 more figures