Applications of modular inclusions in quantum field theory and quantum gravity

In the last few decades, quantum information theory has expanded our understanding of the entanglement structure of space-time. The information theory quantities, namely the quantum relative entropy, a distinguishability measure between two quantum states, have played a key role in providing new insights into both gravitational systems and ordinary quantum field theory in flat space. What makes relative entropy a unique information measure is its well-defined nature for both discrete quantum systems and continuum field theories alike. One important result that the study of relative entropy has led to is a rigorous proof of Quantum Null Energy Condition (QNEC), a lower bound on the energy density of a local region in space (null components of the stress-energy tensor) by the second derivative of entanglement entropy with respect to shape variations in the light-cone direction. Although previous proofs of QNEC relied on the assumption of separable Hilbert spaces and the use of density matrices \cite{Balakrishnan_2019, Bousso_2016}, which are ill defined in QFT, a theory-independent proof requires recasting of QNEC in terms of relative entropy. To that end, in this thesis, we use the abstract machinery of operator algebras and modular theory, which properly account for infinite dimensions, to prove QNEC. We demonstrate the role played by the modular theory of von Neumann algebras and their inclusion properties in explaining this energy inequality and also draw a connection to the field of quantum error correction. Moreover, we present ongoing work on the role of modular theory in understanding correlation measures in Conformal Field Theories such as mutual information, which is a specific instance of relative entropy, and its connection to the thermal partition function and its breakdown. While our previous work proving QNEC focuses on applications of spacetime inclusions that are light-like, new work on mutual information focuses on inclusions of algebras (correspondingly regions) that are space-like.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Physics

Grantor

University of Illinois Urbana-Champaign

Year dc:date

2025

Author dc:creator

• Ceyhan, Fikret Ali

Contributors dc:contributor

• Faulkner, Thomas
• Leigh, Robert G
• Draper, Patrick I
• Pfaff, Wolfgang

• Entanglement
• Relative Entropy

Statement dc:rights

• Copyright 2025 Fikret Ceyhan

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/130016

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F130016