Comments on Defining Entanglement Entropy

TL;DR

This paper analyzes various definitions of entanglement entropy in quantum theories, providing precise path integral formulas and exploring gauge theories without regularization, ultimately linking to holographic methods in conformal theories.

Contribution

It introduces a general algebraic framework for defining multiple types of entanglement entropy and connects different approaches, including those for gauge theories without regularization.

Findings

Path integral expressions for full, distillable, and gauge-invariant entropies.

Extension of Hilbert space approach to gauge theories like Chern-Simons.

Conjecture linking full entanglement entropy to holographic calculations in conformal theories.

Abstract

We revisit the issue of defining the entropy of a spatial region in a broad class of quantum theories. In theories with explicit regularizations, working within an elementary but general algebraic framework applicable to matter and gauge theories alike, we give precise path integral expressions for three known types of entanglement entropy that we call full, distillable, and gauge-invariant. For a class of gauge theories that do not necessarily have a regularization in our framework, including Chern-Simons theory, we describe a related approach to defining entropies based on locally extending the Hilbert space at the entangling edge, and we discuss its connections to other calculational prescriptions. Based on results from both approaches, we conjecture that it is always the full entanglement entropy that is calculated by standard holographic techniques in strongly coupled conformal theories.