Comments on Defining Entanglement Entropy

TL;DR

The paper tackles how to define and compute entanglement entropies for spatial regions in theories where Hilbert-space factorization fails, notably in gauge theories. It develops an algebraic framework linking subregion algebras and their centers to boundary conditions on entangling edges, giving explicit path-integral replica-trick expressions for full, distillable, and gauge-invariant entropies. It then discusses the extended-Hilbert-space approach as a complementary method, showing its correspondence to open-edge boundary conditions and its success in reproducing topological terms like the Kitaev–Kitaev topological entanglement entropy and Kabat’s edge contributions. The authors conjecture that holographic entanglement entropy computes the full entanglement entropy of the maximal regional algebra, and they illuminate how edge modes and boundary data shape universal terms across dimensions and theories, including Chern–Simons and conformal field 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.

Figures (4)

• Figure 1: Depictions of generating operators for various choices of algebras for spin chains discussed in the text. Algebras generated by these sets are all supported on the entire lattice, i.e. there are no sites on which all operators act trivially. Red indicates locations of central generators.
• Figure 2: Generators of fermionic algebras discussed in the text, presented using their bosonized equivalents. Each algebra commutes with the product of all $Z$'s, i.e. with the total fermion parity $(-1)^F$. Locations of all additional central generators are red. In the case of Majorana boundary conditions, the removal of $Z_1$ is taken to mean that $(-1)^F$ is still a central generator, but that an overall constraint is imposed on superselection sector weights such that $(-1)^F = 0$ hold as an operator equation.
• Figure 3: A $\mathbb Z_2$ gauge theory on a lattice $\mathbb M$ with boundary $\partial \mathbb M$ (drawn in blue). There is a central generator $G_v$ at every vertex of this lattice, including on edge sites $v \in \partial \mathbb M$. Boundary plaquettes are those faces that contain boundary links; for instance $f_1$ contains a boundary link $\ell_1 \in \partial \mathbb M$. When $W_{f_1}$ is removed from the algebra, $X_{\ell_1}$ becomes another central generator. It corresponds to the electric field operator parallel to the boundary.
• Figure 4: A side view of the path integral boundary conditions on the $\tau = 0$ slice, illustrating eq. \ref{['def rhoVk']}. At all other values of $\tau$, the integral is unconstrained; even at $\tau = 0$, there is a sum over all values of fields $\vartheta$ in $\bar{\mathbb{V}}$, i.e. outside the two red circles. The setup shown here calculates $\langle\widetilde{\phi}^{(k)}|\rho^{(k)}_{\mathbb{V}}|\phi^{(k)}\rangle$ when central generators are at or near $\partial \mathbb{V}$; their locations are shown in red. The values of central generators are denoted $k$ and are the same at $\tau = 0^+$ and $\tau = 0^-$. The replica trick calculates $\textrm{Tr} [\rho^{(k)}_{\mathbb{V}}]^n$ by taking $n$ copies of this setup and constraining $\phi^{(k)}$ on the replica $l$ to equal $\widetilde{\phi}^{(k)}$ on replica $l + 1$, with $n + 1 \equiv 1$. The fields in the red region are then eigenstates of central generators with eigenvalue $k$ on each replica.