Mutual information and the structure of entanglement in quantum field theory

TL;DR

This paper investigates mutual information between spatial subsystems in scale-invariant quantum field theories as a finite, continuum probe of entanglement structure. It shows that mutual information carries universal singularities when regions collide, reflecting the underlying degrees of freedom, and validates these ideas through holographic calculations and Fermi-liquid considerations. A central proposal is the generalization of twist operators to higher dimensions, enabling computation of entanglement entropy, Renyi entropy, and mutual information via a massless (d-1)-form field framework. The work also discusses the potential for a renormalization group monotone based on entanglement per scale, highlighting both supporting holographic evidence and important caveats in higher dimensions.

Abstract

I study the mutual information between spatial subsystems in a variety of scale invariant quantum field theories. While it is derived from the bare entanglement entropy, the mutual information offers a more refined probe of the entanglement structure of quantum field theories because it remains finite in the continuum limit. I argue that the mutual information has certain universal singularities that are a manifestation of the idea of "entanglement per scale". Moreover, I propose a method, based on an ansatz for higher dimensional twist operators, to compute the entanglement entropy, Renyi entropy, and mutual information in a general quantum field theory. The relevance of these results to the search for renormalization group monotones, to holographic duality, and to entanglement based simulation methods for many body systems are all discussed.

Figures (5)

• Figure 1:
• Figure 2: Strip geometry for the calculation of mutual information in the translation invariant case. We assume the $L \gg w \gg x$ so that the minimal surface problem reduces to a single variable problem. A singularity in the mutual information develops as $x$ approaches zero.
• Figure 3: A sketch of the two bulk minimal surfaces relevant for the calculation of the mutual information. The translation invariant spatial coordinate is suppressed along with the time. In the top panel, the minimal surface for two widely separated strips is simply two copies of the minimal surface for a single strip. In the bottom panel, when the two regions come close, a new minimal surface appears which connects the inner and outer boundaries of the two regions. In this case, there is a non-zero holographic mutual information.
• Figure 4: The analog of the strip geometry, but with one flat side of each strip replaced by an arc of a circle with large radius of curvature. The resulting regions are now nearly translation invariant in the $y$ direction, but also only collide at a point as $x(y)$ approaches $0$.
• Figure 5: An example of the replica method with $n=4$ copies. The region whose entanglement entropy we are calculating is in blue while the rest of the system is in red. The $t$ axis is imaginary time. The copies are glued together so that one passes from copy $\alpha$ to copy $\alpha+1$ when passing through $t=0$ from below in a blue region, while in the red regions no such transition occurs. The blue region at $t=0$ is thus a "branch surface" that terminates on a spacetime codimension $2$ conical singularity given by the boundary between red and blue at $t=0$. The twist operator lies along this $1$ dimensional locus in spacetime.