Entwinement as a possible alternative to complexity

TL;DR

This paper introduces entwinement as a gauge-invariant measure of entanglement in 2D CFTs with $ ext{Z}_N$ symmetry, offering an alternative to holographic complexity for describing bulk geometries like conical defects and BTZ black holes.

Contribution

It defines entwinement in a gauge-invariant way for general CFTs with $ ext{Z}_N$ symmetry and links it holographically to non-minimal geodesics, providing a new tool for bulk geometry reconstruction.

Findings

Entwinement can describe the full bulk geometry of conical defects.

Proposed a gauge-invariant definition of entwinement for thermal states.

Entwinement offers an alternative to holographic complexity.

Abstract

Unlike the standard entanglement entropy considered in the holographic context, entwinement measures entanglement between degrees of freedom that are not associated to a spatial subregion. Entwinement is defined for two-dimensional CFTs with a discrete $\mathbb{Z}_N$ gauge symmetry. Since the Hilbert space of these CFTs does not factorize into tensor products, even the entanglement entropy associated to a spatial subregion cannot be defined as the von Neumann entropy of a reduced density matrix. While earlier works considered embedding the density matrix into a larger, factorizing Hilbert space, we apply a gauge invariant approach by using a density matrix uniquely defined through its relation to the local algebra of observables. We furthermore obtain a fully gauge invariant definition of entwinement valid for general CFTs with $\mathbb{Z}_N$ gauge symmetry in terms of all observables acting on the degrees of freedom considered. Holographically, entwinement is dual to the length of non-minimal geodesics present for conical defects or black holes. In this context, we propose a definition of entwinement for thermal states dual to the BTZ black hole. Our results show that "entwinement is enough" to describe the full bulk geometry for the conical defect and provide strong hints that the same holds true for the BTZ black hole. Thus, it provides an alternative to holographic complexity for the theories considered.