Entanglement measures and their properties in quantum field theory
Stefan Hollands, Ko Sanders
TL;DR
The paper develops a rigorous, operator-algebraic framework to quantify entanglement between causally disjoint regions in quantum field theory, where the standard vN entropy fails for mixed, type III algebras. It introduces the relative entanglement entropy ER and related measures (EI, EN, EM, ED) within algebraic QFT, and derives general upper and lower bounds across broad settings including free fields, integrable models, and conformal theories, leveraging BW- and modular-nuclearity and Tomita– Takesaki theory. A key theme is how entanglement scales with region separation: near-touching corridors yield area-law-type lower bounds, while large separations in massive theories give sub-exponential or exponential decay of ER, with explicit bounds in terms of operator dimensions, spins, cross-ratios, and charges. The work also contrasts ER with replica-trick results, highlights the role of nuclearity and split properties, and connects to charged sectors and DHR/DR theory, offering a comprehensive toolkit for quantitative entanglement in QFT with potential links to holographic concepts.
Abstract
An entanglement measure for a bipartite quantum system is a state functional that vanishes on separable states and that does not increase under separable (local) operations. It is well-known that for pure states, essentially all entanglement measures are equal to the v. Neumann entropy of the reduced state, but for mixed states, this uniqueness is lost. In quantum field theory, bipartite systems are associated with causally disjoint regions. There are no separable (normal) states to begin with when the regions touch each other, so one must leave a finite "safety-corridor". Due to this corridor, the normal states of bipartite systems are necessarily mixed, and the v. Neumann entropy is not a good entanglement measure in the above sense. In this paper, we study various entanglement measures which vanish on separable states, do not increase under separable (local) operations, and have other desirable properties. In particular, we study the relative entanglement entropy, defined as the minimum relative entropy between the given state and an arbitrary separable state. We establish rigorous upper and lower bounds in various quantum field theoretic (QFT) models, as well as also model-independent ones. The former include free fields on static spacetime manifolds in general dimensions, or integrable models with factorizing $S$-matrix in 1+1 dimensions. The latter include bounds on ground states in general conformal QFTs, charged states (including charges with braid-group statistics) or thermal states in theories satisfying a "nuclearity condition". Typically, the bounds show a divergent behavior when the systems get close to each other--sometimes of the form of a generalized "area law"--and decay when the systems are far apart. Our main technical tools are of operator algebraic nature.