Remarks on the entanglement entropy for disconnected regions

TL;DR

This paper investigates the properties of entanglement entropy for disconnected regions in quantum field theory, revealing limitations of existing models and uncovering connections with renormalization group irreversibility.

Contribution

It characterizes the structure of entropy functions under extensivity assumptions and disproves the exact extensivity of several prominent models, including holographic and free field theories.

Findings

Mutual information is extensive for free massless fermions in 2D.

Holographic and free field models do not exhibit exact extensivity.

The entropy function relates to renormalization group irreversibility.

Abstract

Few facts are known about the entanglement entropy for disconnected regions in quantum field theory. We study here the property of extensivity of the mutual information, which holds for free massless fermions in two dimensions. We uncover the structure of the entropy function in the extensive case, and find an interesting connection with the renormalization group irreversibility. The solution is a function on space-time regions which complies with all the known requirements a relativistic entropy function has to satisfy. We show that the holographic ansatz of Ryu and Takayanagi, the free scalar and Dirac fields in dimensions greater than two, and the massive free fields in two dimensions all fail to be exactly extensive, disproving recent conjectures.