Positivity, negativity, and entanglement

TL;DR

This paper investigates the behavior of universal entanglement measures in 4d conformal field theories, revealing conditions under which they can be negative and how they relate to central charges and surface topology.

Contribution

It demonstrates that the sign of universal entanglement entropy in 4d CFTs is indeterminate and links negativity to the difference in central charges and surface genus.

Findings

Universal entanglement entropy can be negative if and only if a > c.

Negative contributions depend on the genus of the entangling surface.

Logarithmic negativity does not always surpass entanglement entropy in a > c theories.

Abstract

We explore properties of the universal terms in the entanglement entropy and logarithmic negativity in 4d CFTs, aiming to clarify the ways in which they behave like the analogous entanglement measures in quantum mechanics. We show that, unlike entanglement entropy in finite-dimensional systems, the sign of the universal part of entanglement entropy is indeterminate. In particular, if and only if the central charges obey $a>c$, the entanglement across certain classes of entangling surfaces can become arbitrarily negative, depending on the geometry and topology of the surface. The negative contribution is proportional to the product of $a-c$ and the genus of the surface. Similarly, we show that in $a>c$ theories, the logarithmic negativity does not always exceed the entanglement entropy.