Multipartite entanglement structure of fibered link states

We study the patterns of multipartite entanglement in Chern-Simons theory with compact simple gauge group <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>G</a:mi></a:mrow></a:math> and level <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>k</c:mi></c:math> for states defined by the path integral on “link complements,” i.e., compact manifolds whose boundaries consist of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>n</e:mi></e:math> topologically linked tori. We focus on link complements which can be described topologically as fibrations over a Seifert surface. We show that the entanglement structure of such fibered link complement states is controlled by a topological invariant, the monodromy of the fibration. Thus, the entanglement structure of a Chern-Simons link state is not simply a function of the link, but also of the background manifold in which the link is embedded. In particular, we show that any link possesses an embedding into some background that leads to Greenberger–Horne–Zeilinger state (GHZ)-like entanglement. Furthermore, we demonstrate that all fibered links with periodic monodromy have GHZ-like entanglement, i.e., a partial trace on any link component produces a separable state. These results generalize to any three dimensional topological field theory with a dual chiral rational conformal field theory.