Holographic Entropy Relations

Abstract

We develop a framework for the derivation of new information theoretic quantities which are natural from a holographic perspective. We demonstrate the utility of our techniques by deriving the tripartite information (the quantity associated to monogamy of mutual information) using a set of abstract arguments involving bulk extremal surfaces. Our arguments rely on formal manipulations of surfaces and not on local surgery or explicit computation of entropies through the holographic entanglement entropy prescriptions. As an application, we show how to derive a family of similar information quantities for an arbitrary number of parties. The present work establishes the foundation of a broader program that aims at the understanding of the entanglement structures of geometric states for an arbitrary number of parties. We stress that our method is completely democratic with respect to bulk geometries and is equally valid in static and dynamical situations. While rooted in holography, we expect that our construction will provide a useful characterization of multipartite correlations in quantum field theories.

The Holographic Entropy Arrangement

“…However, once obtained, these quantities can be examined for their utility more broadly. This philosophy was exploited in our recent framework where we used the holographic setting to propose new information quantities . The specifics of the construction relied on the tools of holographic entanglement entropy using the RT and HRT proposals, which resulted in measures given by specific linear combinations of entropies.…”

“…To identify potentially useful measures, two conditions were required in []. The first requirement was that it should be possible for any reasonable quantity to vanish, so that one can examine situations in which a particular type of correlations is absent.…”

“…As the number of parties increases, the situation gets much more intricate, and in general it is an open question how to find all the primitive quantities. A first step in this direction however was already presented in []. Under a certain restriction on the topology of the allowed field theory configurations, one can show that the problem simplifies dramatically, and one can in fact find all the primitive quantities for an arbitrary number of parties.…”

“…While one might suspect that the topological restriction on field theory configurations that led to the ‐theorem was quite special, and that it should be easy to find other configurations that generate new quantities, this is far from true. A remarkable property of the framework of [] is the relatively small number of quantities that get generated as the number of parties increases. While the present work will still mostly focus on a relatively small , we will show how one is forced to consider rather fine‐tuned classes of configurations to circumvent the result of the ‐theorem and generate new information quantities.…”

“…As explained in [], each primitive quantity can naturally be associated to a hyperplane in entropy space. The set of all primitive quantities for a given number of parties is then an arrangement of hyperplanes – the holographic entropy arrangement .…”