The holographic entropy cone from marginal independence

Abstract

The holographic entropy cone characterizes the relations between entanglement entropies for a spatial partitioning of the boundary spacetime of a holographic CFT in any state describing a classical bulk geometry. We argue that the holographic entropy cone, for an arbitrary number of parties, can be reconstructed from more fundamental data determined solely by subadditivity of quantum entropy. We formulate certain conjectures about graph models of holographic entanglement, for which we provide strong evidence, and rigorously prove that they all imply that such a reconstruction is possible. Our conjectures (except only for the weakest) further imply that the necessary data is remarkably simple. In essence, all one needs to know to reconstruct the holographic entropy cone, is a certain subset of the extreme rays of this simpler “subadditivity cone”, namely those which can be realized in holography. This recasting of the bewildering entanglement structure of geometric states into primal building blocks paves the way to distilling the essence of holography for the emergence of a classical bulk spacetime.

A holographic inequality for N = 7 regions

“…It would be interesting to study in detail how the work of [15] intersects the HCAE conjecture and, by extension, the content of this paper. Another way to organize and predict the structure of holographic entropy inequalities appeared in [12].…”

“…Relation to prior work. The recent reference [15] formulated a program for finding holographic entropy inequalities, which uses the subadditivity of entanglement entropy to bootstrap one's way from lower-N to higher-N inequalities. That program has some -thus far unexplored -relation with the HCAE conjecture because subadditivity is a measure of bipartite correlations.…”

“…Two exceptions include the monogamy of mutual information (1.1), which was interpreted in [10] as quantifying perfect tensor-like entanglement of a four-partite state, and the infinite class of inequalities reported in [7], which discretize a measure of entanglement called differential entropy [11]. For other attempts to interpret known inequalities and/or identify future ones, see [12][13][14][15]. Moreover, known facts about the holographic entropy cone are subject to several important provisos.…”

On the relation between the subadditivity cone and the quantum entropy cone

“…As the number of parties N increases, the search for new inequalities quickly becomes computationally unfeasible due to the fact that the combinatorics governing the number of inequalities typically grows doubly exponentially as a function of N. Furthermore, fixing N is immaterial in QFT, since one can always imagine further partitioning the N regions into smaller subregions. For these reasons, [1] took a different approach to the characterization of the HEC. Rather than looking for the explicit expression of the inequalities at some given N, * templehe@caltech.edu † veronika@physics.ucdavis.edu ‡ maxrota@ucdavis.edu [1] attempted to provide a more implicit description of the HEC for an arbitrary number of parties by relating it to the quantum entropy cone (QEC) [22], and to distill the essential information that would allow for its reconstruction (at least in principle).…”

“…For these reasons, [1] took a different approach to the characterization of the HEC. Rather than looking for the explicit expression of the inequalities at some given N, * templehe@caltech.edu † veronika@physics.ucdavis.edu ‡ maxrota@ucdavis.edu [1] attempted to provide a more implicit description of the HEC for an arbitrary number of parties by relating it to the quantum entropy cone (QEC) [22], and to distill the essential information that would allow for its reconstruction (at least in principle). Drawing from the ideas of [13,15], [1] suggested that this essential information is the solution to the holographic marginal independence problem (HMIP) [23].…”