We show that every holographic entropy inequality can be recast in the form: "some entanglement wedges reach deeper in the bulk than some other entanglement wedges." When the inequality is saturated, the two sets of wedges reach equally deep. Because bulk depth geometrizes CFT scales, the inequalities enforce and protect the holographic Renormalization Group.
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20 pages plus appendices, 3 figures. v2: typos in two equations corrected
We show that every holographic entropy inequality can be recast in the form: "some entanglement wedges reach deeper in the bulk than some other entanglement wedges." When the inequality is saturated, the two sets of wedges reach equally deep. Because bulk depth geometrizes CFT scales, the inequalities enforce and protect the holographic Renormalization Group.
We explore the structure of holographic entropy relations (associated with ‘information quantities’ given by a linear combination of entanglement entropies of spatial sub-partitions of a CFT state with geometric bulk dual). Such entropy relations can be recast in multiple ways, some of which have significant advantages. Motivated by the already-noted simplification of entropy relations when recast in terms of multipartite information, we explore additional simplifications when recast in a new basis, which we dub the K-basis, constructed from perfect tensor structures. For the fundamental information quantities such a recasting is surprisingly compact, in part due to the interesting fact that entropy vectors associated to perfect tensors are in fact extreme rays in the holographic entropy cone (as well as the full quantum entropy cone). More importantly, we prove that all holographic entropy inequalities have positive coefficients when expressed in the K-basis, underlying the key advantage over the entropy basis or the multipartite information basis.
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.
Abstract We explore the structure of holographic entropy relations (associated with ‘information quantities’ given by a linear combination of entanglement entropies of spatial sub-partitions of a CFT state with geometric bulk dual). Such entropy relations can be recast in multiple ways, some of which have significant advantages. Motivated by the already-noted simplification of entropy relations when recast in terms of multipartite information, we explore additional simplifications when recast in a new basis, which we dub the K-basis, constructed from perfect tensor structures. For the fundamental information quantities such a recasting is surprisingly compact, in part due to the interesting fact that entropy vectors associated to perfect tensors are in fact extreme rays in the holographic entropy cone (as well as the full quantum entropy cone). More importantly, we prove that all holographic entropy inequalities have positive coefficients when expressed in the K-basis, underlying the key advantage over the entropy basis or the multipartite information basis.
<!--HTML-->This talk will explain recent puzzling revelations in the ongoing efforts to obtain a useful characterization of entanglement structure of geometric states in a holographic CFT, via the so-called holographic entropy cone (HEC). The relations between subsystem entanglement entropies which delimit this cone are known explicitly for only a rather coarse subdivision of the system (specified by N spatial regions, for up to N = 5). We argue that, subject to a certain graph theoretic conjecture, the task of finding the HEC for arbitrary N can be recast in terms of a much simpler combinatorial one which effectively reduces to the connectivity of entanglement wedges. More specifically, the N-party HEC can be reconstructed by solving the holographic marginal independence problem (HMIP) for a finer subdivision N′ ≥ N, which technically amounts to identifying which extreme rays of this subadditivity cone are realizable holographically. Curiously, despite the fact that subadditivity is a universal property which states that total correlation cannot be negative, the non-trivial facets of the HEC constructed therefrom nevertheless cannot be recast as correlation measures.
We prove two new infinite families of holographic entropy inequalities. A key tool is a graphical arrangement of terms of inequalities that is based on entanglement wedge nesting. It associates the inequalities with tessellations of the torus and the projective plane, which reflect a certain topological aspect of entanglement wedge nesting. The inequalities prove a prior conjecture about the holographic entropy cone. We discuss their relation to black hole physics and differential entropy, and sketch applications to quantum error correction, quantifying randomness of quantum states, and others. Published by the American Physical Society 2024
In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal field theory on the conformal boundary of AdS_{d+2} is, when the AdS radius is appropriately related to the parameters of the CFT, equal to 1/4G times the area of the d-dimensional minimal surface in the AdS bulk which has the junction of those complementary regions as its boundary, where G is the bulk Newton constant. We point out here that the RT-equality implies that, in the quantum theory on the bulk AdS background which is related to the boundary CFT according to Rehren's 1999 algebraic holography theorem, the entanglement entropy between two complementary bulk Rehren wedges is equal to 1/4G times the (suitably cut off) area of their shared ridge. (This follows because of the geometrical fact that, for complementary ball-shaped regions, the RT minimal surface is precisely the shared ridge of the complementary bulk Rehren wedges which correspond, under Rehren's bulk-wedge to boundary double-cone bijection, to the complementary boundary double-cones whose bases are the RT complementary balls.) This is consistent with the Bianchi-Meyers conjecture -- that, in a theory of quantum gravity, the entanglement entropy, S, between the degrees of freedom of a given region with those of its complement is S = A/4G (+ lower order terms) -- but only if the phrase 'degrees of freedom' is replaced by 'matter degrees of freedom'. It also supports related previous arguments of the author -- consistent with the author's 'matter-gravity entanglement hypothesis' -- that the AdS/CFT correspondence is actually only a bijection between just the matter (i.e. non-gravity) sector operators of the bulk and the boundary CFT operators.