Geometric aspects of holographic bit threads

Abstract

We revisit the recent reformulation of the holographic prescription to compute entanglement entropy in terms of a convex optimization problem, introduced by Freedman and Headrick. According to it, the holographic entanglement entropy associated to a boundary region is given by the maximum flux of a bounded, divergenceless vector field, through the corresponding region. Our work leads to two main results: (i) We present a general algorithm that allows the construction of explicit thread configurations in cases where the minimal surface is known. We illustrate the method with simple examples: spheres and strips in vacuum AdS, and strips in a black brane geometry. Studying more generic bulk metrics, we uncover a sufficient set of conditions on the geometry and matter fields that must hold to be able to use our prescription. (ii) Based on the nesting property of holographic entanglement entropy, we develop a method to construct bit threads that maximize the flux through a given bulk region. As a byproduct, we are able to construct more general thread configurations by combining (i) and (ii) in multiple patches. We apply our methods to study bit threads which simultaneously compute the entanglement entropy and the entanglement of purification of mixed states and comment on their interpretation in terms of entanglement distillation. We also consider the case of disjoint regions for which we can explicitly construct the so-called multi-commodity flows and show that the monogamy property of mutual information can be easily illustrated from our constructions. arXiv:1811.08879v3 [hep-th]

Towards a bit threads derivation of holographic entanglement of purification

“…(14) itself, using the construction involving black holes and subsequent row reduction of the basis states. That construction suggests that the number of terms in the superposition (14) will scale at most as dim H (AB) c . Because the total set of states for the entire boundary CFT scales as dim H AB(AB) c ∼ e O(c) , there will be a suppression in the entropy-of-mixing term relative to the leading term that goes like…”

“…We will simply take bit threads to be boundary-anchored one-dimensional objects whose density is at most 1/4G N , defining thread density, as in Ref. [14], as the length of threads within some small neighborhood divided by the volume of that neighborhood. This generalization was proposed in Ref.…”

“…Figure 4: Starting with a thread configuration that saturates the number of threads intersecting γ AB , γ A , and γ B , as shown in (a) and described in Ref. [14], the goal is to construct a configuration like (c), which also saturates Γ. (The parts of threads that cross the exterior component of γ AB and that lie outside the entanglement wedge have been suppressed in these diagrams.)…”

“…5.2.2 of Ref. [14], via either cutting-and-gluing or combing. There, Agón et al construct a configuration of threads that saturates the number of threads crossing γ AB , but not Γ (see their Fig.…”

“…purifications that saturate this bound comprise a class of optimal purifications or completions of the state in question. The E P = E W conjecture has been further extended, studied, and generalized in recent work [14][15][16][17][18][19][20][21][22].…”