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.
Highlights
Simplicity, entanglement entropy has become a powerful tool to investigate some of the most fundamental aspects of the holographic correspondence, from bulk reconstruction to the emergence of spacetime
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
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
Summary
In [13] it was proposed that the entanglement entropy of a holographic theory associated to a spatial boundary region A is given by the maximum flux of a bounded, divergenceless, bulk vector field through the region in question. Since the vector field must be divergenceless, the integral curves should start and end at the boundary These three conditions are completely generic, since the choice of integral curves is highly non-unique we will impose a further constraint to reduce the arbitrariness in the possible vector fields, without eliminating it completely: 4. The final non-trivial check is to verify that the magnitude of the vector field respects the upper bound away from the minimal surface, |V | ≤ 1 If this is not the case the choice of integral curves should be modified and the process should be repeated until the bound is achieved. The critical/essential step of our construction relies on the choice of what we call “good” integral curves
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