A bstract 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.
A bstract 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.
Abstract Quantum corrections to holographic entanglement entropy require knowledge of the bulk quantum state. In this paper, we derive a novel dual prescription for the generalized entropy that allows us to interpret the leading quantum corrections in a geometric way with minimal input from the bulk state. The equivalence is proven using tools borrowed from convex optimization. The new prescription does not involve bulk surfaces but instead uses a generalized notion of a flow, which allows for possible sources or sinks in the bulk geometry. In its discrete version, our prescription can alternatively be interpreted in terms of a set of Planck-thickness bit threads, which can be either classical or quantum. This interpretation uncovers an aspect of the generalized entropy that admits a neat information-theoretic description, namely, the fact that the quantum corrections can be cast in terms of entanglement distillation of the bulk state. We also prove some general properties of our prescription, including nesting and a quantum version of the max multiflow theorem. These properties are used to verify that our proposal respects known inequalities that a von Neumann entropy must satisfy, including subadditivity and strong subadditivity, as well as to investigate the fate of the holographic monogamy. Finally, using the Iyer-Wald formalism we show that for cases with a local modular Hamiltonian there is always a canonical solution to the program that exploits the property of bulk locality. Combining with previous results by Swingle and Van Raamsdonk, we show that the con- sistency of this special solution requires the semi-classical Einstein’s equations to hold for any consistent perturbative bulk quantum state.
Abstract We apply the bit thread formulation of holographic entanglement entropy to reduced states describing only the geometry contained within an entanglement wedge. We argue that a certain optimized bit thread configuration, which we construct, gives a purification of the reduced state to a full holographic state obeying a precise set of conditional mutual information relations. When this purification exists, we establish, under certain assumptions, the conjectured E P = E W relation equating the entanglement of purification with the area of the minimal cross section partitioning the bulk entanglement wedge. Along the way, we comment on minimal purifications of holographic states, geometric purifications, and black hole geometries.
A bstract We give a scheme to geometrize the partial entanglement entropy (PEE) for holographic CFT in the context of AdS/CFT. More explicitly, given a point x we geometrize the two-point PEEs between x and any other points in terms of the bulk geodesics connecting these two points. We refer to these geodesics as the PEE threads , which can be naturally regarded as the integral curves of a divergenceless vector field Vxμ V x μ , which we call PEE thread flow . The norm of Vxμ V x μ that characterizes the density of the PEE threads can be determined by some physical requirements of the PEE. We show that, for any static interval or spherical region A , a unique bit thread configuration can be generated from the PEE thread configuration determined by the state. Hence, the non-intrinsic bit threads are emergent from the intrinsic PEE threads. For static disconnected intervals, the vector fields describing a divergenceless flow is no longer suitable to reproduce the RT formula. We weight a PEE thread with the number of times it intersects with any homologous surface. Instead, the RT formula is perfectly reformulated by the minimization of the summation of PEE threads with all possible assignment of weights.
We apply the bit thread formulation of holographic entanglement entropy to reduced states describing only the geometry contained within an entanglement wedge. We argue that a certain optimized bit thread configuration, which we construct, gives a purification of the reduced state to a full holographic state obeying a precise set of conditional mutual information relations. When this purification exists, we establish, under certain assumptions, the conjectured EP = EW relation equating the entanglement of purification with the area of the minimal cross section partitioning the bulk entanglement wedge. Along the way, we comment on minimal purifications of holographic states, geometric purifications, and black hole geometries.
A bstract The concept of the generalized entanglement wedge was recently proposed by Bousso and Penington, which states that any bulk gravitational region a possesses an associated generalized entanglement wedge E ( a ) ⊃ a on a static Cauchy surface M in general gravitational spacetimes, where E ( a ) may contain an entanglement island I ( a ). It suggests that the fine-grained entropy for bulk region a is given by the generalized entropy S gen ( E ( a )). Motivated by this proposal, we extend the quantum bit thread description to general gravitational spacetimes, no longer limited to the AdS spacetime. By utilizing the convex optimization techniques, a dual flow description for the generalized entropy S gen ( E ( a )) of a bulk gravitational region a is established on the static Cauchy surface M , such that S gen ( E ( a )) is equal to the maximum flux of any flow that starts from the boundary ∂M and ends at bulk region a , or equivalently, the maximum number of bit threads that connect the boundary ∂M to the bulk region a . In addition, the nesting property of flows is also proved. Thus the basic properties of the entropy for bulk regions, i.e. the monotonicity, subadditivity, Araki-Lieb inequality and strong subadditivity, can be verified from flow perspectives by using properties of flows, such as the nesting property. Moreover, in max thread configurations, we find that there exists some lower bounds on the bulk entanglement entropy of matter fields in the region E ( a ) \ a , particularly on an entanglement island region I ( a ) ⊂ ( E ( a ) \ a ), as required by the existence of a nontrivial generalized entanglement wedge. Our quantum bit thread formulation may provide a way to investigate more fine-grained entanglement structures in general spacetimes.
Motivated by the kinematic-space description of gravitational subregions \cite{Basu:2026hbg}, we allow partial entanglement entropy (PEE) threads to be sourced not only from the asymptotic boundary, but also from boundaries of subregions. Based on this setup, we develop a framework to construct configurations for PEE threads and bit threads sourced from surfaces in the AdS bulk. The situations we have analyzed include the Poincaré AdS3 and the planar BTZ black brane with an end-of-the-world (EOW) brane, and the entanglement wedges of boundary multi-intervals. We construct novel configurations of PEE threads and bit threads in these situations, which shed new light in our understanding of the AdS3/BCFT2 correspondence and the holography defined in the entanglement wedge motivated by surface/state correspondence. A notable feature we found is that, although the elementary PEE threads are geodesics, the integral curves of the coarse-grained bit-thread current are generically not geodesics. Across the explicit constructions, the integrated source currents reduce to endpoint distance-difference potentials, which make divergencelessness, the norm bound and bottleneck saturation geometrically transparent. Our results clarify how boundary, brane, horizon and RT-surface degrees of freedom participate in holographic entanglement and provide a concrete link between PEE, bit threads, subregion kinematic space and the surface/state correspondence.