Bit threads, Einstein's equations and bulk locality

Abstract

In the context of holography, entanglement entropy can be studied either by i) extremal surfaces or ii) bit threads, i.e., divergenceless vector fields with a norm bound set by the Planck length. In this paper we develop a new method for metric reconstruction based on the latter approach and show the advantages over existing ones. We start by studying general linear perturbations around the vacuum state. Generic thread configurations turn out to encode the information about the metric in a highly nonlocal way, however, we show that for boundary regions with a local modular Hamiltonian there is always a canonical choice for the perturbed thread configurations that exploits bulk locality. To do so, we express the bit thread formalism in terms of differential forms so that it becomes manifestly background independent. We show that the Iyer-Wald formalism provides a natural candidate for a canonical local perturbation, which can be used to recast the problem of metric reconstruction in terms of the inversion of a particular linear differential operator. We examine in detail the inversion problem for the case of spherical regions and give explicit expressions for the inverse operator in this case. Going beyond linear order, we argue that the operator that must be inverted naturally increases in order. However, the inversion can be done recursively at different orders in the perturbation. Finally, we comment on an alternative way of reconstructing the metric non-perturbatively by phrasing the inversion problem as a particular optimization problem.

51 pages, 7 figures and 2 appendices. v2: Matches published version

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• Deriving the PEE proposal from the Locking bit thread configuration
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• More of the Bulk from Extremal Area Variations
• Geometrizing the Partial Entanglement Entropy: from PEE Threads to Bit Threads
• Bulk reconstruction of metrics inside black holes by complexity
• Quantum bit threads and holographic entanglement
• Quantum information probes of charge fractionalization in large-$N$ gauge theories
• Bending the Bruhat-Tits Tree I:Tensor Network and Emergent Einstein Equations
• Thread/State correspondence: from bit threads to qubit threads
• Diving inside holographic metals
• Holographic thermal entropy from geodesic bit threads
• Towards bit threads in general gravitational spacetimes
• First Law and Quantum Correction for Holographic Entanglement Contour
• Holographic geometry/real-space entanglement correspondence and metric reconstruction
• The landscape of complexity measures in 2D gravity
• Bit Threads: From Entanglement to Geometric Entropies
• Illuminating entanglement shadows of BTZ black holes by a generalized entanglement measure
• Entanglement entropy in cubic gravitational theories