Quantum bit threads and holographic entanglement

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.

Bath deformations, islands and holographic complexity

“…Further, recently a formulation of the subregion volume has been given in terms of bit threads [107,108]. The exciting open question is to understand whether bit thread formulation can provide a more vivid picture of the discontinuity of the subregion complexity in the doubly holographic braneworld model, perhaps along the line of [109,110].…”

Balanced Partial Entanglement and Mixed State Correlations

“…In the purview of holography, one can consider the finer description offered by the Ryu-Takanayagi prescription to relate points on the minimal surface to the corresponding boundary points[23,48]. Explicit constructions of the entanglement contour and the PEE in terms of bit threads[49] are provided in[50][51][52][53] (also see[23,24,47,48]).…”

Local measures of entanglement in black holes and CFTs

“…as in (9). Writing the proposal this way makes it clear that the CMI proposal gives partial entanglement entropies that depend only on the state on A, and not how the state is purified by degrees of freedom in Ā.…”

“…The approach taken in the literature has been to reduce the space of possible formulas by specifying additional physically-motivated properties for the entanglement contour to satisfy, with the hope that with enough sufficiently constraining requirements the entanglement contour will be uniquely defined. This search for uniqueness is undermined somewhat by the fact that in holographic theories the boundary flux density of a flux-maximising bit thread configuration is a density function for the boundary subregion's von Neumann entropy, so is a natural entanglement contour candidate, and yet those thread configurations and their boundary flux densities are highly non-unique [6][7][8][9]. In examples where there exists a special set of bit threads based on geodesics [10] the boundary bit thread flux density has been calculated and shown to equal the entanglement contour calculated with the formula used in this paper [7,11].…”