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arXiv:2001.10170

A note on the kinematic space associated with a subregion

Xing Huang

Published Jan 28, 2020• 0 citations
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Abstract

We discuss the issue in constructing the kinematic space of the geodesics lying partially inside the entanglement wedge associated with a single interval. We then resolve the problem by working with the equivalent kinematic space of the reflected geodesics. We also show that the length of a reflected geodesic corresponds to the (generalized) reflected entropy, which can be computed using entirely the information obtained from the reduced density matrix, satisfying the requirement of the subregion-subregion duality.

DOI
10.48550/arxiv.2001.10170

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We discuss the issue in constructing the kinematic space of the geodesics lying partially inside the entanglement wedge associated with a single interval. We then resolve the problem by working with the equivalent kinematic space of the reflected geodesics. We also show that the length of a reflected geodesic corresponds to the (generalized) reflected entropy, which can be computed using entirely the information obtained from the reduced density matrix, satisfying the requirement of the subregion-subregion duality.
A bstract We study the holographic duality between the reflected entropy and the entanglement wedge cross section with the first order correction. In the field theory side, we consider the reflected entropy for ρABm {\rho}_{AB}^m ρABm​ ρ AB m , where ρ AB is the reduced density matrix for two intervals in the ground state. The reflected entropy in the 2d holographic conformal field theories is computed perturbatively up to the first order in m − 1 by using the semiclassical conformal block. In the gravity side, we compute the entanglement wedge cross section in the backreacted geometry by cosmic branes with tension T m which are anchored at the AdS boundary. Comparing both results we find a perfect agreement, showing the duality works with the first order correction in m − 1.
It is proposed that spatial subregions in quantum gravity can be assigned pure states, rather than mixed reduced density matrices. The state is prepared by a partially frozen gravitational path integral, in which a spacetime subregion containing the spatial subregion is fixed while the field configurations and ambient geometry are summed over. In the semiclassical regime, we further propose a holographic prescription for the entanglement entropy of bipartitions of this state, with a frozen-region analogue of the homology constraint. The prescription satisfies nontrivial self-consistency conditions, including strong subadditivity, complementarity, and entanglement wedge nesting, and reproduces several known entropy formulas in holography and gravity as special cases. The construction suggests an observer-dependent entanglement wedge labeled by the frozen subregion.
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This paper investigates the entanglement entropy inequality and explores the presentation of mutual information and conditional mutual information in kinematic space. Specifically, we examine the regions within kinematic space responsible for computing these physical quantities, enabling a more intuitive understanding of the entanglement entropy inequality. Building upon this, we employ the concept of double holography to analyze the properties of the entanglement inequality in any given region. By utilizing kinematic space, we calculate the contribution of the bulk to the holographic entanglement entropy in double holography. In conclusion, we establish that kinematic space substantiates a conjecture, namely that the entanglement entropy of an entire region can be expressed as a linear combination of the entanglement entropy of a single interval within the entangled region.
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_33​ 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_33​/BCFT2_22​ 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.
A bstract In holographic theories, the reflected entropy has been shown to be dual to the area of the entanglement wedge cross section. We study the same problem in random tensor networks demonstrating an equivalent duality. For a single random tensor we analyze the important non-perturbative effects that smooth out the discontinuity in the reflected entropy across the Page phase transition. By summing over all such effects, we obtain the reflected entanglement spectrum analytically, which agrees well with numerical studies. This motivates a prescription for the analytic continuation required in computing the reflected entropy and its Rényi generalization which resolves an order of limits issue previously identified in the literature. We apply this prescription to hyperbolic tensor networks and find answers consistent with holographic expectations. In particular, the random tensor network has the same non-trivial tripartite entanglement structure expected from holographic states. We furthermore show that the reflected Rényi spectrum is not flat, in sharp contrast to the usual Rényi spectrum of these networks. We argue that the various distinct contributions to the reflected entanglement spectrum can be organized into approximate superselection sectors. We interpret this as resulting from an effective description of the canonically purified state as a superposition of distinct tensor network states. Each network is constructed by doubling and gluing various candidate entanglement wedges of the original network. The superselection sectors are labelled by the different cross-sectional areas of these candidate entanglement wedges.