Holographic entanglement contour, bit threads, and the entanglement tsunami

TL;DR

The paper develops a universal entanglement contour for 1+1d systems and extends it holographically to higher dimensions via bit threads, weaving connections to entanglement density, kinematic space, and holographic measures like LN and EoP. It shows how a contour decomposition $S(A)=\frac{1}{2}\sum_i\big[S(A_i|\cdots)+S(A_i|\cdots)\big]$ captures diverse static CFT structures (vacuum, thermal, defect, black-hole microstates, massive deformations) and elucidates dynamical entanglement through global and local quenches, clarifying the distinction between local contour propagation velocity $v_c$ and nonlocal tsunami growth velocity $v_E$. The construction leverages a max-flow formulation where $s_A(x)=|v(x)|$ and, in holographic setups, can be restricted to the entanglement wedge to define LN and EoP contours; this yields explicit expressions in several geometries and ties the contour to kinematic space via Crofton forms and entanglement density. Overall, the work provides a unified, geometrically motivated framework linking fine-grained entanglement structure, holography, and non-equilibrium dynamics, with potential extensions to covariant flows, higher-curvature gravity, gauge theories, tensor networks, and generalized negativity/purification contours.

Abstract

We study the entanglement contour, a quasi-local measure of entanglement, and propose a generic formula for the contour in 1+1d quantum systems. We use this formalism to investigate the real space entanglement structure of various static CFTs as well as local and global quantum quenches. The global quench elucidates the spatial distribution of entanglement entropy in strongly interacting CFTs and clarifies the interpretation of the entanglement tsunami picture. The entanglement tsunami effectively characterizes the non-local growth of entanglement entropy while the contour characterizes the local propagation of entanglement. We generalize the formula for the entanglement contour to arbitrary dimensions and entangling surface geometries using bit threads, and are able to realize a holographic contour for logarithmic negativity and the entanglement of purification by restricting the bulk spacetime to the entanglement wedge. Furthermore, we explore the connections between the entanglement contour, bit threads, and entanglement density in kinematic space.

Figures (4)

• Figure 1: In the black hole geometry (subspace shown), there are two configurations for the minimal entanglement wedge cross section, the standard minimal surface (short dashed lines) and the disconnected surface reaching the horizon (longer dashes). For the entanglement entropy, bit threads can terminate on the black hole horizon, so both the green and blue bit threads contribute. For LN/EoP, only the blue bit threads will contribute.
• Figure 2: In kinematic space, the conditional mutual information is computed by the "bulk" volumes. The left and right shaded lavender regions are $I(A_2, \bar{A} | A_1)$ and $I(A_2, \bar{A} | A_3)$ respectively. Interpreting MERA as kinematic space, the entanglement contour is computed by the number of isometries (green triangles) in the shaded lavender regions.
• Figure 3: The entanglement contour following a global quench with $c = 1$ and $\beta = 2$. After initial quadratic growth, the contour waves propagate at $v_c = 2$ and cross one another at $t = l/4$, only to halt at $t = l/2$. The contour for the interval saturates at its thermal value.
• Figure 4: (left) The entanglement contour following a local (Calabrese-Cardy) quench for semi-infinite intervals with $c = 1$ and $\epsilon = 1/10$. (right) Local heavy operator quench with central charge $c = 1$, $\alpha_{\psi} = 1/2$, and $\delta = 1$. Now, $v_c = 1$. Once the wave front passes, the contour relaxes to its ground state value.