Temporal Entanglement from Holographic Entanglement Entropy

TL;DR

The paper advances a Lorentzian framework for temporal entanglement entropy by analytically continuing spatial entangling regions into timelike ones, regularized near the light cone. In holography, it prescribes selecting complex extremal surfaces anchored to timelike subregions via a minimal real-area criterion after continuation, ensuring UV-IR consistency and correct vacuum limits. The authors test the prescription in holographic CFTs on R^{1,2} and on R×S^1, revealing how multiple complex saddles organize and how null singularities emerge from the continuation when combined with homology constraints. This approach clarifies when timelike entanglement is well-defined and highlights the necessity of a careful ordering between analytic continuation and saddle minimization. The framework promises broader applications to temporal quantum-information structures and may illuminate phenomena where timelike correlations play a central role.

Abstract

Recently, several notions of entanglement in time have emerged as a novel frontier in quantum many-body physics, quantum field theory and gravity. We propose a systematic prescription to characterize temporal entanglement in relativistic quantum field theory in a general state for an arbitrary subregion on a flat, constant-time slice in a flat spacetime. Our prescriptions starts with the standard entanglement entropy of a spatial subregion and amounts to transporting the unchanged subregion to boosted time slices all the way across the light cone when it becomes in general a complex characterization of the corresponding temporal subregion. For holographic quantum field theories, our prescription amounts to an analytic continuation of all codimension-two bulk extremal surfaces satisfying the homology constraint and picking the one with the smallest real value of the area as the leading saddle point. We implement this prescription for holographic conformal field theories in thermal states on both a two-dimensional Lorentzian cylinder and three-dimensional Minkowski space, and show that it leads to results with self-consistent physical properties of temporal entanglement.

Figures (16)

• Figure 1: (a) Geometrical analytic continuation of the boundary region $\mathcal{R}$. Starting from a region that lies on a constant time slice (1), the light cone is crossed by slightly complexifying the angle $\theta$ when still in the spacelike regime (2). Once the timelike regime is reached (3), the angle can be increased further to attain purely temporal separations (4). (b). The trajectory followed by the angle $\theta$ in the analytic continuation. The light cone is crossed by evading the divergence associated with the proper size of the subregion going to $0$ when $\theta=\frac{\pi}{4}$ with a circle of arbitrarily small radius $\varepsilon$ in the complex $\theta$-plane.
• Figure 2: Geometry of a strip boundary subregion $\mathcal{R}$ in three-dimensional Minkowski space. The strip is rotated in the $t$-$x$ plane keeping the coordinate extent $\Delta r$ fixed. See Fig. \ref{['fig:rotation']} for the case of a general subregion.
• Figure 3: Left panel: width of the strip $\Delta r$ as a function of the position of the entangling surface tip $z_t$ in the bulk spacetime. Right panel: regularized area density of the strip $\mathcal{A}_\textrm{reg}$ as a function of the strip width $\Delta r$.
• Figure 4: For a strip with $\theta=\frac{\pi}{2}$, $z_t$ for all the known complex extremal hypersurfaces in an AdS$_4$-Schwarzschild black brane. Blue (green) curves correspond to vacuum-connected (vacuum-disconnected) solutions. Horizons understood as roots of $f(z) = 0$, see Eq. \ref{['metric']}, are represented as black stars, and critical extremal surfaces as red crosses. This plot appeared earlier in Ref. Heller:2024whi.
• Figure 5: Top panel: width of the strip $\Delta r$ as a function of the tip of the extremal surface $z_t$ for various values of $\theta = \cot^{-1}(1+\eta)$. Bottom panel: regularized area density $\mathcal{A}_\textrm{reg}$ as a function of $\Delta r$ for various angles. Each $\mathcal{A}_\textrm{reg}$ has been shifted by $\log(\frac{\pi}{4} - \theta)^2$ to prevent the curves from overlapping at large $\Delta r$.