Timelike entanglement entropy

TL;DR

<3-5 sentence high-level summary> The paper introduces timelike entanglement entropy (EE), a complex-valued generalization of entanglement entropy obtained by analytically continuing spacelike subsystems to timelike ones, and shows it is naturally interpreted as a pseudo entropy. It provides dual viewpoints: field-theoretic definitions (replica method and Wick-rotation of coordinates) with explicit 2d results and numerical checks, and a holographic framework where timelike EE is computed from a stationary combination of spacelike and timelike extremal surfaces in AdS, including BTZ, shock waves, and local quenches. The work extends to AdSd+1/CFTd and discusses holographic pseudo entropy in dS/CFT, highlighting a deep connection between emergent time and holographic information, while outlining open issues in higher dimensions and multiple saddles.

Abstract

We define a new complex-valued measure of information called the timelike entanglement entropy (EE) which in the boundary theory can be viewed as a Wick rotation that changes a spacelike boundary subregion to a timelike one. An explicit definition of the timelike EE in 2d field theories is provided followed by numerical computations which agree with the analytic continuation of the replica method for CFTs. We argue that timelike EE should be correctly interpreted as another measure previously considered, the pseudo entropy, which is the von Neumann entropy of a reduced transition matrix. Our results strongly imply that the imaginary part of the pseudo entropy describes an emergent time which generalizes the notion of an emergent space from quantum entanglement. For holographic systems we define the timelike EE as the total complex valued area of a particular stationary combination of both space and timelike extremal surfaces which are homologous to the boundary region. For the examples considered we find explicit matching of our optimization procedure and the careful implementation of the Wick rotation in the boundary CFT. We also make progress on higher dimensional generalizations and relations to holographic pseudo entropy in de Sitter space.

Figures (27)

• Figure 1: Holographic Timelike Entanglement Entropy in AdS$_3/$CFT$_2$ (left) and Holographic Pseudo entropy in dS$_3/$CFT$_2$ (right). The green curves and red curves describe the spacelike and timelike geodesic, whose lengths give the real and imaginary part of the entropy. Each of blue intervals is the subsystem A.
• Figure 2: Definition of standard entanglement entropy (left) and timelike entanglement entropy (right) in two dimensional field theories.
• Figure 3: Timelike entanglement entropy via a Wick rotation of coordinates.
• Figure 4: Timelike entanglement for free scalar theory for a connected region. The $t$-direction is infinite and the $x$-direction is compactified as $x\sim x+R$ and we also put the regulator $\delta$. The left panel where the data of different color lie on the top of each other corresponds to fixed $R$ and different $\delta$. As expected modulo some numerical defects for large sugberions the timelike EE does not depend on the ratio of $R / \delta$. The right panel is showing results for different $R$ with fixed $R/\delta$. The structure of the fit function perfectly matches with our analytic expectation. As $R$ is increasing larger subregions are considered and the difference of data sets is encoded in different UV cut-offs which is reflected in the constant term in our fit functions.
• Figure 5: Timelike EE for free scalar theory for a connected region. We consider $t$ to be compactified as $t\sim t-i\beta$ and $R=\infty$. The data points correspond to different values for $\beta$ and the solid curves are fitting functions up to an irrelevant constant of the form $a\log\left(\frac{\beta}{\pi\epsilon}\sinh\frac{\pi T_0}{\beta}\right)$ which lead to $a=0.324$, $a=0.367$, and $a=0.376$ decreasing the temperature.