Pseudo Entropy in dS/CFT and Time-like Entanglement Entropy

TL;DR

Faced with complex holographic entanglement entropy in dS/CFT, the paper argues that these quantities are better understood as pseudo entropy. It defines pseudo entropy via reduced transition matrices and shows the dS/CFT and time-like entanglement entropy in AdS/CFT are related by analytic continuation. The imaginary part of pseudo entropy is interpreted as signaling emergent time in the holographic description, with explicit 2D and higher-dimensional calculations and numerical checks supporting the claim. This framework links non-Hermitian boundary dynamics to a time-geometry interpretation in holography.

Abstract

We study holographic entanglement entropy in dS/CFT and introduce time-like entanglement entropy in CFTs. Both of them take complex values in general and are related with each other via an analytical continuation. We argue that they are correctly understood as pseudo entropy. We find that the imaginary part of pseudo entropy implies an emergence of time in dS/CFT.

Figures (5)

• Figure 1: The reduced density matrix $\rho_A$ of a CFT in dS/CFT gets non-hermitian, which leads to pseudo entropy. Here we consider a CFT on $\mathbb{S}^d$, i.e. the boundary of de Sitter space.
• Figure 2: Definition of Time-like Entanglement Entropy
• Figure 3: The left panel shows the space-like geodesic (green curve) in the Poincaré coordinate, which is embedded in the global coordinate with an additional time-like geodesic (red line) in the right panel.
• Figure 4: Space and time-like geodesics whose length gives the time-like Holographic Entanglement Entropy in the BTZ geometry. For time intervals not symmetric about the origin (right panel) the space-like geodesics intersect the past and future singularities in different locations.
• Figure 5: Numerical results for time-like entanglement and the corresponding fit functions for free scalar and Dirac theories.