Timelike entanglement entropy Revisited

TL;DR

Provides a rigorous operator-algebraic definition of timelike entanglement entropy in QFT via the timelike envelope and the timelike tube theorem, defining S(T)=S(E_T) and arguing it is real-valued. Uses the split property and a canonical intermediate Type I factor to define S(O) and shows S(T)=S(V) for a timelike interval with V a spacelike ball. In 1+1D CFT, recovers S(V)=(c/3) log(T/ε) and extends to finite temperature and finite size via S(T)=(c/3) log((β/π ε) sinh(π T/β)) and S(T)=(c/3) log((L/π ε) sin(π T/L)), with S(T)=S(V) for general d+1 where V is a d-ball of radius R=T/2. In holography, demonstrates S(T)=S(T_h) under a conformal map and argues the holographic dual for S(T_h) is a single spacelike geodesic, ensuring a real entropy and clarifying the status of entanglement across time in Rindler wedges and light-cone regions.

Abstract

We present an operator-algebraic definition for timelike entanglement entropy in CFT. This rigorously defined timelike entanglement entropy is real-valued due to the timelike tube theorem. We further demonstrate why the timelike entanglement entropy should be real-valued from both path integral argument and holography perspective.

Figures (4)

• Figure 1: The timelike envelope $\mathcal{E}_{\mathcal{T}}$ (the gray shaded region) consists of all points that can be reached by deforming the timelike interval $\mathcal{T}$ (the purple line) to a family of timelike curves (black dashed lines), which is equivalent to a causal diamond $\mathcal{O}$. All points at $t=0$ constitutes a spacelike ball $V$ (the blue region).
• Figure 2: The density matrix for disjoint symmetric segments $A$ and $B$. The usual adjacent configurations with divergent entanglement entropies $c/3 \log \ell/\epsilon$ or $c/6 \log \xi/\epsilon$ are obtained by taking simple limits.
• Figure 3: Penrose diagrams of Poincaré patch of AdS$_{3}$ spacetime and holographic duals of $S(\mathcal{T})$, $S(\mathcal{T}_{\text{h}})$. In the left panel, the blue line represents a finite timelike interval $\mathcal{T}$ on the conformal boundary of AdS$_{3}$, the red line denotes a timelike geodesic, and the two green lines denote spacelike geodesics. In the right panel, the blue line represents the temporal half-axis $\mathcal{T}_{\text{h}}$, and the green line denotes a spacelike geodesic.
• Figure 4: The left/right Rindler wedges and the future/past light-cones are illustrated, respectively.