Realization of "ER=EPR"

TL;DR

This work offers a concrete realization of ER=EPR by deriving an Einstein-Rosen bridge from entanglement in a two-CFT thermofield double state and identifying the wormhole entropy with a disjoint entanglement entropy. By formulating a geometric entanglement measure $\chi = (1/2) S_{disj}^2$ and showing that $S_{disj}$ is UV-finite, the authors extract a dual metric that features a wormhole throat and relate the entanglement wedge cross-section to the wormhole geometry via the Ryu-Takayanagi prescription. They further connect the entanglement entropy of disjoint regions to the Bekenstein-Hawking entropy, $S_{BH} = A/(4 G_N)$, confirming the horizon area interpretation of wormhole entropy. Finally, they provide a quantitative verification of Van Raamsdonk's conjecture by analyzing the beta-dependence of disjoint entropies, demonstrating that spacetime connectivity arises from entanglement and its disentanglement collapses the wormhole while length scales diverge.

Abstract

We provide a concrete and computable realization of the $ER=EPR$ conjecture, by deriving the Einstein-Rosen bridge from the quantum entanglement in the thermofield double CFT. The Bekenstein-Hawking entropy of the wormhole is explicitly identified as an entanglement entropy between subsystems of the thermofield double state. Furthermore, our results provide a quantitative verification of Van Raamsdonk's conjecture about spacetime emergence.

Figures (8)

• Figure 1: The density matrix $\rho=\vert\text{TFD}\rangle\langle\text{TFD}\vert$, the blue shaded region represents the Euclidean path integral. The two gray infinite slits are states living in left and right Hilbert space respectively. We have set $x\in(-\infty,+\infty)$ and $\tau\in[0,\beta)$.
• Figure 2: At $t=0$ slice, we calculate the disjoint entanglement entropy between the disconnected segments $A$ and $B$.
• Figure 3: Penrose diagram of the eternal AdS black hole. Two vertical lines represent two CFTs living on the left and right boundaries respectively. The blue line represents the time slice $T=0$.
• Figure 4: The red line represents the horizon. The blue line indicates the RT surface of the entanglement entropy between $A$ and $B$.
• Figure 5: Segments $C$, $D$ and $A$, $B$ represent two different disjoint configurations in the entangled TFD state.