Algebraic ER=EPR and Complexity Transfer

TL;DR

This work reframes ER=EPR within an algebraic boundary framework, tying bulk connectivity to the type of boundary operator algebras in the $G_N\to0$ limit. It defines precise classical and quantum spacetime notions and introduces a rigorous concept of quantum wormholes via type I vs III$_1$ algebras, extending to multipartite states with canonical purification. The authors argue that Page-time physics is governed by a dynamical transfer of high-complexity Type III$_1$ operators between subsystems, explaining transitions in QES dominance and connectivity, and illustrate the framework with evaporating black holes and crossed-product examples. The approach aims to sharpen the connection between entanglement structure and spacetime geometry, offering a unified language for classical, quantum volatile, and dynamically evolving spacetimes with potential extensions beyond AdS/CFT.

Abstract

We propose an algebraic definition of ER=EPR in the $G_N \to 0$ limit, which associates bulk spacetime connectivity/disconnectivity to the operator algebraic structure of a quantum gravity system. The new formulation not only includes information on the amount of entanglement, but also more importantly the structure of entanglement. We give an independent definition of a quantum wormhole as part of the proposal. This algebraic version of ER=EPR sheds light on a recent puzzle regarding spacetime disconnectivity in holographic systems with ${\cal O}(1/G_{N})$ entanglement. We discuss the emergence of quantum connectivity in the context of black hole evaporation and further argue that at the Page time, the black hole-radiation system undergoes a transition involving the transfer of an emergent type III$_{1}$ subalgebra of high complexity operators from the black hole to radiation. We argue this is a general phenomenon that occurs whenever there is an exchange of dominance between two competing quantum extremal surfaces.

Figures (15)

• Figure 1: The bulk duals of two copies of a holographic CFT in a thermofield double state. (a) For $T > T_{HP}$, the two sides are connected by an Einstein-Rosen bridge. (b) For $T < T_{HP}$, we have two classically disconnected spacetimes with entangled quantum fields between them.
• Figure 2: (a) The Page curve for an evaporating AdS black hole describes the time evolution of the von Neumann entropy of the system $B$ dual to the evaporating black hole and the system $R$ consisting of the radiation evaporating into a reservoir. The puzzle pointed out in EngFol22 considers two times $t_1 < t_P < t_2$ that have the same von Neumann entropy of order ${{\mathcal{O}}} (1/G_N)$. (b) The Penrose diagram of coupled black hole and reservoir systems. Cauchy slices for the black hole spacetime at $t_1$ and $t_2$ are shown, with the red regions, labeled by $\Sigma_B$, corresponding to slices of the entanglement wedge of the black hole. At $t_1$, the entanglement wedge of the black hole consists of the full Cauchy slice while that at $t_2$ only the region exterior to $\chi_1$. Dashed lines represent the positive energy shocks created by coupling the black hole and reservoir.
• Figure 3: (a) At $t_1$, Cauchy slices of the entanglement wedge ${\mathcal{W}}_B$ of the black hole system are inextendible, and ${\mathcal{W}}_B$ disconnected with the entanglement wedge of the radiation. (b) At $t_2$, the entanglement wedge of the black hole is given by the part labeled as ${\mathcal{W}}_B$, while region $I$ is a subset of the entanglement wedge ${\mathcal{W}}_R$ of the radiation. They are now classically connected at the quantum extremal surface (QES). In the Penrose diagram, ${\mathcal{W}}_B$ and ${\mathcal{W}}_R$ appear to also be connected at the boundary at all times after coupling, but this is just an artifact of the Penrose diagram: ${\mathcal{W}}_B$ and ${\mathcal{W}}_R$ are separated by an infinite proper distance there and are not gravitationally interacting.
• Figure 4: A cartoon of the dynamical transition at the Page time. A type III$_1$ factor consisting of operators of high complexity (Python's lunch BroGha19EngPen21aEngPen21b) is transferred from the black hole system to the radiation system at the Page time.
• Figure 5: (a) cartoon picture for the entanglement between the interior of the black hole and the radiation. A Cauchy slice is shown with $B$ denoting the boundary, $H$ the horizon, $K$ the QES that dominates after the Page time, and the spacetime smoothly capping off at $C$. (b) The example of Araki-Woods where two groups of $N \to \infty$ spins are pairwise entangled with each pair in a generic entangled state.

Theorems & Definitions (2)

• Lemma 1
• proof