Algebraic ER=EPR and Complexity Transfer

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 $\mathcal{O}(1\mathrm{/}{G}_N)$ 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.