The entanglement entropy of typical pure states and replica wormholes

In a 1+1 dimensional QFT on a circle, we consider the von Neumann entanglement entropy of an interval for typical pure states. As a function of the interval size, we expect a Page curve in the entropy. We employ a specific ensemble average of pure states, and show how to write the ensemble-averaged Renyi entropy as a path integral on a singular replicated geometry. Assuming that the QFT is a conformal field theory with a gravitational dual, we then use the holographic dictionary to obtain the Page curve. For short intervals the thermal saddle is dominant. For large intervals (larger than half of the circle size), the dominant saddle connects the replicas in a non-trivial way using the singular boundary geometry. The result extends the 'island conjecture' to a non-evaporating setting.

Automated summary

In a 1+1 dimensional QFT on a circle, the von Neumann entanglement entropy of an interval for typical pure states is studied, with an expectation of a Page curve in the entropy. By employing a specific ensemble average of pure states, the ensemble-averaged Renyi entropy is written as a path integral on a singular replicated geometry. Assuming a QFT with a gravitational dual, the holographic dictionary is used to obtain the Page curve, showing dominance of the thermal saddle for short intervals and connection of replicas using a singular boundary geometry for larger intervals. This result extends the 'island conjecture' to a non-evaporating setting.