Abstract
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 Rényi 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.
Highlights
For black hole evaporation [6, 7]
One reason is that thermal entropies can be written in field theory as a Euclidean path integral over some geometry
We propose that a holographic calculation can be made by taking all the field-theory geometries as possible asymptotic boundaries for the gravitational path integral
Summary
We will focus here on Euclidean two-dimensional field theories on S21π × R but everything can be generalized to theories in a general dimension d on Sd−1 ×R. Our ensemble of pure states is the Euclidean evolution of this basis. Our favorite choice for the operators O would be all the field operators φ(θ) , which correspond to the field state basis |α ≡ |φα for every field configuration φα(θ).4 For this choice of basis the ensemble states (sandwiched with a field-state |φ0 ) can be written as a Euclidean path integral φ0|ψα =. For any other basis a similar path integral can be written, only the identification (the green circle) at τ. Given a quantity Wα calculated on each ensemble state |ψα , the averaged quantity can be written formally as W =. Like the field basis |φα , we need to specify a measure on the formal sum, and the value of W will depend on that choice.
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