Probing Hawking radiation through capacity of entanglement

Abstract

We consider the capacity of entanglement in models related with the gravitational phase transitions. The capacity is labeled by the replica parameter which plays a similar role to the inverse temperature in thermodynamics. In the end of the world brane model of a radiating black hole the capacity has a peak around the Page time indicating the phase transition between replica wormhole geometries of different types of topology. Similarly, in a moving mirror model describing Hawking radiation the capacity typically shows a discontinuity when the dominant saddle switches between two phases, which can be seen as a formation of island regions. In either case we find the capacity can be an invaluable diagnostic for a black hole evaporation process.

Capacity of entanglement in random pure state

“…This agrees with the result in [6] obtained from the Marchenko-Pastur distribution. Our replica computation reveals the importance of the sub-leading contribution f (n, α) to the capacity of entanglement.…”

“…We can see that C A vanishes for d A = 1 and approaches zero at large d A d B . This is qualitatively similar to the result of the planar limit found in [6]. We emphasize that our result (4.9) is exact at finite d A , d B and (4.9) includes all the non-planar corrections.…”

“…C A in the opposite regime α > 1 can be obtained by sending α → α −1 using the symmetry (2.10). The computation of C A in this limit (3.1) has been already done in [6] using the planar eigenvalue density of the Wishart-Laguerre ensemble, known as the Marchenko-Pastur distribution. Here we will use the replica method to compute C A , which clarifies the role of replica wormholes in C A .…”

“…As discussed in [3], S A in the random pure state model exhibits a similar behavior as the Page curve for the Hawking radiation from an evaporating black hole. In a recent paper [6], it is argued that the capacity of entanglement C A introduced in [7] is a useful quantity to diagnose the phase transition around the Page time. C A is defined by…”

Grey-body Factors, Irreversibility and Multiple Island Saddles

“…This motivates us to state the following island-in-thestream decoupling rule: if two intervals are separated by a region large enough to be dominated by an island saddle, then they are uncorrelated. 13 It is easy to extend this result to prove that, on the other hand, two intervals are correlated, IpA 1 , A 2 q ą 0, if the entropy of A 1 Y A 2 is dominated by an island saddle which image contains both A 1 and A 2 , A 1 Y A 2 Ď Ĩ.…”