Canonical Purification of Evaporating Black Holes

TL;DR

The paper investigates how spacetime connectivity emerges in evaporating black holes within AdS/CFT by employing the canonical purification framework and the quantum extremal surface. It shows that after the Page time the canonical purification becomes a short Lorentzian wormhole between the black hole and radiation boundaries, realizable by a unitary on the radiation alone, thereby giving a concrete ER=EPR realization without modifying the black hole state. Before the Page time, the entanglement wedge lacks a nontrivial QES, and the canonical purification remains disconnected, illustrating that bipartite entanglement alone does not guarantee semiclassical connectivity. Using a concrete JT gravity with conformal matter example, the authors compute the post-Page dilaton profile, QES location, and shock structure, and discuss broader implications for multipartite entanglement, code subspaces, and observer experiences in the context of spacetime emergence.

Abstract

We show that the canonical purification of an evaporating black hole after the Page time consists of a short, connected, Lorentzian wormhole between two asymptotic boundaries, one of which is unitarily related to the radiation. This provides a quantitative and general realization of the predictions of ER=EPR in an evaporating black hole after the Page time; this further gives a standard AdS/CFT calculation of the entropy of the radiation (without modifications of the homology constraint). Before the Page time, the canonical purification consists of two disconnected, semiclassical black holes. From this, we construct two bipartite entangled holographic CFT states, with equal (and large) amount of entanglement, where the semiclassical dual of one has a connected ERB and the other does not. From this example, we speculate that measures of multipartite entanglement may offer a more complete picture into the emergence of spacetime.

Figures (12)

• Figure 1: To the left we see the entanglement wedge of $\rho$, and in the center we see $\mathcal{W}_E[\rho]$ glued to its CPT conjugate across the QES/HRT surface $\chi$. To the right we display the final evolution of the data on $\Sigma \cup \tilde{\Sigma}$, giving the spacetime dual to $\left| * \right\rangle{\sqrt{\rho}}$. Shockwaves (red) are present when including quantum corrections.
• Figure 2: The three relevant non-gravitational systems in the microscopic picture of the evaporating two-sided black hole at a moment of time. The blue circles supports holographic CFTs, while the orange plane supports the reservoir system.
• Figure 3: The relevant subregions and states for evaporating one-sided (a) and two-sided (b) black holes in AdS.
• Figure 4: On the left we see the QES $\chi$ of an evaporating black hole for some time $t>t_P$. On the right, we see the canonical purification, dual to the state $\left| * \right\rangle{\sqrt{\rho_{BH}(t)}}$.
• Figure 5: On the left we see the bulk picture of an AdS black hole evaporating into a reservoir, with $\chi$ the QES for some time $t>t_P$. On the right, we see the microscopic picture at the fixed time $t$.