Random Circuits in the Black Hole Interior

We thank the referee for pointing out this similarity. In footnote 14 we added a comment comparing the average geometry with the setup of Fig. 7 of Ref. [14].

We thank the referee for the good questions that have led to an improvement of the manuscript. Here are our detailed responses:

1. Yes, the energy variance is different from the canonical ensemble in general. This is the main difference in so far as the energy distribution is well peaked and thus described just by its mean and variance. As to whether this is a significant difference, it depends. Taking the example of energy eigenstates, we know via ETH that local observables will be approximately thermal to a good approximation, and moreover, being intensive quantities, they will be insensitive to the precise ensemble in the thermodynamic limit. By contrast the energy variance is identically zero. Yet both the canonical ensemble and energy eigenstates are thought to have the same coarse grained black hole geometry, so in this sense the exterior geometry does not depend on having a particular energy variance, at least to leading order in $N$. The outcome of a physical measurement of the heat capacity obtained by coupling the system to a reservoir is a distinct issue in general, although of course we agree that the variance is tied to the heat capacity (at constant volume) in the canonical ensemble. If a particular value of the energy variance is crucial for the application of interest, there are methods available for adjusting it, for example, by making the amount of imaginary time evolution have some random component. We discuss several of these points and attempt to further clarify the issues raised by the referee in improved text in section 2.2, 2.4, and appendix C.

2. In page 11, when introducing the ensemble of states before (2.8) we have included an explanation for our choice of how to apply the finite-temperature random circuit.

3. If the equilibrium state overlaps substantially with the thermal state in the thermodynamic limit, we do not expect that there is classical matter outside of the horizon even for single realizations. One reason is that in this case the average geometry is that of Schwarzschild and any classical matter must have a positive (null) energy, so its backreaction effects cannot average out to zero. However, as shown in Ref. [61], typically for this to be the case one needs perturbations $\mathcal{O}_\alpha$ which are relevant enough. In higher dimensions these would only act deep in the bulk, and thus they would not spread out substantially to the exterior. However, as we explain in section 2.2., 2.4 and appendix C, this is not generally the case. If this is not the case, as the referee points out, we do expect deviations from thermality which could arise at leading order, signaled by classical matter outside of the black hole. We have added a clarification of this in the caption of Fig. 6.

4. We have modified the caption of Fig. 6 to correct that $\exp(-\tau H_{\text{eff},1})|{\text{TFD}}\rangle$ is the Euclidean evolution of a state, as the referee points out. We have also added the relation between the coordinate $\tau$, the total boundary time $t$ and the onset time $t_\star$ of the approximate time-translation symmetry. We have also clarified that $M$ has the geometry of an Euclidean wormhole locally in this region, while, as the referee points out, the global topology of $M$ is that of a disk (times a sphere).

5. We have added footnote 11 to clarify how it is possible to take $K \sim N^2$ perturbations in AdS/CFT without leaving the supergravity regime. As the referee points out, a primary with conformal dimension $O(N^2)$ is beyond supergravity. The point is that it is not necessary that the drive operators $\mathcal{O}_\alpha$ are different primaries. They could all be coming from the same primary operator, applied at different spatial points on the sphere, with independent Brownian couplings at each point. They could also be non-local operators on the sphere which create particles deep in the bulk. With this choice, one creates a large number of perturbations which are well described by supergravity but which together backreact on spacetime. As we mention in footnote 14, constructing these solutions explicitly in higher dimensional AdS/CFT is an interesting open problem. Our symmetry-based argument suffices for the purposes of this paper, and it relies on the assumption outlined on page 13. Such an assumption is supported by general microscopic studies such as Ref. [99], where they argue that $H_{\text{eff},1}$ has a unique GS similar to the TFD under very general ETH-like ansatze for the $\mathcal{O}_\alpha$. Thus, as long as the assumption is correct for choices of the $\mathcal{O}_\alpha$, the results of this paper should apply in higher dimensional AdS/CFT as well.

6. Above (2.21) we have clarified that the Lorentzian eternal traversable wormhole relies on the exact isometry being present in the Euclidean section. If this symmetry is broken, even weakly as it is for individual instances, then this analytic continuations generally leads to a complex metric. Perhaps the complex parts of the metric are small, but naively one loses this interpretation.

7. We have expanded Sec. 3.5 to include the role of the logarithmic corrections to the scaling. We have made more transparent that we are using a leading asymptotic form of this relation in the introduction and conclusions. We have improved the presentation of the relation in the introduction and conclusions accordingly.