The Mechanism behind the Information Encoding for Islands

1. Interesting idea

1. Shoddy execution, ignoring many important counterarguments.

First of all, let me apologise for the delay. I spent some time doing background reading.

I think this paper is at best mis-named and at worst completely wrong.

The author claims to have uncovered "the" mechanism for the non-local encoding of islands into a bath. But the actual result of the work is merely to exhibit an operator that is roughly localised to an island region which has a non-zero commutator with the bath Hamiltonian. There is no evidence provided that this is the operator that is encoded in the bath. Without this evidence, how can the paper be called "The mechanism?"

Let me back up a bit to explain this point. Given an operator $O(x)$ in a non-gravitational QFT there are many dressed operators $O_{dr} (x)$ in semiclassical gravity such that $\lim_{G_N \to 0} O_{dr} (x) = O(x)$. The author has taken $O(x)$ to be an operator in the island and found one possible dressed operator $O_{dr,G} (x) = O(x + V)$.

Let us call the operator encoded in the bath $O_{dr,I} (x)$. The question of whether $O_{dr,G} \overset{?}{=} O_{dr,I}$ is, I believe, not even considered by the author. (There might not be a unique $O_{dr,I}$; the refined question is whether there is any $O_{dr,I}$ that agrees with $O_{dr,G}$.) This is why they cannot claim that they have uncovered the mechanism for non-local encoding. If the answer to the questio is plausibly yes, it is only the name of the paper that is wrong.

However, I don't believe that $O_{dr,G}$ is a plausible candidate for $O_{dr,I}$. The reason is that we actually know somehting about $O_{dr,I}$: it is whatever is constructed by the Petz map. But, since the Petz map involves replica wormholes, I expect that $[O_{dr,I}, H_{bath}] = O (e^{-1/G_N})$ whereas $[O_{dr,G}, H_{bath}] = O (G_N^\alpha)$ (See e.g. 3.18,4.74 which assert that $\alpha = 0$).

The absolute minimum this work needs, then, is (a) a change of title and (b) a qualitative argument that $O_{dr,G}$ is plausibly a candidate for $O_{dr,I}$. The appendix added does not come even close to making this argument.

As a stretch goal, the author may try to genuinely show that these operators are the ones we recover from entanglement wedge reconstruction, thereby proving my concerns wrong; this would make this paper amazing. Apart from the Petz map, https://arxiv.org/abs/1912.02210 had a relatively concrete way of reconstructing island operators.

Another clarification is the role of state-dependence. $O_{dr,G} = O(x + V)$, where $V$ is an operator. If one takes $x$ inside the island, $O_{dr,G} (x) |\psi\rangle = \sum_i \langle V_i | \psi \rangle O(x+V_i) | V_i \rangle$, where $V_i$ are eigenvalues. It could be that some of the values $x+V_i$ do not lie in the island! This is a standard subtlety one has to worry about when working with dressed operators.

The other referee correctly pointed out that $V_i = O (\sqrt{G_N})$ and so one does not need to worry about it for that reason. The author doesn't seem to have made even a rudimentary effort to understand either my comment or the referee's reply (as evidenced by the author comments in the resubmission). The author's words were

Referee 1 was wondering how to see that the Goldstone boson dressed operator is state-dependent. Referee 2 pointed out that the dressed operator is for some reason state-dependent. [Emphasis mine]

I would suggest that the author spend a little more time thinking about referee comments. While I admit that my description of the question was a bit quick, referee 2 was quite clear about the question as well as the answer.

I still don't understand eq 3.18. If I understand correctly $z \neq \epsilon$, $V^\mu (x,z) \neq \epsilon^{d+2} U^\mu (x)$. But 3.18 asserts that $[V^\mu (x,z), H_{bath}] = [\epsilon^{d+2} U^\mu (x), H_{bath}]$. 3.18 thus implies $[V^\mu (x,z) - \epsilon^{d+2} U^\mu (x), H_{bath}] = 0$. This can either be true if $z \neq \epsilon$, $V^\mu (x,z) \neq \epsilon^{d+2} U^\mu (x)$ (but this is not the case) or there is some non-trivial reason that the difference commutes with $H_{bath}$. Shouldn't this non-trivial reason be explained, if it exists?

If this non-trivial reason doesn't exist, then I have the following objection. If $(x,z)$ is in the exterior of the black hole, then it is clear (via HKLL) that $[V^\mu (x,z), H_{bath}] = O(G_N^0)$ as in 3.18. However, if $(x,z)$ is inside the black hole, then it seems to this reviewer that the $G_N$ scaling of $[V^\mu (x,z), H_{bath}]$ needs to be found via a non-trivial calculation.

The absolute minimum this work needs is (a) a change of title and (b) a qualitative argument that $O_{dr,G}$ is plausibly a candidate for $O_{dr,I}$. The appendix added does not come even close to making this argument.

As a stretch goal, the author may try to genuinely show that these operators are the ones we recover from entanglement wedge reconstruction, thereby proving my concerns wrong; this would make this paper amazing. Apart from the Petz map, https://arxiv.org/abs/1912.02210 had a relatively concrete way of reconstructing island operators.

Ask for major revision

Publish (easily meets expectations and criteria for this Journal; among top 50%)