Measurement-induced phase transition in teleportation and wormholes

The authors study a phase transition from a non-traversable TFD state to an eternally traversable wormhole in JT gravity/SYK via two different mechanisms: a Gao-Jafferis-Wall type coupling and a Kitaev-Yoshida style post-selection (both turned on at $t=0$). In both cases, it is shown that at a large enough coupling strength/projection rate, the wormhole can be traversed at arbitrarily late times.

Overall, I find this to be an interesting contribution, the broad strokes of which are well explained and motivated in the introduction. There are, however, a few points in the main body of the text that could use some clarification/minor editing, listed below. Once these are addressed, I would recommend the manuscript for publication.

Conceptual questions/suggestions:
1. As a general comment for section 3, I think there could be better clarification as to which comments/formulas hold in the original Maldacena-Qi (MQ) case (when the coupling is turned on infinitely far in the past) and which hold only in this case (when the coupling is turned on at a finite $u=0$. E.g. most of the discussion up to 3.12 holds for both, but the matching to 3.5/TFD is the new step here?
2. In this vein, a comparison of 3.13 with the MQ analogue could be useful to discuss the finite $\mu$ comment made in the last paragraph of page 4.
3. (Less important question/suggestion) Is it straightforward to compare this to the MQ time-reversed problem of turning off the coupling at $u=0$ to go from a past-eternal to TFD solution?
4. Is there anything more that can be said about the Lorentzian bulk in the KY projection case? Is the inserted energy complex only in the case of continuous projections, or is this also true for just a single projection? Is the geometry better understood in this case?.
5. In a similar vein, can more be said about the lack of revivals in the KY case? Should I roughly think of this as the projection just eventually deleting the inserted signal or is something more complicated happening? (If the former, is it easy to see why it happens after the signal crosses exactly once?)
6. (Very minor) For clarity, I would suggest upgrading footnote 4 to a comment in the main body of the text.


Minor grammatical comments:
7. In many places there is a missing "the". E.g. in front of Sachdev-Ye-Kitaev/SYK model, TFD state, GJW/KY protocol, Maldacena-Qi (MQ) wormhole, Einstein-Rosen bridge, Schwarzian throughout. Also specifically, "the NLTS...,""the intrinsic sign problem...,""the maximal value of Im$G_{LR}$...,""The goal is to make the anti-commutator...,""Wormhole length can be diagnosed by the correlator...,""where the bar denotes the time-reversal operation,""the operators $U_{ab}$...," etc.
8. Wrong quotation marks for Latex: e.g. in "sharp projections'' and "weak projections'' later on
9. Pg 4. Get rid of we in "so we this problem does not arise...''

The authors show that in the Schwarzian sector of SYK, the traversable wormhole protocols of Gao-Jafferis-Wall and Kitaev-Yoshida can be applied continuously in such a way that there is a phase transition between creating a finite-time or eternal wormhole. The phase transition depends on the rate at which the measurement is applied.

I don’t think this is a surprising result, but it is interesting and it is well explained. The calculations are impressive and, as far as I can tell, correct. I think this paper is a good contribution to making connections between high energy physics and the recent developments in statistical properties of hybrid classical/quantum dynamics. I am happy to recommend it for publication in SciPost.

Before publication, there are a few things I would like the authors to address. These should only require small changes to the manuscript. In addition, I have included below a list of typographical/grammatical errors that I noticed, which I ask the authors to correct.

Important changes:
1. On page 2, the authors say “exponentially many measurements.” Exponentially many in what? This sentence is not clear to me.
2. On page 4, the authors note that in the strong coupling phase, the GJW eternal wormhole has revivals, while the KY wormhole does not. In the KY wormhole, while a signal can pass through the wormhole starting at any time, it never bounces back. It would be great if the authors could give some interpretation of this phenomenon, as it is not at all clear to me how to understand this from a holographic pictures.
3. On page 4, it would be good to clarify what you mean by “it requires a finite \mu coupling.” Does this mean the effect can’t be seen in perturbation theory in \mu?
4. Around equations 2.7-2.10 it isn’t clear what the authors mean by the “plus and minus parts,” and when they should be expected to “cancel out.” It would be good to add a few sentences clarify, since there are likely readers interested in this paper who are not intimately familiar with the details of the Schwinger-Keldysh formalism.

Typos/Grammar:
1. “Sachdev-Ye-Kitaev model” -> “the Sachdev-Ye-Kitaev model” (this appears several times in the paper)
2. “in proving [23] NLTS (no low energy trivial states) conjecture” -> “in proving [23] the NLTS (no low energy trivial states) conjecture”
3. “naive application of local measurements lead to” -> “naive application of local measurements leads to” (the subject of the clause is the word ‘application,’ which is singular)
4. “just projection, we will use these terms” -> “just projection. We will use these terms”
5. “an Euclidean” -> “a Euclidean” (even though the word ‘Euclidean’ starts with a vowel, it is pronounced with a hard ‘y’ sound)
6. “the goal is to make anti-commutator” -> “the goal is to make the anti-commutator” (this appears twice)
7. “allowing the information transfer” -> “allowing information transfer”
8. “denote the projection rate $\kappa$” -> “denote the projection rate by $”\kappa$”
9. “this is essentially GJW computation” -> “this is essentially the GJW computation”
10. “Maximal value of” -> “The maximal value of”
11. “a novel type of a wormhole” -> “a novel type of wormhole”
12. “even if the L, R subsystems initially unentangled” -> “even if the L, R subsystems are initially unentangled”
13. “that does not include interaction between these two subsystems” -> “which does not include interaction between these two subsystems” (this is sort of an archaic grammar rule — when to use ‘that’ vs’ which’ — but I did notice it)
14. “with unnormalized density matrix” -> “with an unnormalized density matrix”
15. “by Schwarzian action” -> “by the Schwarzian action”
16. “equations of motion of Schwarzian” -> “equations of motion of the Schwarzian”
17. “(dimensionful) Schwarzian coupling” -> “the (dimensionful) Schwarzian coupling”
18. “will leave physical Green function” -> “will leave the physical Green function”
19. “Quantity $\ell = - \phi$” -> “The quantity $\ell = - \phi$”
20. “that could be understood as a simple” -> “which could be understood as a simple”
21. Equation 3.11 should have a period and the following sentence should start with a capital letter.
22. “perform projection at $u=0$” -> “perform a projection at $u=0$”
23. “would result at non-zero anti-commutator” -> “would result in a non-zero anti-commutator”
24. “real, that requires” -> “real, which requires”
25. “from the equations of motions” -> “from the equations of motion”
26. “any continuous QFT.It would be interesting” -> “any continuous QFT. It would be interesting”