Generalized Lindblad Master Equation for Measurement-Induced Phase Transition

1) The paper is interesting and scientifically sound
2) The topic is extremely timely
3) Needed calculations are carried out explicitly

- Initial sections are just a re-derivation of well known results and in doing so some misleading error have been made (see report)
- not very clear connections of the results with MIPT physics

In this paper the author propose a generalized Lindblad equation to compute the seconde Rényi entropy in systems subject to the combined action of unitary evolution and measurement.

The paper is interesting, scientifically sound and the topic is extremely timely. However I feel that some modifications needs to be done before the publication. My main concern is about Sec.2 where some well known results are re-derived and some confusion is made in Sec.4 . I formulate here below the points that need to be addressed :

1) The Lindblad ME (6) arises naturally when considering a system being subject to both unitary dynamics and measurement. Indeed it is well known that the average dynamics of this kind of systems gives rise to a Lindblad ME both in case of a projective measurement and weak measurement. The derivation that the author propose (from Eq.1 to Eq.6) requires that $\sum_{\alpha=1}^n L^\dagger_\alpha L_\alpha = \mathbb{I}$ . This fact imply that the anti commutator part of the Lindblad ME is completely trivial and amounts to an identity matrix.

However, when using an operator-sum representation [Eq.(9,10)] to derive the Lindblad ME one can notice that the equality $\sum_{\alpha=1}^n L^\dagger_\alpha L_\alpha = \mathbb{I}$ does NOT need to be satisfied in general [in Eq.(10) one can simply keep $M_0$ and $M_1$ as done in the references I listed below]: the operator-sum representation is still well defined since $\sum_{\alpha=0}^N M_\alpha^\dagger M_\alpha = \mathbb{I}$ and the result is a Lindblad ME with a non-trivial anticommutator (giving rise to the so called non-Hermitian effective Hamiltonian). This is the typical situation arising from a weak measurement approach (coupling the system to an ancilla and then performing a projective measurement on it).
I thus suggest the author to clarify the difference between Eq.(6) [that strongly rely on the assumption $\sum_{\alpha=1}^n L^\dagger_\alpha L_\alpha = \mathbb{I}$ and the “weak measurement” scenario where$\sum_{\alpha=1}^n L^\dagger_\alpha L_\alpha \neq \mathbb{I}$. Maybe the author can find useful the paradigmatic cases studied in the list of references.

[1] https://quantum-journal.org/papers/q-2021-08-19-528/
[2] https://journals.aps.org/prb/abstract/10.1103/PhysRevB.103.224210
[3] https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.2.033512

2) I do not understand the connection of the numerical results with the MIPT. What I mean is that in order to claim that a system undergoes (or not) a MIPT one needs to study how the entanglement entropy scales as the size of the system is increased. Typically this requires to simulate large L and to perform a scaling analysis. I suggest the author to reshape Sec.7 and be more prudent about how the results are supportive or not of a MIPT. What we can say is that, as expected, we observe a reduction of the entropy as \gamma/J is increased.

3) As a sanity check of the generalized Lindblad equation I suggest the author to compare the results in Fig.2 with a standard quantum trajectory evolution performed on a Lindblad ME (defined in the standard physical Hilbert space and not in the doubled space) with jump operators as in Eq.(24) and Hamiltonian as in Eq.(23). Then, at each time the entropy can be computed for each trajectory and the average should correspond to their results. This is a cheap simulation that would clarify the results. This “direct” method would directly give the correct contribution of the entropy as in Eq.(15).

4) Some typos need to be fixed as, e.g., MIFT is written instead of MIPT in different points in the manuscript

1) reorganize the initial Sections of the manuscript
2) explain better the results and their connections with MIPT
3) check the correctness of the results with a "direct" quantm trajectory approach
4) fix the typos