Full counting statistics as probe of measurement-induced transitions in the quantum Ising chain

- interesting quantity at study in the contect of MIPT

- the paper seems rushed out with no clear conclusions or connections to previous works/other observables

The authors want to propose the cumulants of magnetic fluctuations as a different observable to detect MIPT. The idea is nice, and it could also be that this quantity performs better than others (entanglement entropy, purification time, etc). The problem is that they do not do any comparison. The critical gamma ~ 4 they find is not confirmed by any other piece of literature or any other method, for example entanglement entropy scaling or connected correlator. So as it stands the result is not trustable and could be just a crossover.

I suggest to the authors to consider a model where MITP is established (see for example https://arxiv.org/abs/2303.12216, https://arxiv.org/abs/2302.12820) and benchmark their approach there.
Otherwise, to simply compare with other observables known to detect MIPT. Only later the paper can be considered for publication.

1. Very active research topic
2. The chosen system allows a powerful combination of analytic and numerical approaches
3. Derivations and computations are clearly presented and easy to understand

1. The physical meaning of the computed quantity is not clarified
2. Conclusions are sketchy, implications relevant to ongoing research are not stated
3. Simplicity of model excludes examination of effects of quantum interactions

This paper considers measurement-induced phase transitions. Instead of focusing on the dynamics of entanglement, the authors consider the effect of continuous measurement on full counting statistics. They chose the Ising quantum spin chain with a zero transverse magnetic field, which is a simple enough system to allow the application of a powerful combination of analytic and numerical tools.
The main weakness of the paper is that the physical (whether theoretical or experimental) significance of the quantity they consider is unclear. Instead of a straightforward construction of full counting statistics, which, as the authors observe, would lead to a rather trivial result due to its linear dependence on the density matrix, the authors chose to average the cumulants of magnetisation over the quantum trajectories which are nonlinear in the density matrix, and then use these results to reconstruct a distribution.
It is entirely unclear what the significance of this distribution is, and how it is related to eventual physics, even at the theoretical level. The paper completely lacks a discussion of this important point.
The second main issue, which seems to be related to the first one is that the conclusions are rather sketchy and descriptive, no deeper physical consequences are drawn from the results. It is not at all clear how these results may be relevant for subsequent work in the field.
In my opinion, clearly, none of the expectations for acceptance in Scipost Physics (groundbreaking discovery; breakthrough on a stumbling block; opening a new pathway; a novel and synergetic link between different research areas) is met. As a result, I cannot recommend the paper for publication in Scipost Physics in its present form, and suggest transferring the paper to Scipost Physics Core, where it can be published once the authors addressed the requested changes.

1. The authors should clarify the physical content of the full counting distribution they constructed.
2. The conclusions must be stated clearly, highlighting their significance for subsequent research.
3. It would also be useful if the authors could discuss how the inclusion of interactions (such as a transverse magnetic field) would affect the results. Does the integrability of such interaction matter?