Measurement induced transitions in non-Markovian free fermion ladders

1. The idea is interesting
2. The setup is neat

The current work tries to understand the interplay between non-Markovianity of the bath and the steady-state entanglement of the system. To facilitate the discussion, the authors consider a one-dimensional complex fermion ladder with particle-number conservation, where one of the two chains is measured and serves as a non-Markovian bath. The dynamics is designed to preserve the Gaussainity of the state so that the problem is numerically tractable. The authors examine two different cases, the persistent measurement where the measurement probability is 1, and the sporadic measurement where the measurement probability is less than 1. In both regimes, non-area law behaviors are observed when the bath dynamics is fast (t2 being large). The authors also invoke a quantitative measure of the non-Markovianity.

I found both the question and the setup in this manuscript interesting. Attempt to provide a quantitative characterization of the non-Markovianity in a many-body system is also very nice. However, I found that some of the results are not clear to me and not conclusive enough. The quality of the numerics needs be improved.

I would like to hear the author's responses to the following questions first before making any decision.

1. Is it possible to draw some phase diagrams for the two cases of persistent and sporadic measurement?

2. For the persistent measurement, since the bath has no entanglement after the measurement, it seems to me that the entanglement of the system fermion is the same as the entanglement of the system plus the bath. Namely, it is effectively probing a single-chain entanglement transition. And then there is no volume-law phase in this case. Is it possible to make some connection between the current setup and that of Ref 53 to get a more clear conclusion? E.g. whether the non-area law exists in the thermodynamic limit, and is the transition also of KT type?

3. From appendix A, the half-chain entanglement (both the von Neumann entropy and the negativity) exhibits a significant fluctuation, e.g. Fig 11 and 14. The label of vertical axis is the same as eqn7, which implies it is averaged over trajectory although the text says it is for a single trajectory. In any case, the results in the main text, e.g. Fig 2 and 6, have no visible error bar. This is surprising to me. I am wondering how the error bar can be invisible given the fact that the single-trajectory result has a strong fluctuation.

4. The result in Fig 7 is obtained for L=16, which is much smaller than the system sizes used in the previous plots. I am curious what happens at larger system sizes?

5. In Fig 8, is it possible to do some finite-size scaling for subfig b,d,f, as the authors have done in appendix C, to infer what happens in the thermodynamic limit?

6. The dynamics has a periodicity in t_{12}, which can be seen through most of the plot. However, the plots of the measure of non-Markovianity N(\phi), Fig 9 and 10, do not have such a structure. What is the reason for that?

7. When t_{12} = \pi, the two chains are decoupled and the system chain undergoes unitary dynamics. Thus, I would expect N(\phi) to have a dip near t_{12} = \pi? However, this is not the case from the plot. Do I miss something and how to properly understand the result?

8. The authors mention two difficulties of calculating N(\phi), and thus only present the result for L=4. I am curious whether one can still utilize the Gaussianity to compute it for a slightly larger system size, e.g. L=10. Specifically, the Gaussianity of a density matrix \rho implies that it maps Slater determinant states to a linear superposition of Slater determinant states, i.e. e^{-c^\dag M c} \ket{n} = \sum_m A_{n,m}\ket{m}, where \ket{m} are Slater determinant states that form a complete orthonormal basis, M is determined via the fermion two-point function and A_{n,m} can also be calculated by utilizing Gaussianity. This way we can write down \rho_1 - \rho_2 as a matrix directly, e.g. in the basis of Fock states, without calculating the state of the two chains. I am wondering whether this is helpful in increasing the accessible system size and curious of the authors’ thoughts.

9. N(\phi) is computed for the system. The introduction emphasizes that previous studies focus mainly on Markovian bath. Therefore, I am wondering whether we should instead measure the non-Markovianity of the bath, which addresses the effect of the bath memory on the system dynamics more directly. In this case, one may also need to modify the protocol a bit to actually control the memory of the bath. Maybe I misunderstand the main question the introduction posts, but could the authors clarify my confusion?

1 - the paper considers an interesting and novel aspect of measurement-induced phenomena, namely the effect of non-Markovianity
2 - the model studied is interesting and the phenomenology uncovered appears to be rich

1 - the results are inconclusive in terms of establishing a phase diagram, phase transition, or different universal properties of the phases
2 - considering the fermionic Gaussian nature of the model, the sizes studied seem small relative to standards in the field, which may contribute to weakness (1)
3 - some lack of clarity in the meaning of "(non)-Markovianity" in the present context

I think the idea of the paper is interesting and has potential to lead to insightful results or new phenomena. The results as presented in this version seem, to me, a bit preliminary: there is numerical evidence for different behaviors, but not enough resolution to locate a transition or to distinguish a logarithmically-entangled phase from a volume-law.

The I_{L/8} diagnostic seems to rule out the volume-law but that is a bit indirect, as the "volume-law" ansatz (I_{L/8}->0) is based on the random circuit model, from what I can tell; whereas this particular free-fermion ladder could in principle realize a volume-law entangled phase with different universality (say, one based on "rainbow" states). Absent any hints of saturation in the entropy or negativity it is hard to rule out a volume-law phase.

I think the difficulty in interpreting the numerics comes in part from the fact that the system sizes studied are a bit small (L<=256, counting both legs of the ladder); in this field I think simulating ~1000 fermions should be fairly straighforward, and that could help sort out some of the finite-size issues. Other diagnostics, such as tripartite mutual information and purification of a probe qubit, could also be helpful.

For the non-Markovianity diagnostic, the size limit (L=4) is particularly problematic. I would suggest a modified measure based on the relative entropy instead of the trace distance. The relative entropy is also monotonic under channels but has the advantage of being efficiently computable for fermionic Gaussian states (since log(rho) is a quadratic Hamiltonian). I don't know if any desirable properties get lost but it seems like this should capture the desired property and greatly alleviate the size limit.

In addition, I have some questions.
1. It's not clear to me exactly how the diagnostic of non-Markovianity Eq.(22) is computed here. Starting from different initial states, there are different probability distributions for outcomes, etc; are these drawn independently for the two states? Is the measure computed between pairs individual realizations, and then averaged? Or are the states averaged over realizations first? It would be good to clarify this better. Depending on how this is done, I think it may return non-zero values even in purely Markovian cases, since the dynamics under consideration is not a channel. A good sanity check would be to apply this to the p=1 model but also *reset* all bath modes to some fixed configuration after measurement (this eliminates all memory from the bath and makes the dynamics Markovian).
2. I think the volume-law ansatz for I_{L/4} ~ L^{1/3} [Table I] is based on the case of two contiguous intervals (i.e. touching at one endpoint). I think for the case of antipodal intervals the scaling should be O(1), based on the configuration of minimal membranes.

If the authors can address these questions and improve the paper accordingly then I am in principle in favor of publication.

1 - Extend size of simulations and/or try other diagnostics (e.g. tripartite mutual information) to better control finite-size effects
2 - Consider alternative non-Markovianity diagnostic based on relative entropy (see Report)
3 - Check comments 1 and 2 from report and correct as needed