General quantum-classical dynamics as measurement based feedback

1. Clarity of writing.

2. Handling stochastic master equations in a very friendly way for physicists, yet making statements that could be made rigorous.

3. Clearly drawn connection to known results in quantum optics that help in linking different communities.

1. Connection to the literature, both in view of the possible physical applications and in view of the general formalism, is not adequate.

2. The focus of the paper should be more clearly stated.

3. In the presentation it is not always clear what is taken for granted and what is derived.

The paper provides in a very readable way the connection between the stochastic master equation for continuous measurement with diffusive noise and the hybrid classical-quantum Fokker-Planck equations considered in Ref.[5]. The idea is to start from the stochastic master equation for continuous measurement and use the measured signal as noise source for the stochastic evolution of classical variables, on which the Hamiltonian and the measuring operators can be made dependent. The connection is clearly outlined, so that I find the paper deserves publication in that it provides useful links between different formulations. I find however that a better discussion of the potential physics that could be described and that motivated e.g. Ref.[5] would be in place. Indeed the formalism itself has been discussed also in previous publications, e.g. the quoted Ref.[22], where also jump noise has been considered. The neglect of jump measurements should be clarified. Combining the two kinds of noise does not necessarily lead to a trivially additive effect.

A better introduction to the addressed physical background.

Clearer motivation for not dealing with jump noise and the possible new implications in considering it.

The abstract mentions considering non-Markovian feedback, but apparently, the point is not further discussed.

The wording "without additional positivity constraints" in the abstract is somehow deceitful to me. The stochastic differential equation and the corresponding Fokker-Planck are equivalent.

Summations over $k$ appear and disappear and should be made more consistent in notation to avoid confusing the reader.

Ask for minor revision

1. It provides a clear and mathematically consistent connection between the dynamics of quantum systems under continuous measurement and feedback, and the recent results on the form of the more general hybrid quantum-classical dynamics (Ref 5).

2. The treatment exploits the intuition provided by the known results on the dynamics of measured quantum systems in order to clarify the typical features of a generic hybrid classical-quantum dynamics and the related decoherence and diffusion effects.

3. The paper, adopting its "diagonal form" , has the merit that complete positivity of the dynamics is naturally obtained and one has a simplified form of the general approach of Ref. [5].

4. The conclusions, even though somehow very general, provide a broad picture of the challenges one has to afford in trying to derive fundamental hybrid quantum-classical theories.

5. The paper provides a quite direct connection between hybrid models and collapse models, which in some approaches does not seem so obvious.

1. The alternative approach in which jump stochastic processes (such as photodetection) replace weak continuous measurements (i.e. homodyne/heterodyne like measurements, with a strong local oscillator) is mentioned only briefly. It is not evident if this at the end does not change too much the physics or not. A more explicit comment on what can be achieved considering jump processes and eventually a more explicit comparison with Refs. 10 and 22 could be extremely useful to the reader.

2. The derivation of the Markovian limit is a bit misleading in my opinion (see below)

In my opinion the paper deserves publication in SciPost, as it perfectly fulfills its scope, that is, "Provide a novel and synergetic link between different research areas.". Such a synergetic link can be really fruitful for both fields, as already pointed out in the list of strengths provided above. In my opinion the paper only deserves to clarify few points in order to become readable by a broader set of scientists.

1. The alternative approach based on jump stochastic processes should be discussed in more detail and not only almost incidentally as it is done in the present version. I understand that the mathematics can be different and one cannot exploit the effectiveness of the Ito calculus, but a more detailed discussion of the approach of Ref. 10 and 22 would provide a broader and clearer view.

2. The derivation of Sec IV of the Markovian limit in which the dynamics of the classical variables is adiabatically eliminated should be adjusted in my opinion. In fact, The equations, starting from eq. 24 seem to be written only for a single feedback term, i.e., a single feedback operator b_k. Instead I expect that, in order to be more general, we have many operators b_k, one for each measured signal r_k. However in most of the equations below the terms with b_k are always outside the sum over the index k.
This makes the derivation misleading and I think it would be proper to adjust it, since the beginning, by assuming a general sum over k in the fluctuating potential of Eq. (24).

Ask for minor revision

1. Very well written.
2. Provides a synergetic link between different research areas.
3. Reviews continuous measurement and feedback well.

1. Does not review in sufficient detail the recent work of Oppenheim and co-workers, and its relation to previous work on semiclassical gravity.

I think this paper serves a very useful purpose to bridge the divide between: (a) some recent, very prominent, claims about the promise of a supposedly new approach to semiclassical general relativity; (b) a body of literature on semiclassical gravity in the Newtonian regime which is already well linked to work on spontaneous collapse; and (c) a large body of literature on continuous measurement and feedback. Therefore I think it could be valuable enough to publish in SciPost. However, I think the author should devote more space to explaining the research claims and results in these recent papers by Oppenheim and co-workers. Many of the potential readers of this paper may not be familiar with that work, beyond the popular discourse on it. What is novel in those works compared to the Newtonian-regime papers, and what are the short-comings (see also requested change 9 below)?

In addition to the major weakness discussed above, I have these requests for changes:

1. Equation (1) is only one model of continuous measurement -- the other is a jump unravelling of a master equation. This is eventually mentioned, but only 2 pages later. It would be better not to hold the knowledgeable reader is suspense for so long.

2. Equation (1) uses the superoperator notation of the textbook [1] for the deterministic part but not for the stochastic part. Is there any particular reason to replace H with M?

3. It is claimed (correctly) that Eq. (9) is the most general form of feedback, but it is not explained why this is so.

4. The statement “but all objects are now implicitly z dependent” may be misleading. It is exactly the 3 objects (or sets of objects) mentioned in the preceding sentence that can have z-dependent.

5. The paper needs a hyphens in compound adjectives, e.g. in z-dependent, measurement-induced etc. It also needs an n-dash in the adjectival phrase "hybrid quantum--classical".

6. The phrase "the “diagonal” form" is used on page 4 a few paragraphs before it is explained what this means.

7. Below Eq.(32), what follows “noting that” is well-known in previous literature. If the author can find where this was first noted, at least in this context, and reference it, that would be of service to the community.

8. Regarding “Further, since the continuous measurement interpretation disappears, it is unclear if such models can be used as a basis for consistent hybrid dynamics”, could the author say something stronger here? Or at least expand upon the problem for those unfamiliar with it.

9. With regard to the major issue mentioned in the report, here is an example of where more information would be highly desirable: “However, because the problematic quantum measurement part is left unaddressed ...” Why and how was this left unaddressed by Oppenheim and co-workers?

Ask for major revision