Monitoring continuous spectrum observables: the strong measurement limit

1. The paper is mostly extremely clear and pedagogical.
2. The new results are clearly highlighted and exemplified.
3. A conjecture is made for the limit of strong monitoring for a particle in a general potential, which should motivate further mathematical work.

1. Because the paper represents a synthesis of ideas from measurement theory and stochastic processes, some technical ideas are not described in sufficient detail for a reader with background in only one of these fields.

This paper provides a rigorous formulation of the challenging problem of continuous monitoring of a quantum observable taking continuous values (e.g. particle position) in the strong measurement limit.

In the case of discrete observables this limit leads to a classical Markov process on the possible measurement outcomes. In the continuous case one naturally expects a diffusion-type process. This paper shows how to derive this process in a controlled way for two important cases -- where the strong monitoring competes with (1) dissipative and (2) Hamiltonian evolution. The general case of dissipative evolution is resolved, but the authors treat only the case of harmonic oscillator evolution in the Hamiltonian case, making a conjecture about strong monitoring in the presence of evolution in a smooth potential.

1. The quantity $\gamma$ is not defined in Eq. (1)!

2. A sketch of the derivation of Eq. (1) and (2) is given in Section 2. It would be a shame not to flesh this out (it would also show where $\gamma$ comes from).

3. I'm not sure that introducing the probability space $(\Omega, \mathcal{G}, p)$ helps the clarity of the theorem stated in Section 2.1, as none of these quantities appear in the subsequent statement of the theorem (I know $\mathcal{G}$ appear in the proof: maybe it could be defined there?).

4. In the statement of that theorem: would it correct to say that $W_t$ is a Brownian motion (dropping the $\mathcal{H}_t$-adapted?

5. End of Section 2.2: "for any bounded function.". Perhaps should add "of compact support"?

1- An important problem within the area.
2- Clear exposition.
3- Timely results that will generate further discussion of the topic.

1-Discussion of relevance of the models / the scaling limits considered is missing.

The paper develops further the perturbation theory for non-demolition measurements in quantum mechanics. The case of discrete observables is now reasonably well understood. In this article the authors do a very natural (and also important) step to study the theory for observables with continuous spectrum. This step is not only technically more difficult but also presents a conceptual challenge due to the non-existence of long time limit of posterior states for the unperturbed dynamics.

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The authors study two situations: (a) Dissipative perturbation, (b) Hamiltonian perturbation. In both cases, the authors provide a convincing analysis leading to the leading order effective dynamics (in appropriate rescaling). In the case (a) they prove a theorem covering large class of systems. In the case (b) they analyze a particular example in details and give good arguments for a general conjecture. Both results are of high quality and important for the current developments in the area.

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The paper is well written. For example, I really like the metaphor of the cheater that is nicely connected to different filterings and conditional expectation values.

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I was wondering about the following two questions while reading the article. The first one is how much of the results can be inferred from the perturbation theory of Lindbladians (e.g. Appendix of https://arxiv.org/pdf/1512.05801.pdf) without any discussion of the unraveled equations. (Or maybe more precisely how much this can be decoupled). For example in the case (a) in the Heisenberg picture the Lindbladian associated to the average evolution in Eq.(9) has for $D=0$ a kernel consisting of any functions of the position operator $f(X)$. The perturbation theory for Lindbaldians then gives that for large $\gamma$ (in the leading order) the evolution inside the kernel is generated by the projected perturbation, which is exactly the generator found above Eq.(14). Similarly (I believe) a semi classically analysis of the Lindbaldian in (b) would lead to the generator (and manifold) of the Markov process in (25), (26).

The second question is how relevant are the models and the scaling limits? For example is the model in 4.1. a good description of some particle detector? If yes, is the classical limit, the relevant physics limit for the detector?

1-Should there be hat on rho in Eq. (10)?
2-What is the meaning of bar on Y in Eq.(12)?
3-Definition of $\mathbb{E}_{\mu_0}$ is missing.
4-It took me some time to decipher the meaning of $\mathcal{D} \cdot$ maybe some more explanation of which operators acts on what objects might help.
5- $\Pi_t^{\alpha}$ below Eq.(14) should be probably $\Pi_t^{\gamma}$.
6-The discussion of reverse martingales is intriguing but cryptic. For example can you provide a reference for the claim that "reverse martingales are the right generalization for means of partial sums ...".
7-What is the hat on psi below equation (20)
8-On page 19 above third displayed Eq. from the bottom, what does "comparing" refer to? Also in that equation should B be S?