Monitoring continuous spectrum observables: the strong measurement limit

We thank both referees for their positive comments on our manuscript.

— Let us first answer the first question by referee 1:

The first question asks "how much of the results can be inferred from the
perturbation theory of Lindbladians (…) without any discussion of the unraveled
equations. (…)?". The Lindblad operator codes linearly for the mean evolution of the
density matrix. This perturbative approach can of course be done but it will only
give information on the behaviour of functions linear in the density matrix. The
statement we have proved concerns the behavior of all polynomials (or convergent
series) in the density matrix, which cannot be deduced from the perturbative theory
of the Lindladian. Nevertheless, the proof we have given is close to this suggested
perturbative approach but it applies to operators acting on any polynomials in the
density matrix.

The second question asks "how relevant are the models and the scaling limits?". For
case (a), the limit is simply the limit of large information extraction rate without
any scaling of the other coupling constants. The case (b) is more delicate because
the Zeno effect takes place at large extraction rate and freezes the dynamics unless
appropriate scalings of the couplings constants is chosen, as is well known. As
recalled in the introduction, the case (b) possesses three different regimes
depending on the time scale: (a) a collapse regime, (b) a classical regime in which
the localised wave function moves in space according to classical dynamics, and (c)
a diffusive regime in which the wave function diffuses randomly. As explained in the
text (in the introduction and at the beginning of section 4), the scalings we have
chosen iare adapted to describe the cross-over from regime (b) to (c), which is
therefore within the semi-classical approximation.

Whether the model in 4.1. is a good description of some particle detector is a
question worth asking. It is well documented that the model of section 4, which is
at the basis of the theory of quantum trajectories, is a good description of the
quantum back-action induced by the monitoring of an observable with continuous
spectrum. Nevertheless, as exemplified by the famous (but yet unsolved) Mott track
problem, it remains an open question whether such quantum trajectory equations are
adapted to describe the detection of particle trajectories in say bubble chambers
(or in any other similar particle detection devices).

— Let now answer the referee 2 comments:

As pointed by the referee our paper "represents a synthesis of ideas from
measurement theory and stochastic processes". Although we have tried our best to
introduce and explain the objects and the techniques we used, we necessary had to
assume that the reader has some (basic) knowledge on both quantum measurement and
probability theory. Making this assumption is unavoidable and it is part of the
difficulties (but also part of beauty) of this scientific topics. But, besides
references [1,2] to books already included, we have added a reference to our lecture
notes on this topic.

Referee 1
We have implemented all the requested changes:
- points 1, 2, 3, 5 and 7 concern misprint that we corrected.
- we didn’t implement point 4 because, even if there are more intrinsic formulations
of the action of the operator $\mathcal{D}$), there are not as explicit as the
straightforward one given in the text and less useful.
- point 6: we have given a more explicit formulation of the statement "reverse
martingales are the right generalization for means of partial sums" and added a
reference.
- point 8: we have labeled the appropriate equation to which the comparison refers.

Referee 2
We have implemented all the requested changes:
- point 1 concerns a misprint that we corrected.
- point 2: we think giving the derivation will render the text too heavy (because it
is bit long) but we have added a reference to our lecture notes on this topic.
- point 3: as suggested by the referee, we change the formulation, to make simpler,
and we remove the reference to an explicit probability space.
- point 4: The referee is right that the theorem remains valid if « H_t adapted » is
dropped. The theorem simply becomes less precise. However keeping the full statement
(including H_t adapted) is crucial because H_t is the information collected by
observing only the outcomes of the indirect measurements : W_t is accessible to the
observer whereas B_t is'nt. We have introduced different filtrations and the main
point of this theorem is to explain the interplay between these distinct
filtrations.
- point 5: the referee points correctly that we have to better characterise the
function for which the statement is correct. We did it in the text.