## SciPost Submission Page

# Monitoring continuous spectrum observables: the strong measurement limit

### by M. Bauer, D. Bernard, T. Jin

#### - Published as SciPost Phys. 5, 037 (2018)

### Submission summary

As Contributors: | Tony Jin |

Arxiv Link: | https://arxiv.org/abs/1805.07162v3 |

Date accepted: | 2018-10-01 |

Date submitted: | 2018-09-05 |

Submitted by: | Jin, Tony |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Mathematical Physics |

Approach: | Theoretical |

### Abstract

We revisit aspects of monitoring observables with continuous spectrum in a quantum system subject to dissipative (Lindbladian) or conservative (Hamiltonian) evolutions. After recalling some of the salient features of the case of pure monitoring, we deal with the case when monitoring is in competition with a Lindbladian evolution. We show that the strong measurement limit leads to a diffusion on the spectrum of the observable. For the case with competition between observation and Hamiltonian dynamics, we exhibit a scaling limit in which the crossover between the classical regime and a diffusive regime can be analyzed in details.

### Ontology / Topics

See full Ontology or Topics database.Published as SciPost Phys. 5, 037 (2018)

### Author comments upon resubmission

— 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.

### List of changes

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.