Heat current and fluctuations between a dissipative qubit and a monitor under continuous measurement and feedback

The authors investigate heat current and fluctuations in a problem where a qubit is coupled to reservoirs and is also monitored by continuous
measurement and feedback.

The problem is timely and the results are interesting. I can recommend the article for publication after the authors address the following points:

1. Presentation of the model: Several reservoirs are indicated in Fig.1 and in section 2.1. However, nothing is mentioned afterwards about the
role of them. Do they have the same temperature? Since the results focus on a single bath, the authors should mention that.

2. In Fig. 2 the authors describe m and n as monitor and feedback states, respectively. It would be useful if this is also mentioned in Section 3.1.

3. Even more important, the results depend on the temperature of the thermal bath. I think that the authors can further stress this point. In Fig.2, no
Information is provided on the temperature of the bath.

4. In Section 3.3 they conclude that the temperature of the thermal bath affects the condition of zero current between the qubit and the monitor, but
nothing is said about the current between the qubit and the bath. Is the qubit thermalized with this bath or is there heat flowing through
qubit between the monitor and the thermal bath?

5. In section 4 I do not find the explicit definition of J_c(t). In the discussion of the results, it would be nice to find some discussion on how the features
of the Poisson noise and backaction noise are related to the behavior of the heat current discussed in fig. 2.

Ask for major revision

- Addresses a timely topic of interest in quantum thermodynamics.
- Goes beyond a description in terms of average quantities by considering the heat current fluctuations.

- Only the steady-state configuration is addressed, while a previous study by the authors on a similar system also considered the transient dynamics.

The authors study a dissipative qubit in the presence of continuous measurement and feedback by a monitoring system, extending a previous study by the same authors (Ref. [42]), where only the measurement process was included.
They employ a master equation approach to obtain the steady-state heat exchange between qubit and monitor. They assume that the qubit is coupled to a collection of bosonic baths, that are treated as an Ohmic environment. The analysis is complemented by a quantum trajectory analysis allowing them to study the stochastic evolution of the monitored qubit.

The key results are:
- the simultaneous presence of measurement and feedback allows to selectively heat or cool the qubit;
- the measurement-induced backaction produces fluctuations in the heat current that deviate from the Poissonian limit, and the system is often found in the sub-Poissonian regime.

I believe this work is of good quality and deserves publication. However, I am not confident that publication in SciPost Physics is appropriate, as the journal's acceptance criteria are not met in my opinion. Specifically, the authors indicate that their work provides a novel and synergetic link between different research areas, but I fail to see that this is the case.
To my understanding, the only connection mentioned by the authors with a different research field (other than quantum thermodynamics/energetics) is with electronic transport. This connection is in my opinion rather faint: it is indeed based on the analysis of the Fano factor - which is also used in electronic transport - but I do not think that this fact by itself constitutes a novel and synergetic link between different research areas.

Instead, I believe this works meets the acceptance criteria of SciPost Physics Core and therefore recommend publication in that journal.

1. There is a slightly misleading statement about the Poissonian noise in electronic transport (in the introduction and below equation 19). Noninteracting electrons in a nanowire exhibit Poissonian noise in the weak transmission limit only. Otherwise, the noise obeys a well-known $T(1-T)$ behavior (where $T$ is the nanowire transmission), while the current is just proportional to $T$. Consequently, there can be deviation from the Poissonian noise limit. I suggest that the authors correct the statements by explicitly indicating that the Poissonian regime in electric transport is fulfilled at weak transmission.

2. The following sentence in the introduction is unclear. "By considering continuous quantum measurement and feedback, we can address the steady-state properties of the heat exchange, it’s a steady-state heat current, when accounting for dissipation, which is inevitable but ubiquitous in nature.". I would suggest to reformulate it.

3. The authors state that their model can find application as a measurement/feedback-based quantum refrigerator. Given that they have already calculated the heat fluctuations, could they also provide the efficiency of the cooling process and compare it with standard performance quantifiers, such as the thermodynamic uncertainty relation?

4. I would like to encourage the authors to upload the data for the trajectory simulations to an established data repository (e.g., Zenodo), to facilitate the validation and reproducibility of their analysis.

5. Grammar can be improved. For instance:
- page 4: "the steady-state heat current holds the following inequality" $\to$ "fulfils/satisfies the following inequality";
- page 5: "the measurement to the ground state and the feedback to the excited state" $\to$ "the measurement projects the system in the ground state..."?;
- page 5: "the qubit energy is heated by the measurement" $\to$ either "the qubit is heated" or "the qubit energy is increased";
- caption of Fig. 2: "the heat current never takes negative" $\to$ "never takes negative values";
- page 7: "the correlation function is decomposed of the delta function... and the time non-local correlator" $\to$ "is decomposed in a delta function... and a time non-local correlator";
- page 9: "the heat change by the quantum jump and the excess heat are opposite signs" $\to$ "... are of opposite signs" or "have opposite signs";
- Page 11: "It shows zero at four points" $\to$ "it vanishes at four points" or "it crosses zero at four points" or simply "it is zero at four points".

Accept in alternative Journal (see Report)