QMetrology from QCosmology: Study with Entangled Two Qubit Open Quantum System in De Sitter Space

Dear Editor,

Authors have answered some of the questions raised by me. It can be accepted for publication.

The authors have improved a lot on the formatting which makes me satisfied now. However I still need more explanations about the following major parts before this paper going any further:

1. Why is the studied object important? As I also mentioned in the last round of review, dS space does not bring anything noteworthy and therefore the authors must underscore why should we care about this particular problem: does it bring new understanding of quantum metrology (like breaking the no-go theorem of quantum metrology with dephasing, see [1] below), or this particular space is commonly encountered?
2. For such a simplified system, the author should discuss about what is the best way to fully use the Fisher information in order to estimate the phase. For large scale system this remains an open question, but for 2-qubit system this should be derived and included in the text.

Also, as pointed out by other referees, I recommend the authors to clearly distinguish introduction & review with the original works derived in this paper. Current form of this manuscript is still ambiguous to me.

1. The analysis appears to be erroneous.
2. It is not clear why the problem studied here is significant.
3. Very similar studies have appeared in the past.
4. Relevant past works and similar studies have not been cited.
5. The paper does not always clearly distinguish what has been done by others and what is the original contribution of the authors.
6. Presentation lacks clarity.

This paper studies Fisher Information for a particular system: a pair of qubits coupled to a scalar field which acts as a thermal bath. The body of the paper starts with a section on the 'two qubit open system' which (though it was not clearly stated) is a review of previous work. At the end of this section, schematic formulae for the density matrices of the two qubits are given and it is stated that the detailed expressions have been calculated in a different paper (co-authored by some of the authors) in the approximation $2 \pi k \omega >> 1$ and subject to the constraint $\coth (\pi k \omega_0)=0$ where $k, \omega,\omega_0$ are all parameters that enter the density matrices of the qubits. It is stated that the results of this other paper ( not explicitly given here) will be used in obtaining Fisher information.

The original contribution of the paper begins from the section titled 'Estimation of Parameters' and essentially consists of using the density matrices mentioned above to calculate classical and quantum Fisher Information for this system. Variation of CFI and QFI with the different parameters that enter the density matrices is studied.

Now let us elaborate on the issues with the paper:
1. The analysis appears to be erroneous. As per the statement made in the paper, the results for density matrices were obtained in the para regime $k\omega \gg 1 $ subject to the constraint $\coth \pi k \omega_0 =0$. Now there is no real solution to the constraint, the only solution is $k \omega_0 = (n+1) i/2 $ for integer $n$. $\omega_0$ being real, this means $k$ must be purely imaginary. Then it is unclear how $k \omega \gg 1 $ can be fulfilled, since $\omega$ is real again. In any case, in the section on Fisher Information only real values of all the parameters have been considered and further only small values $k\omega$ have been considered. So it appears that neither of the two conditions, under which the paper claims to have derived the expressions for density matrices, were satisfied when the same density matrices were used for the calculation of Fisher Information.

2. It is not clear why the problem studied here is significant: Even if the above issues did not exist, the study here would still need to establish significance. It has not been established why the qubit system is of physical interest, or why metrological considerations are appropriate for this system, or if the calculation of QFI tells us anything non-trivial. There have been interesting applications of Fisher Information in a de Sitter background (for instance, to ask if there are fundamental limitations to measuring cosmological observables of interest), but in this case the study of QFI appears to be arbitrary.

3. Similar studies have appeared in past: Quantum Fisher information for an Unruh-DeWitt detector coupled to a scalar field in a dS background was obtained in arXiv 1806.08922, QFI for a qubit coupled to a scalar field in a dS background was obtained in 'Protecting quantum Fisher information in curved space-time' by Zhiming Huang (EPJP, 2018). Even if significance could be established for this direction of research, the paper would have to be establish how this study is telling us something significantly different from these previous ones.

4. Relevant past works and similar studies have not been cited: the two papers mentioned above which also studied QFI in a very similar context have not been cited. The model of the two qubit system in de Sitter used here first appeared in arXiv 1310.7650 and the formulae given in this paper appear to closely follow the results presented there. While this paper has been cited in previous papers by some of the same authors, it was not cited here. Some other relevant references which studied Fisher information or the dynamics of similar systems in de Sitter space but have not cited: 1812.02345, 1707.09702, 1407.4930, 1605.07350, 1706.0917, 1707.08414.

5. The paper does not always clearly distinguish what has been done previously by others and what is the original contribution of the authors: The authors did not mention where the qubit model used here originated. The fact that it had already been studied in de Sitter space (first in 1310.7650) should have been mentioned. These omissions, together with the authors' referring to the qubit model as 'our model', can be misleading for the reader. More generally, there is no discussion in the introduction about any previous work on open systems in dS (even though some of this research is cited), which can again give an incorrect impression about the originality of the paper.

6. Presentation lacks clarity: Previous results that have been used as the main input in the study of this paper were not presented explicitly nor was a detailed reference given (see point 1 above). Quantities were not defined where they were introduced (for instance, $L$ and $\omega_0$ first appear in eqns 10 and 11 and are not defined till two pages later). There were a number of things that were not clearly explained (such as the distinction between $\omega$ and $\omega_0$, both of which are defined as 'Fourier modes of the Wightman functions' in the paper). Lack of paragraph breaks hampered readability (the whole first column of the second page is a single block of text).