Decoherence and pointer states in small antiferromagnets: A benchmark test

We thank the referees for the detailed and constructive reports. Some important points were raised that were not addressed in our work. We believe that the reports of the referees greatly helped improve the manuscript.
Below, we comment on the remarks made in the reports.

Report 60

(1) I am slightly confused from the introduction of the choice of the initial states. If on one side I see the reason for this choice, on the other side it seems implied that there are sizeable non-Markovian effects. It seems to me that the different degree on this "non-Markovian behaviour" and its differences going from the weak to the strong coupling are not addressed. I guess a discussion of this point is relevant for the analysis (or at least the authors should point out why/if it is not relevant)

Reply: * We agree that this is an important aspect not covered in the work. To address this point we have calculated the trace distance between the reduced density matrices with initial states and |psi_0> and |psi_N> (the non-monotonicity of this quantity expresses non-Markovian behaviour), for both strong and weak coupling (see Figs. 4 & 7). A paragraph, discussing Figs. 4 and 7, is added at the bottom of Secs. 5.1 and 5.2, respectively.

(2) - I was unable to find in the paper more informations on the choice of the random couplings. It seemed in reading that a single choice of the couplings was presented

Reply: * Our paper was indeed not very explicit about the precise nature of the random couplings. An additional paragraph was added in Sec. 3.3 to correct for this.

(3) - I miss to see the reason of choosing 4-sites for the system instead of 2 (for example). Is there something that we learn from this choice?

Reply: * The 2-site central-system is rather constrained due to its limited size. In the S^z_tot=0 subspace, the CS can in fact be mapped onto a two-level system. Therefore, we choose to consider the 4-site central system. We briefly discuss this point at the bottom paragraph of Sec. 3.2.

(4) The presentation may also improve considerably in the introduction and definition of various quantities. For example: the reduced density matrix \rho is introduced in page 2 and defined in page 5, similarly for the Hamiltonian H

Reply: * These suggestions have been implemented.

(5) I think that the discussion on how to draw some general conclusions from the analysis of few cases can be improved.

Reply: * We agree that certain sentences can be interpreted to general if one were to only read the Conclusion. We have revised part of the Conclusion by using a more careful choice of words.

(6) The format of Fig 2 and 4 should be improved, some features discussed in the text are difficult to visualise.

Reply: * To improve readability we have put time t in Figs. 2 & 4 on logaritmic scale and increased the font. In addition, extra simulations have been carried out such that multiple orders of magnitude in time are visible.

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Report 58

(7) The authors studies only very small systems (4 spins 1/2 for the system coupled to 16 spins 1/2 for the environment). The system is far from an experimentally realistic system as studied in [42].
- The emergence of classicality cannot be seriously studied with such small systems.

Reply: * The fact that 16 environment spins is enough to study decoherence processes is indeed counter intuitive, and was not discussed in our work. In the Discussion (Sec. 6) we now stress the importance of the precise details of the environment and refer to the appropriate references.

(8) Please include the values of the parameters studied in the simulation in the captions of Fig 2-5. This would ease the reading of the text.
(9) A few words commenting Fig 3& Fig 5 would be welcome in the caption (or in the text): why |χ> has disappeared for K=1 and K=20 ? When in the ideal case, the entropy is zero, please write it.

Reply: * The requested changes have been implemented.