Is energy conserved when nobody looks?

Dear Editors,

we thank for the report and the helpful remarks by the referee. We have revised the manuscript and kindly ask for further consideration. The details of changes and comments are below.

Yours sincerely, Stanisław Sołtan, Mateusz Frączak, Wolfgang Belzig, and Adam Bednorz ----------------------------------------- Reply/Comments to the Report:

Dear Dr. Kornich,

We first would like to thank you for taking the time to assess our manuscript. We also thank for listing the strengths of our work, on which we agree completely. There are some doubts about the validity of the experiments, and we try to clarify them in the detailed response below. We hope we are able to convince you that with the applied changes the work is a valuable addition to the ongoing discussion about the foundations of quantum theory. We are convinced that the finding will trigger further investigations, e.g. in more concrete experimental setting.

Sincerely yours, Stanisław Sołtan, Mateusz Frączak, Wolfgang Belzig, and Adam Bednorz

Detailed response:

Report: 1) In the end of Section 3 the authors state that indeed, the measurements destroy the translational symmetry of time and thus energy might be not conserved. The argument that the other quantity is also not conserved (angular momentum component in this case) does not solve this problem. Maybe there are more reliable arguments for this? They should be added into the manuscript.

Reply: In the previous version we have not discussed the detection process at all, using only Kraus operators. We decided to add a minimal description of the detection mechanics and a new figure 3 which helps to understand the subtleties of the symmetries and detection, at the end of section 3. In particular, one can construct a model based on a so-called quantum clock, where the Hamiltonian is manifestly time-independent.

Report: 2) Both experiments for the z-component of angular momentum use square capacitors, which violate rotational symmetry of space. Then the angular momentum should not be conserved in any case. Is there some explanation for this?

Reply: Indeed, square capacitors break the rotational symmetry, but it has negligible effect on the system due to the weakness of the measurement. In the revised version we added the detector model with interaction preserving total angular momentum. As we explain in section 5, in a real experiment the symmetry leading to conservation will be probably only approximate and, since the measurement should be weak, a slight asymmetry should not matter. In that section, we derive an operational criterion in the form of a Legget-Garg-typ inequality, which can be used to quantify the violation.

Report: 1) On the page 5, it is written that the magnetic field is produced by the particle itself. Is there some reason for this? Why not simply external magnetic field?

Reply: The magnetic field is of Zeeman type, i.e. orbital movement makes the charge an effective magnet. Since it is only technical and might be confusing, we removed this comment from the text. The actual measurement model of angular momentum is explained on page 7, between (12) and (13).

Report: 2) In Section 5, the authors use macrorealism assumption. Why? The system under consideration is not macroscopic, why does it necessarily have properties (and why particularly q,q',x,y) which can be determined without perturbing the past or future state of the system?

Reply: Indeed, it is not "macro". The term "macroscopic realism" has been coined by Leggett and Garg [20] for a hypothetical macrosopic systems (SQUID) maintaining quantum coherence. However, in practice LG simply used the Hamiltonian of a two-level system and the macroscopic realization is irrelevant. In this sense, our considerations would apply to macroscopic quantum systems as well. Nevertheless, we decided to change "macroscopic" to "objective" which is indeed more appropriate in our opinion. Then, the existence of a positively defined probability of q,q',x,y is equivalent to objectivity. Whether quantum mechanics is consistent with such objectivity is a matter of ongoing debate, to which our discussion of conservation laws hopefully contributes.

Report: 3) In the end of page 7, why the noise D is called large? For the considered limit q->0 it goes to zero.

Reply: We thank for spotting this inconsistency. We meant large variance and corrected the text.