Algebraic Properties of Quantum Reference Frames: Does Time Fluctuate?
Round 1
Reviewer 1 Report
In this work, the authors consider the construction of quantum reference frames (QRF), within system in which the dynamics are imposed by a Dirac constraint, from an algebraic approach using "almost-positive states" with respect to some chosen observable that acts as a reference. Within such construction:
A) The chosen observable does not show quantum fluctuations.
B) Not all observables can be consistently used as a reference, and in particular the possibility of switching from the reference frame of one observable to another depends upon the interaction that may exists between the two.
I find the construction is consistent and provides some interesting insights in the field of QRF, mainly their finding B. Hence, I shall recommend the article for publication, just after the authors satisfactorily clarify the following points:
1) I do not find specially surprising statement A above that, within the construction considered, the observable chosen as reference does not fluctuate and shall be considered as a parameter. This is ultimately the case also in the construction in [2] cited by the authors in page 2 (paragraph before line 49); in that case simply because the reference frame is directly taken away from the physical Hilbert space.
Therefore, I think that the true novelty of the present work is the introduction of restrictions on what can or cannot act as a reference frame, according to the possibility of constructing an algebra of almost-positive states with respect to it. I think it would be important that the authors highlight this fact.
2) The authors consider only models for time reference frames, although claim that the construction could be extended. I wonder if the authors could provide, if not a complete example, just some sketch on how this construction may look like for, say, spatial degrees of freedom, so that the generality of the ideas considered becomes visually more clear.
3) I am not sure to have fully understood the discussion that the authors do about entanglement, mostly in page 12, lines 411-427. My vague understanding is that in previous constructions of QRF entanglement between a physical clock C acting as reference frames and a system S was a necessary ingredient so that one could explain (in the case of time reference frames) evolution of S "in time" as described in C; while in the construction by the authors this is not the case. Is this correct? I shall notice that, in many constructions of QRF (for example [2] again), entanglement is a frame-dependent notion. What entanglement are exactly the authors discussing? Could the role of some reference frames depend on entanglement when considered from the perspective of another reference frame, or from a global "perspective neutral" description of the same problem? Could the authors clarify all this better in the manuscript, and eventually provide some sort of simple example where the necessity of entanglement in the new construction is clearly dropped?
I think that it would be relevant and illustrative in this sense, to compare the construction of QRF by the authors with the recent one in https://arxiv.org/abs/2112.00046, where the entanglement between the reference frame R and the system S, *when considered in the perspective neutral frame*, is strongly related to the capacity of R to act as a good reference frame for S.
In this same direction, I found the discussion of Section 7 on Relational Quantum Mechanics hard to understand. Since it relies to a great extent on the discussion about the role of entanglement, the authors may wish to rephrase Section 7 in light of the changes they introduce into the mentioned discussion on entanglement.
4) In the Introduction (pages 1-2, lines 32-48) the authors emphasize a sharp distinction between "time and space" and "rulers and clocks". I think the philosophy of such distinction goes in the opposite direction to the aim of many of the approaches to QRF. At least on my side, I see QRF from a machian perspective, in which all what physics can talk about are relations between systems. This is why QRF, although through several different approaches, ultimately focus on relational degrees of freedom between quantum systems. Any notion of time or space should ultimately be emerging out of these sets of relations. On the contrary, the authors seem to give some primary ontology to time and space, and then they put quantum systems on top of of these realities. The ideal clocks they build would be then actually *running with the time*. I wonder if I correctly captured the ideas of the authors from their discussion; and if so, if they wished to defend them, namely defend their statement "a clock is not the same as time, and a ruler is not the same as space" (lines 44-45), that seems to inspire their construction, with respect to a genuinely relational understanding of QRF.
Author Response
Thank you for your details comments and suggestions. We have clarified all
points by including new explanations at the following places in the paper:
1) In the last paragraph of the introduction (page 3) and in the first
paragraph of the Discussion section (page 13) we now emphasize the novelty in
our treatment of result A identified by the referee: We go beyond previous
studies of quantum reference frames by constructing a setting in which the
time degree of freedom is not completely removed by going to a physical
Hilbert space. Even though the result of non-fluctuating time is similar to
some (but not all) of the previous statements, we believe that the added
clarity of an explicit treatment of the time degree of freedom is important.
2) We have added an explicit discussion of non-time reference frames in the
bottom half of page 4.
3) We have tried to clarify the role of entanglement, which indeed may appear
in (at least) two different forms in this context. We are not in a position
yet to discuss frame transformations and therefore have to leave the question
of entanglement seen from different frames for future papers. Discussions of
entanglement in our paper refer to the heuristic assumption that any
relational description necessarily implies entanglement (or at least
correlations) in order to constitute a relational description. Our main result
here is that entanglement does not occur if the relational description is with
respect to a redundant degree of freedom. We have addressed this issue in more
detail in the first paragraph of Section 7 (page 12) and (more briefly) in the
first sentence of the same Section on page 13.
4) The distinction between clock and time or ruler and space is now described in
more detail in the top paragraph on page 2. This distinction is indeed based
on the possibility of emergent time and space supported by the referee, rather
than pre-existing time and space.
Reviewer 2 Report
The paper under review provides an interesting contribution to the issue of defining quantum reference frames, namely frames associated to measuring devices with quantum properties. This became recently a very lively area of research, with possible implications for quantum foundations and quantum gravity.
Using the formalism of constrained quantum systems, the authors observe that if reference frames are to be considered to be part of the quantum system, then they are subject to quantum constraints such that the reference variables turn out to lose their quantum properties.
From reading the paper it is not very clear how the notion of quantum reference frames used in this paper compares to the one used in previous literature, such as [1-8]. In particular, it is not clear if the authors are claiming that the previously used notion is inconsistent within the formalism of quantum mechanics or if they are simply proposing an alternative definition. It would be useful if they could spell these issues more clearly.
Author Response
Thank you for this evaluation. We have described the relationship between our
constructions and previous work on quantum reference frames in more detail in
the last paragraph of the Introduction (page 2) and in the first paragraph of
the Discussion section (page 13). Briefly, our new methods make it possible to
clarify quantum properties of clocks and time by using a new notion of states
that can be applied to the clock degree of freedom, which would be completely
removed in traditional treatments based on a physical Hilbert space.
