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An objective collapse model without state dependent stochasticity
by Lotte Mertens, Matthijs Wesseling, Jasper van Wezel
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Submission summary
| Authors (as registered SciPost users): | Lotte Mertens · Jasper van Wezel |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202301_00035v1 (pdf) |
| Date submitted: | 2023-01-26 11:06 |
| Submitted by: | van Wezel, Jasper |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
The impossibility of describing measurement in quantum mechanics while using a quantum mechanical model for the measurement machine, remains one of its central problems. Objective collapse theories attempt to resolve this problem by proposing alterations to Schrodinger's equation. Here, we present a minimal model for an objective collapse theory that, in contrast to previous proposals, does not employ state dependent stochastic terms in its construction. It is an explicit proof of principle that it is possible for Born's rule to emerge from a stochastic evolution in which no properties of the stochastic process depend on the state being evolved. We propose the presented model as a basis from which more realistic objective collapse theories can be constructed.
Author comments upon resubmission
REFEREE 1:
We thank the first referee for their careful reading of our manuscript, and for their supportive comments and outspoken appreciation of the novelty of our approach.
We are happy to adopt both of the constructive suggestions of the referee: 1) In the revised manuscript, we include additional discussion of experiments that may be used to test the proposed theory. 2) In the revised manuscript, we made the paragraph mentioned by the referee into a separate subsection. We do not include a separate figure for the relation between N and epsilon tau_r, because without estimates of the parameter values relevant to any particular implementation of our model, we feel the figure would not add insight beyond the analytical expression already presented in the manuscript.
REFEREE 2:
The report of the second referee, unfortunately, does not address the actual content of our manuscript. Instead, as explained below, it argues that we should not be allowed to ask particular research questions (i.e. to question assumptions made in existing but as-yet untested theories). We believe this is a biased and unprofessional attitude towards the presentation of new scientific results.
Nevertheless, we will respond to the remarks made by the referee:
1) the report begins with: "The authors are critical of the stochastic part of the modified Schrodinger equation depending on the quantum state of the system, in objective collapse theories. It is not clear to me why the authors find this objectionable. I request the authors to look up [PRL 111, 210401 (2013)] for the very general and plausible conditions under which stochastic modification of the Schrodinger equation essentially uniquely leads to collapse models. And this does imply that the stochastic component can depend on the quantum state of the system. I hope the authors will agree that objective collapse means spontaneous collapse, i.e. collapse does not require the presence of a measuring apparatus. Yet, the collapse inducing mechanism can be state dependent; in fact it should be so. Because the most likely explanation of the origin of stochasticity is coarse-graining of an underlying deterministic evolution (which is non-unitary but norm preserving). Randomness then arises as a consequence of ignorance of the fine-grained dynamics, quite like in coin tossing and in Brownian motion. The authors suggest that objective collapse is being caused by coupling of the quantum system to a universal noise field, and in order to preserve universality the coupling must be state dependent. However, we do not know whether there indeed is such a noise field in nature. It is much more reasonable that the noise field is a representation of coarse-graining of a deterministic underlying dynamics. Thus, I am not at all convinced that the coupling to the stochastic component ought to be independent of the state of the quantum system."
-> This paragraph clearly states the opinion of the referee's that existing collapse models having state-dependent noise is not a problem. Our manuscript, however, addresses the separate question of whether state-dependent noise is a necessary assumption for collapse theories in general. Our work shows it is not. We explicitly show that it is possible to construct an objective collapse theory without state-dependent noise.
The referee does not comment on this result, but states their own subjective reasons for feeling satisfied with state-dependent noise. This has no basis yet in either experimental verification of any particular model, nor in theoretical exploration of scientific constraints and possibilities.
The discussion of state-dependent noise in existing objective collapse theories that is included in our manuscript is intended as an illustration of why the question addressed in our work might be considered relevant within the field. It is not intended to rule out or exclude any existing theory as a candidate for describing quantum measurement. Which model, out of all possible flavours of objective collapse theories, best describes nature will have to be determined by experiment. We feel that in the absence of experimental observation, the question which types of noise are possible at all, is a relevant research question, and exploring different types of models is of scientific interest to the broader research community.
2) The referee continues: "Also, the authors have not investigated whether such a requirement on their part is consistent with the nosignaling condition."
-> We agree with the referee on this point, and thank them for pointing out the important open question of whether or not models for quantum measurement other than existing objective collapse theories necessarily allow for faster-than-light signalling. In fact, this is a question we are actively investigating, and which we will report on in the near future. It is however separate from the question posed in the current paper, which is whether or not models for quantum measurement with state-independent noise can be made to reproduce Born's rule.
3) The referee then states: "The other important criticism concerns the derivation of the Born probability rule. The authors’ own derivation of the Born rule by fine-tuning the parameters of their model is not satisfactory at all! A different choice of parameters will contradict experiment! In conventional objective collapse theories, the Born role inevitably follows IF one assumes the non-unitary stochastic evolution to be norm-preserving [for a proof see e.g. Chapter 6 of Stephen Adler’s book `Quantum theory as an emergent phenomenon’]. Thus the real question to ask is: why should norm be preserved? This very important question remains unanswered in collapse models, and also in Adler’s theory of trace dynamics described in the aforesaid book."
-> Our result is that it is possible to construct a theory with state-independent noise that yields Born rule, if the parameters in the model obey a certain relation. The referee is not satisfied by this, but fails to explain why. If a fine-tuning between parameters occurs in physical theories, this invariably means there is an underlying physical relation between the parameters. A famous example is Einstein's use of fluctuation-dissipation to explain why the diffusion and drift terms in Brownian motion must be fine-tuned with respect to one another (that is, they are related by the laws of thermodynamics). The fine-tuning seen in our proposed model should be taken in the same spirit: the fact that these parameter values are related simply implies that the physical observables they represent are related.
Please note that a similar fine-tuning also occurs in "conventional collapse theories": the Girsonov transform employed in the derivation of those theories is carefully fine-tuned to produce conservation of norm and adherence to Born's rule (which in that case can be simultaneously achieved).
4) The referee finishes with: "On the whole, I feel the authors exhibit an inadequate/incorrect understanding of the objective collapse theories developed by, and subsequently to, the seminal work of Ghirardi et al. (1986). The authors do not have any valid / admissible criticism of objective collapse theories. Of course those theories are phenomenological, and need further investigation: what is the origin of spontaneous collapse; what is the noise spectrum actually like (white, colored, …); why is norm preserved despite non-unitarity; is there a deterministic theory underlying objective collapse; is gravity involved; how to make relativistic collapse models, if at all that is possible. I respect the authors’ interest in collapse models; however I think that interest should be directed towards the unresolved issues mentioned here; not towards questioning objective collapse model equations per se – those appear to be built on reasonable physical premises, and if they are to be ruled out, that would be through testing them in the lab."
-> The referee here lists several open problems of "conventional collapse models" and insists that we should work on those, instead of on the scientific question posed in the current manuscript.
We understand that the referee may personally find the open problems they list more interesting than the one we address. However, the question "can state-independent noise give rise to Born's rule" is a valid scientific question, and one that we argue is of interest to a field of researchers working on models of quantum measurement. This is echoed by the supportive and enthusiastic report of referee 1. Besides, exploring different models might lead to the uncovering of new properties of objective collapse theories in general, that can then also be used to address the open problems in conventional models listed by the referee.
We feel that insisting that our results should not be published, solely because the referee prefers to find answers to different questions than the one we pose, is unprofessional.
List of changes
1) In the revised manuscript, we include additional discussion of experiments that may be used to test the proposed theory.
2) In the revised manuscript, we made the paragraph mentioned by referee 1 into a separate subsection.
Current status:
Reports on this Submission
Report #2 by Eric Aspling (Referee 1) on 2023-2-23 (Invited Report)
- Cite as: Eric Aspling, Report on arXiv:scipost_202301_00035v1, delivered 2023-02-23, doi: 10.21468/SciPost.Report.6788
Report
I still really enjoyed this article. I find that the authors may have over-corrected to appease the earlier referee. I have made some suggestions and would highly recommend publication if these recommendation are made.
Requested changes
I have read the most recent reviewers remarks and highly suggest the authors follow their suggestions. Furthermore I have a new list of suggestions, mostly grammatical.
1.) First page, second paragraph, last sentence: "large molecule" should be "a large molecule".
2.) Second page, second column, third full paragraph: Out of place citation and period.
3.) Second page, last sentence: extra comma.
4.) Third page, left column: equation at the bottom is split. Perhaps this would be better off in an equation environment or with some added tildes.
5.) Third page, last paragraph: repeated phrase "with vanishingly small \epsilon" in back-to-back sentences.
6.) Fourth page, second column, first full paragraph: missing period and funky citations. It should read "[23,37]."
7.)Appendix, directly below the equality in (A.4): "fo" should be "for".
My last comment is that in the first sentence of section II the authors assert, "all objective collapse theories propose an alteration or correction to Schrodinger’s equation". They then outline their theory in a manner that resembles the equation (1). However, in section V the authors state, "We therefore do not argue that the current model presents a realistic objective collapse theory". Which seems contradictory to the sentence in section II. Clarification is required here.
Report #1 by Anonymous (Referee 4) on 2023-2-7 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202301_00035v1, delivered 2023-02-07, doi: 10.21468/SciPost.Report.6699
Report
An objective collapse model without state dependent stochasticity
by Lotte Mertens, Matthijs Wesseling, Jasper van Wezel
I am writing this review as a condensed matter theoretician with an interest in fundamental problems, a certain reading background in the subject, but little own publication record in this field.
Generally, I find the present manuscript very well written and largely understandable even to a non-expert such as myself. It avoids the mistakes of many other such manuscripts which often use very specialist jargon. I enjoyed reading the manuscript. And I finished reading it. With a sense of having understood its main points. I largely only have a few technical comments. Should these be taken into account by the authors, I would enthusiastically recommend publication. I have also had read of the rebuttal provided by the authors following the original round of reviewer comments.
1. First page, first paragraph. Why do the authors state that macroscopic realm of objects is comprised of about 10 to the 18 atoms or more? I would have normally expected to see a number such as 10 to the 23. If this is not a typo, then maybe the 10 to the 18th should be explained.
2. Page 1. Left hand column, second paragraph. I think it would be helpful to support some of the discussion of interpretations of quantum mechanics by citing a few relevant books, and not just the single review article as is presently done. As such a book, I can for example suggest:
- Jonathan Allay, Quantum reality: Theory and philosophy, second edition, as published by Taylor and Francis.
3. First page. Second paragraph of the right tend column. When speaking about the “existing objective collapse theories” … with a non-linear state dependent factor.” relevant citations should be provided Beyond the simple citation of the review article.
4. Page 1. Right and column. The authors state that “origin of the coupling remains unexplained.” Perhaps the use of the word “unexplained” explains some of the antagonism of reviewer 2. Perhaps a reformulation can help. For example using “unclear”, “controversial” or “mysterious” instead of “unexplained”.
5. Page 1 right and column last paragraph. The authors state that their model distinguishes from “existing collapsed models … without … a state dependent probability distribution function.” Perhaps here one could add the comment regarding “fine tuning” discussed by reviewer 2 and in the rebuttal. Only a small comment needs to be made here. And a discussion similar to the one in the rebuttal could be added later in the discussion section.
6. Page 2 bottom of the left-hand column. “The first part of this process.... the latter part.” I'm not quite sure what the “first” and the “latter” part refer to. This should be clarified.
7. Page 2, right hand column, third paragraph. “... all of the predictions of quantum mechanics are recovered.” Is it true to say “all” the predictions are recovered? A “time-dependent norm”, as used in the following sentence, is for example not a prediction of quantum mechanics. Perhaps it is simply best to remove “all of” Or to reformulate more precisely.
8. Page 5. Right and column top paragraph. I presume the discussion of the microscopic region corresponds to N epsilon ->0 as in the equation 4. It would be good to use the mathematical expression here.
9. Page 5 section B first paragraph, centre of. “pi in orange “ should use the symbol for Pi.
10. Page 6 just before the start of section C. That cos^2 (phi0/2) Is indeed the desired Born rule becomes much clearer if equation (6) was cited again here.
11. Figure 2. The legend. In the Figure 4 be not equals 1.0. Is near unreadably faint. I suggest to simply use black for that label.
12. Page 7. Right Hand column. The authors assume that “the infinitesimal symmetry breaking field is beyond the control of any experiment.” Doesn't that, however, introduce a stochastic coupling for the state of the system? One could imagine an experimenter who does control the state well enough and for then it would be a coupling to the state with some stochasticity. I feel that this assumption might very well be the weak point of the authors approach here. At least it should be discussed.
13. Appendix equation A1. The symbol G used here as a general time evolution operator clashes with G introduced an equation 1 where it is the collapse-generating operator. I suggest using an operator such as F.
14. Appendix just after equation A5. Why do the overall phase theta and normalisation n not appear in the equations for phi-dot and theta-dot (sorry, no greek). There should be a physics-based explanation.
15. When reading the rebuttal to reviewer 2 I find the authors’ explanations largely convincing. However, I also find that the issue of non-signalling and the fine tuning may very well be something that a reader might find puzzling. Hence, I would like to suggest including the discussion of non-signalling and fine tuning like what is done in the rebuttal also in the main text of the manuscript.
Author: Lotte Mertens on 2023-02-27 [id 3406]
(in reply to Report 1 on 2023-02-07)
I am writing this review as a condensed matter theoretician with an interest in fundamental problems, a certain reading background in the subject, but little own publication record in this field.
Generally, I find the present manuscript very well written and largely understandable even to a non-expert such as myself. It avoids the mistakes of many other such manuscripts which often use very specialist jargon. I enjoyed reading the manuscript. And I finished reading it. With a sense of having understood its main points. I largely only have a few technical comments. Should these be taken into account by the authors, I would enthusiastically recommend publication. I have also had read of the rebuttal provided by the authors following the original round of reviewer comments.
--> We thank the referee for taking the time to consider the manuscript and for their support of publication. We will address the comments point by point below.
-
First page, first paragraph. Why do the authors state that macroscopic realm of objects is comprised of about 10 to the 18 atoms or more? I would have normally expected to see a number such as 10 to the 23. If this is not a typo, then maybe the 10 to the 18th should be explained. --> The power $10^{18}$ is not a mistake but we will include a sentence in the manuscript to explain this. Above $10^{18}$ there have been experiments confirming the laws of classical physics to predict their collective dynamics. But below this number (and above $10^6$), we do not have any experimental data on quantum measurements. Therefore it is usual to classify the region $10^6 - 10^{18}$ as the 'mesoscopic' regime. We added the appropriate citations to the sentence as well.
-
Page 1. Left-hand column, second paragraph. I think it would be helpful to support some of the discussion of interpretations of quantum mechanics by citing a few relevant books, and not just the single review article as is presently done. As such a book, I can for example suggest: - Jonathan Allay, Quantum reality: Theory and philosophy, second edition, as published by Taylor and Francis. --> We will add this citation and thank the referee for suggesting it.
-
First page. Second paragraph of the right tend column. When speaking about the “existing objective collapse theories” … with a non-linear state dependent factor.” relevant citations should be provided Beyond the simple citation of the review article. --> We agree with the referee and added the appropriate citations.
-
Page 1. Right and column. The authors state that “origin of the coupling remains unexplained.” Perhaps the use of the word “unexplained” explains some of the antagonism of reviewer 2. Perhaps a reformulation can help. For example using “unclear”, “controversial” or “mysterious” instead of “unexplained”. --> We will adjust the wording and thank the referee for noting this.
-
Page 1 right and column last paragraph. The authors state that their model distinguishes from “existing collapsed models … without … a state dependent probability distribution function.” Perhaps here one could add the comment regarding “fine tuning” discussed by reviewer 2 and in the rebuttal. Only a small comment needs to be made here. And a discussion similar to the one in the rebuttal could be added later in the discussion section. --> We thank the referee for the suggestion and included a sentence in the introduction and a paragraph in section IV of the manuscript to comment on the fine-tuning of the model.
-
Page 2 bottom of the left-hand column. “The first part of this process.... the latter part.” I'm not quite sure what the “first” and the “latter” part refer to. This should be clarified. --> We clarified this in the text.
-
Page 2, right hand column, third paragraph. “... all of the predictions of quantum mechanics are recovered.” Is it true to say “all” the predictions are recovered? A “time-dependent norm”, as used in the following sentence, is for example not a prediction of quantum mechanics. Perhaps it is simply best to remove “all of” Or to reformulate more precisely. --> As the norm of a quantum state can never be measured there is no prediction on the time-dependence, which is the main-point we try to convey in this sentence. We will clarify the sentence.
-
Page 5. Right and column top paragraph. I presume the discussion of the microscopic region corresponds to N epsilon ->0 as in the equation 4. It would be good to use the mathematical expression here. --> The referee is right and we will add $N$ into the manuscript. We thank the referee for noting this.
-
Page 5 section B first paragraph, centre of. “pi in orange “ should use the symbol for Pi. --> We thank the referee for finding this typo and will adjust it.
-
Page 6 just before the start of section C. That $\cos^2 (\phi_0/2)$ Is indeed the desired Born rule becomes much clearer if equation (6) was cited again here. --> We thank the referee for this suggestion and will implement it.
-
Figure 2. The legend. In the Figure 4 be not equals 1.0. Is near unreadably faint. I suggest to simply use black for that label. --> We chose this colour because it is the same as the colour of the line and we prefer to keep it this way.
-
Page 7. Right Hand column. The authors assume that “the infinitesimal symmetry breaking field is beyond the control of any experiment.” Doesn't that, however, introduce a stochastic coupling for the state of the system? One could imagine an experimenter who does control the state well enough and for then it would be a coupling to the state with some stochasticity. I feel that this assumption might very well be the weak point of the authors approach here. At least it should be discussed. --> Just as in the standard theory of spontaneous symmetry breaking, the field required to break unitarity goes as $1/N$. Applied to macroscopic systems the field required is fundamentally out of reach, and will never be reached. Recal for example the classic illustration of a table being localised by a flap of the wings of a butterfly on the other side of the world.
-
Appendix equation A1. The symbol G used here as a general time evolution operator clashes with G introduced an equation 1 where it is the collapse-generating operator. I suggest using an operator such as F. --> We adjusted it.
-
Appendix just after equation A5. Why do the overall phase theta and normalisation n not appear in the equations for phi-dot and theta-dot (sorry, no greek). There should be a physics-based explanation. --> This is an indication of the fact that overall phases and normalisation can not be measured in experiment. If they would appear in the equations for $\dot{\phi}$ and $\dot{\theta}$ this would not be true anymore. We will include a sentence on this in the Appendix.
-
When reading the rebuttal to reviewer 2 I find the authors’ explanations largely convincing. However, I also find that the issue of non-signalling and the fine-tuning may very well be something that a reader might find puzzling. Hence, I would like to suggest including the discussion of non-signalling and fine-tuning like what is done in the rebuttal also in the main text of the manuscript. --> As mentioned above we included a paragraph and sentence on the fine-tuning and will add a sentence about non-signalling as well.
We again thank the referee for their supportive comments and careful consideration and hope to have answered their comments to satisfaction.
Author: Lotte Mertens on 2023-02-27 [id 3405]
(in reply to Report 2 by Eric Aspling on 2023-02-23)We again thank the referee for the support of the manuscript. We are happy to include the suggestions and implement the following proposed corrections to the manuscript.
1.) First page, second paragraph, last sentence: "large molecule" should be "a large molecule". 2.) Second page, second column, third full paragraph: Out of place citation and period. 3.) Second page, last sentence: extra comma. 4.) Third page, left column: equation at the bottom is split. Perhaps this would be better off in an equation environment or with some added tildes. 5.) Third page, last paragraph: repeated phrase "with vanishingly small $\epsilon$" in back-to-back sentences. 6.) Fourth page, second column, first full paragraph: missing period and funky citations. It should read "[23,37]." 7.)Appendix, directly below the equality in (A.4): "fo" should be "for".
--> We understand the confusion between the sentences. What we meant in section V is that even if it indeed is a valid objective collapse model, it is not a realistic theory of nature. We will adjust this sentence to make it more clear.
We again thank the referee for taking the time to consider our work carefully.