PONE-D-22-13566On the use of discrete-time quantum walks in decision theory PLOS ONE

Dear Dr. Chen,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

Please submit your revised manuscript by Aug 04 2022 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:

• A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.
• A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.
• An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: https://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols. Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols.

We look forward to receiving your revised manuscript.

Kind regards,

Salvatore Lorenzo

Academic Editor

PLOS ONE

Journal Requirements:

When submitting your revision, we need you to address these additional requirements.

1. Please ensure that your manuscript meets PLOS ONE's style requirements, including those for file naming. The PLOS ONE style templates can be found at 

https://journals.plos.org/plosone/s/file?id=wjVg/PLOSOne_formatting_sample_main_body.pdf and 

https://journals.plos.org/plosone/s/file?id=ba62/PLOSOne_formatting_sample_title_authors_affiliations.pdf

2. Thank you for stating the following financial disclosure: 

   "No"

At this time, please address the following queries:

a) Please clarify the sources of funding (financial or material support) for your study. List the grants or organizations that supported your study, including funding received from your institution. 

b) State what role the funders took in the study. If the funders had no role in your study, please state: “The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.”

c) If any authors received a salary from any of your funders, please state which authors and which funders.

d) If you did not receive any funding for this study, please state: “The authors received no specific funding for this work.”

Please include your amended statements within your cover letter; we will change the online submission form on your behalf.

3. Thank you for stating the following in your Competing Interests section:  

   "No" 

Please complete your Competing Interests on the online submission form to state any Competing Interests. If you have no competing interests, please state "The authors have declared that no competing interests exist.", as detailed online in our guide for authors at http://journals.plos.org/plosone/s/submit-now 

 This information should be included in your cover letter; we will change the online submission form on your behalf.

4. We note you have included a table to which you do not refer in the text of your manuscript. Please ensure that you refer to Table 1 in your text; if accepted, production will need this reference to link the reader to the Table.

5.  Please review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice.

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

********** 

2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: N/A

********** 

3. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

********** 

4. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

********** 

5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: In this work, the authors study quantum walks and present a qualitative comparison between their behavior and some (known) results from decision theory. They analyze how different parameters associated with the initial state and the coin-flip operator affect the final distribution of the walker.

The main results presented in the manuscript seem to contribute to the literature and be technically correct. However, I do have some comments and clarifications that I would like the authors to address. Most of them are related to mathematical and physical concepts since they are my area of expertise.

1. To start, I would like to mention a few typos I identified across the manuscript.

- In line 78, “This serve” should read as “This serves.”

- In lines 148 and 149, it should be observed that the word scissors does not have a singular form. Then, “scissor” should be replaced by “scissors” or “a pair of scissors.”

- There is a missing comma at the end of Eq. (25).

- In line 501, “taking measurement from Eq (31) on the position bases” should read as “taking the measurement from Eq (31) at the locations.”

- In line 519, “taking measurement Eq (31)” should read as “taking the measurement from Eq (31).”

- In lines 654 and 656, “decision maker” should read as “decision-maker.”

- In line 656, “makes” should be replaced with “make.”

- In line 681, “results” should read as “result.”

3. At least the first time the term Hilbert space is mentioned (lines 160 and 161), the fact that the space is a complex Hilbert space should be emphasized. It does not help that the example mentioned in line 165 is the Euclidian (real) space.

4. In lines 164 and 165, it is said that <s_i| complex="" conjugate="" is="" of="" the="">. In reality, the former is the adjoint of the latter. In the case of finite vector spaces, the adjoint is the conjugate transpose.

5. The terms “basis” and “bases” are not properly used in the manuscript. Every time the authors refer to the element of a basis (for instance, |x>), they call it a basis. Meanwhile, the collection of all elements of a basis form a basis (and not a bases). In a vector space of dimension d over the real or complex numbers, an infinite bases (with each basis containing d elements) can be constructed. This problem appears in multiple places. A (possibly non-exhaustive) list of them is lines 162, 166, 167, 169, 170, 192, 351, 357, 362, and 413.

6. In lines 166 and 167, the sentence “Without loss of generality, we will use the term vector for the bases and states in the Hilbert space” is unnecessary. It is also slightly problematic in the sense that the correct statement would be: “Pure quantum states are represented by vectors in the Hilbert space.”

7. When talking about density matrices in lines 174 and 175, the authors state that it “is an ensemble of pure states.” This is not always the case because density matrices can also be associated with quantum systems entangled with other quantum systems via a partial trace. Because of it, even the term “statistical uncertainty” in line 179 may generate disagreement among physicists.

8. In line 180, when mentioning that the evolution of a system is unitary, it should be mentioned that the evolution of a closed system is unitary. In general, this is not the case for the dynamics of open quantum systems.

9. The form for the unitary U in line 182 only holds if H is time-independent. This should be explicitly mentioned.

10. In the subsection “Measurement and observable,” it should be stated that only projective measurements are being described. The most general measurement, not discussed there, is represented by a positive operator-valued measure (POVM).

11. In line 194, I believe the authors meant to write “pure state” instead of “quantum state.”

12. In line 196, when stating that a quantum state collapses, it is prudent to add between parenthesis “(or is updated)” since the word “collapse” might suggest specific interpretations about the measurement process and the ontology of quantum states.

13. In the subsection “The essential ingredient: quantum interference,” I suggest that the authors explicitly add the expression that defines q after Eq. (18). Moreover, I have an additional comment that does not need to be taken into consideration for the current manuscript but may help the authors in future works. More precisely, I want to briefly mention why I have issues with the approach they used to discuss interference in this subsection. First, to me, Eq. (18) can be read as a proof that quantum systems do not admit joint probability (even though I understood the specific way “joint probability” was defined in this work). Second, the current presentation might give the impression that interference is associated with measurements alone when, in fact, it is a characteristic of the dynamics of quantum systems. Observe that one of the main distinctions between classical statistical computation and quantum computation is the fact that, in the former, probabilities are directly manipulated. At the same time, in the latter, complex amplitudes are the quantities that can be manipulated.

14. In line 408, it should be clarified that although the “position space” under consideration is associated with a single spatial dimension, the dimension of the Hilbert space itself is infinite (since x runs over the integers).

15. In line 427, the phrase “incorporates both quantum and Kolmogorovian probabilities” seems to make more sense to me when referring to density matrices resulting from the statistical mixture of pure states (in fact, the rest of the paragraph suggests that this is what the authors had in mind). However, as already mentioned in this report, density matrices also arise from partial traves of composed entangled systems. In these cases, it is possible to argue that density matrices originating from partial traces do not necessarily have Kolmogorovian probabilities associated with them.

16. In line 453, the authors write that “the state propagates without deformation and independently along the left and right directions.” However, in the case under consideration, the equation for psi_R reads as psi_R(x,t) = - psi_R(x-1, t-1). Does the negative sign not have any effect on the dynamics? Could it be that the probability distribution of the system is what is not affected during the dynamics?

17. Maybe the authors could mention that the operator in Eq. (35) is generally a POVM. In the special case of projective measurements, {M_i} are orthogonal projectors, and the POVM reduces to a projection-valued measure (PVM).

18. In line 614, the sentence “This corresponds to a classical random walk evolving in parallel to a quantum walk” does not make much sense to me. I believe the authors mean that the resulting distribution corresponds, in a sense, to an interpolation between a quantum walk and a classical random walk. While the last sentence is correct, I would not say these two walks are “running in parallel.”

19. In line 617, the sentence “recognizing that the spin state at each x is equivalent to a qubit” can be removed because there is not much content to it. I would not say that the coin system is a “spin at each x.” Instead, it can be thought of as the spin of the system. Moreover, a qubit, by definition, is any two-dimensional quantum system. Then, a spin is a qubit.

20. The sentence “Fig 5 shows that a small value of p_m (0.05, 0.15) already gives a characteristic shape of a quantum random walk, which is very different from the classical random walk recovered for p_m = 1” in lines 638 to 640 does not seem to be correct. For values of p_m between 0.05 and 0.15, the distribution starts having its maximum value at zero. This shows that a bit of decoherence is enough for the quantum walk to approach a classical random walk, in accordance with the analysis of the variance in Fig. 6. Moreover, the authors used the term “quantum random walk.” For consistency across the text, I suggest the replacement of this phrase with “quantum walk.”

21. In lines 654 to 656, the conclusion that “if a decision maker constantly “checks” her/his state of mind, the behavior will shift towards a non-interference mode. This resonates with the evidence that continuous probing and self-assessment makes the decision maker more rational” seems quite surprising. In quantum computing, for instance, decoherence must be avoided at all costs to ensure quantum advantage can be achieved. Here, however, “decoherence” leads to advantages. This is one of the results that makes me question if quantum models are really necessary for decision theory. Can the authors comment on this?

22. In lines 680 to 682, the sentence “If the distance from the starting point and one end (say, left end) is a, and the distance from the other end is b, given infinite time, it is a standard results in classical diffusion theory that the probability that a classical particle hits the left end first is b/(a + b)” seems to be valid only if the coin used for the classical walk is fair. Is this the case?

23. In lines 742 to 745, the authors stated that “Behaviorally, rho can be interpreted as regulating an “internal time constraint”, for example to save computational resources, even when there is no explicit time pressure. The parameter t may be thought of as exogenous (e.g. a time limit imposed by the experimenter or the environment).” Because of the trade-off between rho and the total time of the quantum walk, would it be possible to, alternatively, fix rho and define an effective time for the dynamics? The effective time would be a function of the internal and external time constraints.

24. In Supplemental Information 1, the authors write that “There is one main difference between the standard theory of quantum measurements and QDT: the former considers measurements over passive systems, so there is no preferred quantum state, and any basis can be employed. In QDT, the decision maker is an active entity, represented by her own state of mind, that encodes her personal attributes/preferences.” I would like to comment that, even in quantum mechanics, there is a preferred basis in various scenarios. An example is the so-called computation basis used in quantum circuits and, more generally, in systems used for quantum control, like the coin system in a quantum walk.

25. Is the gain of replacing classical models for quantum ones only conceptual? If it is, one should evaluate if this shift is really worth it because quantum mechanics itself comes with many interpretational issues. What is the position of the authors on this aspect?

I hope the comments/questions presented here help improve the work.</s_i|>

********** 

6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

**********

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email PLOS at figures@plos.org. Please note that Supporting Information files do not need this step.

On the use of discrete-time quantum walks in decision theory

PONE-D-22-13566R1

Dear Dr. Chen,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

An invoice for payment will follow shortly after the formal acceptance. To ensure an efficient process, please log into Editorial Manager at http://www.editorialmanager.com/pone/, click the 'Update My Information' link at the top of the page, and double check that your user information is up-to-date. If you have any billing related questions, please contact our Author Billing department directly at authorbilling@plos.org.

If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org.

Kind regards,

Salvatore Lorenzo

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

Reviewers' comments: