Le, TP

Olaya-Castro, A

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UCL Discovery

 Le and Olaya-Castro Reply: In a recent Comment [1] onour Letter [2], Feller et al. [1] identified a mistake in ourmathematical expression of “strong independence” forsystem-environment states that satisfy spectrum broadcaststructure. We concede that we wrote a mathematical con-dition that is necessary but not sufficient. However, we usedthe original and correct qualitative definition for strongindependence throughout the Letter and in our proofs,therefore the proofs and statements, aside from the afore-mentioned mathematical expression, remain correct.The concept of strong independence was originally,qualitatively, introduced by Horodecki et al. [3] as thesituation where “...the only correlation between the envi-ronments should be the common information about thesystem. In other words, conditioned by the informationabout the system, there should be no correlations betweenthe environments.”We adhered to this qualitative definitionall throughout our Letter and our proofs of the connectionbetween strong quantum Darwinism and spectrum broad-cast structure [2] and, therefore, our proofs in [2] containno error.The mistake in our Letter is in the formal expression(9) in [2] for the written statement of strong independence.As the authors of the Comment [1] indicate, the correctformal expression for Eq. (9) should be given by theconditional multipartite mutual information, rather thanthe conditional pairwise mutual information, for when thereare more than two environments. Therefore, Eq. (9) in [2]shall readIðE1;…; EFjSÞ ¼ 0;which is consistent with the statement we wrote in ourLetter right after Eq. (9): “Strong independence meansthat there are no correlations between the environmentsconditioned on the information about the system”.We also note that the purpose of our original Letter [2]was twofold: to introduce the concept of strong quantumDarwinism and to make the connection with spectrumbroadcast structure [3]; i.e., strong quantum Darwinismwith strong independence is equivalent to spectrum broad-cast structure. Since strong quantum Darwinism is not at allconditioned on strong independence, our proposed strongquantum Darwinism framework remains completelyuntouched by the Comment [1], both conceptually andmathematically. Its importance in establishing the mini-mum conditions for the objectivity of a state thereforeremains true.In conclusion, we are grateful to the authors of theComment [1] for catching the error in Eq. (9) in the originalLetter [2] and replacing it with the correct mathematicalexpression.We thank the authors for communicating with us aheadof the publication of their Comment. T. P. L. acknowledgessupport from the Engineering and Physical SciencesResearch Council. A. O.-C. acknowledges support fromthe Gordon and Betty Moore Foundation (GrantNo. GBMF8820).Thao P. Le 1,2 and Alexandra Olaya-Castro 11Department of Physics and AstronomyUniversity College LondonGower Street, London WC1E 6BT, United Kingdom2School of Mathematical SciencesUniversity of NottinghamUniversity Park Campus, Nottingham NG7 2RDUnited KingdomReceived 28 January 2021; accepted 29 March 2021;published 7 May 2021DOI: 10.1103/PhysRevLett.126.188902[1] A. Feller, B. Roussel, I. Frérot, and P. Degiovanni, precedingComment, Phys. Rev. Lett. 126, 188901 (2021).[2] T. P. Le and A. Olaya-Castro, Phys. Rev. Lett. 122, 010403(2019).[3] R. Horodecki, J. K. Korbicz, and P. Horodecki, Phys. Rev. A91, 032122 (2015).Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.PHYSICAL REVIEW LETTERS 126, 188902 (2021)0031-9007=21=126(18)=188902(1) 188902-1 Published by the American Physical Society

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Le and Olaya-Castro Reply:

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