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Optimizing encoding and decoding to match known channel noise increases secret key rates in two-way deterministic QKD for certain noise classes.

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2026-06-29 12:22 UTC pith:S5SDTDWB

load-bearing objection The paper shows adaptive encoding/decoding can raise key rates for some noise models in three two-way protocols but only when the exact noise is known in advance. the 2 major comments →

arxiv 2605.27902 v1 pith:S5SDTDWB submitted 2026-05-27 quant-ph

Noise adaptive two-way secure deterministic quantum key distribution

Abinash Kar , Ayan Patra , Aditi Sen De , Tamoghna Das This is my paper
classification quant-ph
keywords quantum key distributionnoise-adaptive protocolstwo-way QKDsecret key rateentropic uncertainty relationscollective attacksdense codingPauli channels
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces noise-adaptive QKD protocols in which the honest parties adjust their state preparation and measurement bases according to the noise model affecting the channel. It applies this idea to three two-way protocols: entanglement-based secure dense coding, the LM05 protocol, and a prepare-and-measure BB84 variant. Using entropic uncertainty relations, the authors compute secret key rates under collective attacks and compare adaptive versus fixed strategies. For independent but identical noise on the forward and backward channels, as well as for correlated and non-Markovian environments, they identify channel classes where the adaptive versions produce higher rates. For depolarizing and bit-flip Pauli channels the rates remain unchanged, and the optimal adaptive choices are generally non-unique and distinct from those maximizing dense-coding capacity without security constraints.

Core claim

For independent but identical noise acting on the forward and backward transmission channels, as well as for correlated and non-Markovian environments, noise-adaptive encoding and decoding yield enhanced secret key rates in the three considered two-way protocols, while the same adaptive strategies provide no benefit for depolarizing and bit-flip channels; the optimal sets are generally non-unique and differ substantially from the unitaries that maximize dense-coding capacity in the absence of security constraints.

What carries the argument

Noise-adaptive optimization of encoding states and decoding measurements chosen to maximize the secret key rate given the eavesdropper-induced noise model on the honest subsystems.

Load-bearing premise

The honest parties know or can accurately estimate the noise model induced by the eavesdropper so they can choose optimal encoding and decoding operations.

What would settle it

An experiment or calculation showing identical secret key rates for adaptive and non-adaptive versions of any of the three protocols on a channel from the identified classes under collective attack would falsify the rate-enhancement claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Adaptive schemes produce higher secret key rates than fixed strategies in secure dense coding for the identified noise classes.
  • The same rate improvement holds for the LM05 protocol and the two-way prepare-and-measure BB84 protocol under the same noise conditions.
  • The sets of optimal encoding-decoding operations are generally non-unique.
  • These optimal operations differ substantially from the unitaries that maximize dense-coding capacity without security constraints.
  • No rate improvement occurs when the channel is a depolarizing or bit-flip Pauli channel.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Realistic implementations would require reliable prior or real-time estimation of the noise parameters.
  • The distinction between capacity-maximizing and security-maximizing choices points to a potential trade-off when both goals are pursued simultaneously.
  • The framework could be tested on hardware by varying noise correlation strength while holding other parameters fixed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces noise-adaptive QKD protocols in which honest parties optimize encoding (state preparation) and decoding (measurement basis) operations according to the noise models induced by an eavesdropper on the honest subsystems. It analyzes three two-way protocols—entanglement-based secure dense coding (SDC), entanglement-free LM05, and prepare-and-measure BB84—deriving secret key rates via entropic uncertainty relations under collective attacks. For independent but identical noise on forward/backward channels as well as correlated and non-Markovian environments, it identifies channel classes where adaptive schemes yield higher rates than conventional fixed-strategy schemes; for certain Pauli channels (depolarizing, bit-flip) the adaptive and non-adaptive rates coincide. The optimal operations are shown to be generally non-unique and to differ from those maximizing dense-coding capacity without security constraints.

Significance. If the derivations hold, the work supplies a concrete framework for improving secret-key rates over realistic noisy channels by adapting encoding/decoding to the specific noise model, while also providing the useful negative result that adaptation confers no advantage on standard depolarizing and bit-flip channels. The explicit comparison with capacity-maximizing unitaries and the use of standard entropic uncertainty relations are strengths.

major comments (2)
  1. Abstract and §3 (or wherever the optimization is formalized): the claimed rate improvements rest on the premise that the honest parties possess exact prior knowledge of the eavesdropper-induced noise model (Kraus operators or correlation structure) in order to select the optimal encoding/decoding. The manuscript does not appear to quantify how this model is obtained from finite channel sampling or to analyze the degradation of the adaptive advantage under realistic estimation error; this is load-bearing for the central claim that adaptive schemes are superior for the identified channel classes.
  2. §4 (key-rate derivations): while entropic uncertainty relations are invoked, the manuscript should explicitly state the range of channel parameters (e.g., noise strength, correlation strength) over which the adaptive advantage is demonstrated, together with any numerical verification or plots that confirm the analytic bounds.
minor comments (2)
  1. Notation for the adaptive versus non-adaptive key-rate expressions should be made uniform across the three protocols to facilitate direct comparison.
  2. The statement that optimal sets are 'generally non-unique' would benefit from an explicit example (e.g., two distinct unitaries yielding the same rate for a given channel) in the main text or appendix.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each major comment below in a point-by-point manner and indicate the changes we will incorporate in the revised version.

read point-by-point responses

  1. Referee: Abstract and §3 (or wherever the optimization is formalized): the claimed rate improvements rest on the premise that the honest parties possess exact prior knowledge of the eavesdropper-induced noise model (Kraus operators or correlation structure) in order to select the optimal encoding/decoding. The manuscript does not appear to quantify how this model is obtained from finite channel sampling or to analyze the degradation of the adaptive advantage under realistic estimation error; this is load-bearing for the central claim that adaptive schemes are superior for the identified channel classes.

    Authors: We agree that the analysis assumes the honest parties have exact knowledge of the noise model, which is a standard idealization in theoretical QKD derivations using entropic uncertainty relations. The manuscript does not address practical estimation from finite samples or the effect of estimation errors on the reported advantage. This is a genuine limitation for claims of practical improvement. In the revised manuscript we will add an explicit statement of this assumption in the introduction and a brief discussion in the conclusions noting that robust estimation methods would be needed in practice and that the adaptive gain may degrade under imperfect knowledge. We will not perform a full finite-sample analysis, as that lies beyond the current theoretical scope focused on known noise classes. revision: partial

  2. Referee: §4 (key-rate derivations): while entropic uncertainty relations are invoked, the manuscript should explicitly state the range of channel parameters (e.g., noise strength, correlation strength) over which the adaptive advantage is demonstrated, together with any numerical verification or plots that confirm the analytic bounds.

    Authors: The derivations identify specific channel classes (certain correlated and non-Markovian environments) where adaptation improves the key rate and show coincidence with non-adaptive rates for Pauli channels such as depolarizing and bit-flip. We concur that stating explicit parameter ranges and providing numerical confirmation would strengthen the presentation. In the revised §4 we will add a subsection specifying the ranges of noise strength and correlation parameters for which the advantage holds, together with plots comparing adaptive and non-adaptive rates for representative values to verify the analytic expressions. revision: yes

Circularity Check

0 steps flagged

No circularity; key rates derived from standard entropic uncertainty relations applied to explicit noise models

full rationale

The paper states it derives secret key rates for adaptive vs non-adaptive protocols via entropic uncertainty relations under collective attacks for specified channel classes (independent identical noise, correlated/non-Markovian, Pauli channels). No equations or steps reduce by construction to fitted parameters renamed as predictions, self-citations that bear the central load, or ansatze imported from prior author work. The premise of known noise models is an explicit modeling assumption, not a derived result. The finding of enhanced rates for certain classes and none for others follows directly from applying the cited standard relations, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract provides no explicit free parameters or invented entities; the central claim rests on the standard mathematical tool of entropic uncertainty relations and the domain assumption that noise models are known to the honest parties.

axioms (2)
  • standard math Entropic uncertainty relations can be applied to derive secret key rates under collective attacks for the three protocols
    Invoked to obtain the key rates for both adaptive and non-adaptive cases
  • domain assumption Honest parties can characterize the noise model induced by the eavesdropper on the honest subsystems
    Required for the optimization step that defines the adaptive protocols

pith-pipeline@v0.9.1-grok · 5771 in / 1265 out tokens · 26021 ms · 2026-06-29T12:22:28.419559+00:00 · methodology

0 comments
read the original abstract

We introduce noise-adaptive quantum key distribution (QKD) protocols, in which the honest parties optimize the encoding (state preparation) and decoding (measurement basis) operations according to the noise models affecting the honest subsystems induced by an eavesdropper. This extends conventional QKD schemes that employ fixed encoding and decoding strategies independent of the noise characteristics of the communication channel. We investigate three representative protocols: entanglement-based secure dense coding (SDC), the entanglement-free Lucamarini and Mancini (LM05), and a two-way prepare-and-measure Bennett Brassard (BB84) protocols. Using entropic uncertainty relations, we derive the corresponding secret key rates for both adaptive and conventional non-adaptive scenarios under collective attacks. For independent but identical noise acting on the forward and backward transmission channels, as well as for correlated and non-Markovian environments, we identify classes of channels for which adaptive schemes yield enhanced secret key rates for the considered protocols. In contrast, we also determine Pauli channels, including depolarizing and bit flip channels, for which adaptive strategies provide no benefit. We further show that these optimal sets are generally non-unique and can differ substantially from the unitaries that maximize dense-coding capacity in the absence of security constraints. Our results establish noise-adaptive encoding and decoding as a powerful framework for improving secure communication over realistic noisy quantum channels.

discussion (0)

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Reference graph

Works this paper leans on

78 extracted references · 3 canonical work pages · 1 internal anchor

  1. [1]

    The corre- sponding probabilities are constrained by the complete positive and trace-preserving (CPTP) condition [67]

    General Pauli channel Having established positive results for specific Pauli noise models across all three protocols, we proceed to analyze the most general case, where all three Pauli op- erators, and the qubit identity, act upon the input state with random probabilitiesp i ∈[0, 1][66]. The corre- sponding probabilities are constrained by the complete po...

  2. [2]

    INFOSYS Scholarship for senior students

    key rates. This is due to the fact thatC noisy quan- tifies the total classical information that can be trans- mitted using the shared resource and optimal encod- ing; it does not account for information leakage to the environment due to the noise in the transmission chan- nel. This information can be accessed by any malicious eavesdropper, and she can, i...

    2024

  3. [3]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information:10th Anniversary Edition(Cam- bridge University Press,2010)

    2010

  4. [4]

    Deutsch, Proceedings of the Royal Society of Lon- don

    D. Deutsch, Proceedings of the Royal Society of Lon- don. Series A, Mathematical and Physical Sciences400, 97(1985)

    1985

  5. [5]

    Shor, inProceedings35th Annual Symposium on Founda- tions of Computer Science(1994) pp.124–134

    P . Shor, inProceedings35th Annual Symposium on Founda- tions of Computer Science(1994) pp.124–134

    1994

  6. [6]

    L. K. Grover, inProceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing (STOC)(ACM,

  7. [7]

    Preskill, Quantum2,79(2018)

    J. Preskill, Quantum2,79(2018)

    2018

  8. [8]

    R. L. Rivest, A. Shamir, and L. Adleman, Communica- tions of the ACM21,120(1978)

    1978

  9. [9]

    C. H. Bennett and G. Brassard, inProceedings of the IEEE International Conference on Computers, Systems and Signal Processing(Bangalore, India,1984) pp.175–179

    1984

  10. [10]

    C. H. Bennett and G. Brassard, Theoretical Computer Sci- ence560,7(2014), theoretical Aspects of Quantum Cryp- tography – celebrating30years of BB84

    2014

  11. [11]

    A. K. Ekert, Phys. Rev. Lett.67,661(1991)

    1991

  12. [12]

    Mayers and A

    D. Mayers and A. Yao, inProceedings39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)(1998) pp.503–509

    1998

  13. [13]

    Science283(5410), 2050–2056 (1999) https://doi.org/ 10.1126/science.283.5410.2050

    H.-K. Lo and H. F. Chau, Science283,2050(1999), https://www.science.org/doi/pdf/10.1126/science.283.5410.2050

    work page doi:10.1126/science.283.5410.2050 2050

  14. [14]

    P . W. Shor and J. Preskill, Phys. Rev. Lett.85,441(2000)

    2000

  15. [15]

    Gisin, G

    N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys.74,145(2002). 18

    2002

  16. [16]

    Scarani, H

    V . Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, Rev. Mod. Phys. 81,1301(2009)

    2009

  17. [17]

    J. S. Bell, Physics Physique Fizika1,195(1964)

    1964

  18. [18]

    J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett.23,880(1969)

    1969

  19. [19]

    Bell nonlocality

    N. Brunner, D. Cavalcanti, S. Pironio, V . Scarani, and S. Wehner, Rev. Mod. Phys.86,839(2014), quant- ph/1303.2849

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  20. [20]

    Horodecki, M

    R. Horodecki, M. Horodecki, K. Horodecki, and P . Horodecki, Rev. Mod. Phys.81,865(2009)

    2009

  21. [21]

    Csiszar and J

    I. Csiszar and J. Korner, IEEE Transactions on Informa- tion Theory24,339(1978)

    1978

  22. [22]

    U. M. Maurer, IEEE Transactions on Information Theory 39,733(1993)

    1993

  23. [23]

    A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V . Scarani, Phys. Rev. Lett.98,230501(2007)

    2007

  24. [24]

    Masanes, S

    L. Masanes, S. Pironio, and A. Acín, Nat. Commun.2 (2011),10.1038/ncomms1244

    work page doi:10.1038/ncomms1244 2011

  25. [25]

    C. H. Bennett, Physical Review Letters68,3121(1992)

    1992

  26. [26]

    Bruß, Phys

    D. Bruß, Phys. Rev. Lett.81,3018(1998)

    1998

  27. [27]

    Bechmann-Pasquinucci and N

    H. Bechmann-Pasquinucci and N. Gisin, Phys. Rev. A59, 4238(1999)

    1999

  28. [28]

    Stucki, N

    D. Stucki, N. Gisin, O. Guinnard, G. Ribordy, and H. Zbinden, New Journal of Physics4,41(2002)

    2002

  29. [29]

    Liao, W.-Q

    S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P . Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M. Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, Nat...

    2017

  30. [30]

    Liao, W.-Q

    S.-K. Liao, W.-Q. Cai, J. Handsteiner, B. Liu, J. Yin, L. Zhang, D. Rauch, M. Fink, J.-G. Ren, W.-Y. Liu, Y. Li, Q. Shen, Y. Cao, F.-Z. Li, J.-F. Wang, Y.-M. Huang, L. Deng, T. Xi, L. Ma, T. Hu, L. Li, N.-L. Liu, F. Koidl, P . Wang, Y.-A. Chen, X.-B. Wang, M. Steindor- fer, G. Kirchner, C.-Y. Lu, R. Shu, R. Ursin, T. Scheidl, C.-Z. Peng, J.-Y. Wang, A. Ze...

    2018

  31. [31]

    X.-T. Fang, P . Zeng, H. Liu, M. Zou, W. Wu, Y.-L. Tang, Y.- J. Sheng, Y. Xiang, W. Zhang, H. Li, Z. Wang, L. You, M.-J. Li, H. Chen, Y.-A. Chen, Q. Zhang, C.-Z. Peng, X. Ma, T.- Y. Chen, and J.-W. Pan, Nature Photonics14,422(2020)

    2020

  32. [32]

    J.-P . Chen, C. Zhang, Y. Liu, C. Jiang, W. Zhang, X.-L. Hu, J.-Y. Guan, Z.-W. Yu, H. Xu, J. Lin, M.-J. Li, H. Chen, H. Li, L. You, Z. Wang, X.-B. Wang, Q. Zhang, and J.-W. Pan, Phys. Rev. Lett.124,070501(2020)

    2020

  33. [33]

    Hwang, Phys

    W.-Y. Hwang, Phys. Rev. Lett.91,057901(2003)

    2003

  34. [34]

    Schmitt-Manderbach, H

    T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Wein- furter, Phys. Rev. Lett.98,010504(2007)

    2007

  35. [35]

    C.-Z. Peng, J. Zhang, D. Yang, W.-B. Gao, H.-X. Ma, H. Yin, H.-P . Zeng, T. Yang, X.-B. Wang, and J.-W. Pan, Phys. Rev. Lett.98,010505(2007)

    2007

  36. [36]

    H.-K. Lo, M. Curty, and B. Qi, Phys. Rev. Lett.108, 130503(2012)

    2012

  37. [37]

    Yin, T.-Y

    H.-L. Yin, T.-Y. Chen, Z.-W. Yu, H. Liu, L.-X. You, Y.- H. Zhou, S.-J. Chen, Y. Mao, M.-Q. Huang, W.-J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X.-B. Wang, and J.-W. Pan, Phys. Rev. Lett. 117,190501(2016)

    2016

  38. [38]

    Pirandola, C

    S. Pirandola, C. Ottaviani, G. Spedalieri, C. Weedbrook, S. L. Braunstein, S. Lloyd, T. Gehring, C. S. Jacobsen, and U. L. Andersen, Nature Photonics9,397(2015)

    2015

  39. [39]

    Pironio, L

    S. Pironio, L. Masanes, A. Leverrier, and A. Acín, Phys. Rev. X3,031007(2013)

    2013

  40. [40]

    Vazirani and T

    U. Vazirani and T. Vidick, Phys. Rev. Lett.113,140501 (2014)

    2014

  41. [41]

    Barrett, R

    J. Barrett, R. Colbeck, and A. Kent, Phys. Rev. Lett.110, 010503(2013)

    2013

  42. [42]

    A. Acín, N. Gisin, and L. Masanes, Phys. Rev. Lett.97, 120405(2006)

    2006

  43. [43]

    Masanes, R

    L. Masanes, R. Renner, M. Christandl, A. Winter, and J. Barrett, IEEE Transactions on Information Theory60, 4973(2014)

    2014

  44. [44]

    Winczewski, T

    M. Winczewski, T. Das, and K. Horodecki, Phys. Rev. A 106,052612(2022)

    2022

  45. [46]

    Lucamarini and S

    M. Lucamarini and S. Mancini, Phys. Rev. Lett.94,140501 (2005)

    2005

  46. [47]

    Muhuri, A

    A. Muhuri, A. Patra, R. Gupta, T. Das, and A. Sen(De), Physics Letters A583,131550(2026)

    2026

  47. [48]

    N. J. Beaudry, M. Lucamarini, S. Mancini, and R. Renner, Phys. Rev. A88,062302(2013)

    2013

  48. [50]

    Lu, C.-H

    H. Lu, C.-H. F. Fung, X. Ma, and Q.-y. Cai, Phys. Rev. A 84,042344(2011)

    2011

  49. [51]

    Shadman, H

    Z. Shadman, H. Kampermann, C. Macchiavello, and D. Bruß, New Journal of Physics12,073042(2010)

    2010

  50. [53]

    Shadman, H

    Z. Shadman, H. Kampermann, D. Bruß, and C. Macchi- avello, Phys. Rev. A85,052306(2012)

    2012

  51. [54]

    T. Das, R. Prabhu, A. Sen(De), and U. Sen, Phys. Rev. A 90,022319(2014)

    2014

  52. [55]

    T. Das, R. Prabhu, A. Sen(De), and U. Sen, Phys. Rev. A 92,052330(2015)

    2015

  53. [56]

    Berta, M

    M. Berta, M. Christandl, R. Colbeck, J. M. Renes, and R. Renner, Nature Physics6,659(2010)

    2010

  54. [57]

    Devetak and A

    I. Devetak and A. Winter, Proceedings of the Royal Soci- ety A: Mathematical, Physical and Engineering Sciences 461,207(2005)

    2005

  55. [58]

    ˙Zukowski, A

    M. ˙Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ek- ert, Phys. Rev. Lett.71,4287(1993)

    1993

  56. [59]

    S. Bose, V . Vedral, and P . L. Knight, Phys. Rev. A57,822 (1998)

    1998

  57. [60]

    C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett.70,1895(1993)

    1993

  58. [61]

    S. Bose, M. B. Plenio, and V . Vedral, Journal of Modern Optics47,291(2000)

    2000

  59. [62]

    Ziman and V

    M. Ziman and V . Bužek, Physical Review A67,042321 (2003)

    2003

  60. [63]

    D. Bruß, M. Lewenstein, A. Sen(De), U. Sen, G. M. D’Ariano, and C. Macchiavello, International Journal of Quantum Information4,415(2006)

    2006

  61. [64]

    Hiroshima, Journal of Physics A: Mathematical and General34,6907(2001)

    T. Hiroshima, Journal of Physics A: Mathematical and General34,6907(2001)

    2001

  62. [65]

    Patra, R

    A. Patra, R. Gupta, T. Das, and A. Sen(De), Phys. Rev. A 110,032419(2024)

    2024

  63. [66]

    Watrous,The Theory of Quantum Information(Cam- bridge University Press,2018)

    J. Watrous,The Theory of Quantum Information(Cam- bridge University Press,2018)

    2018

  64. [67]

    Gottesman, Phys

    D. Gottesman, Phys. Rev. A57,127(1998). 19

    1998

  65. [68]

    S. Chen, S. Zhou, A. Seif, and L. Jiang, Phys. Rev. A105, 032435(2022)

    2022

  66. [69]

    Shadman, H

    Z. Shadman, H. Kampermann, D. Bruß, and C. Macchi- avello, Phys. Rev. A84,042309(2011)

    2011

  67. [70]

    Macchiavello and G

    C. Macchiavello and G. M. Palma, Phys. Rev. A65,050301 (2002)

    2002

  68. [71]

    Addis, G

    C. Addis, G. b. u. Karpat, C. Macchiavello, and S. Man- iscalco, Phys. Rev. A94,032121(2016)

    2016

  69. [72]

    Breuer and F

    H.-P . Breuer and F. Petruccione,The Theory of Open Quan- tum Systems(Oxford University Press,2007)

    2007

  70. [73]

    Muhuri, R

    A. Muhuri, R. Gupta, S. Ghosh, and A. Sen(De), Phys. Rev. A109,032616(2024)

    2024

  71. [74]

    Shrikant, R

    U. Shrikant, R. Srikanth, and S. Banerjee, Phys. Rev. A 98,032328(2018)

    2018

  72. [75]

    Shadman, H

    Z. Shadman, H. Kampermann, C. Macchiavello, and D. Bruß, Quantum Measurements and Quantum Metrol- ogy1,21(2013)

    2013

  73. [76]

    Biswas, G

    D. Biswas, G. M. Vaidya, and P . Mandayam, Phys. Rev. Res.6,043034(2024)

    2024

  74. [77]

    Jayashankar and P

    A. Jayashankar and P . Mandayam, Journal of the Indian Institute of Science103,497(2023)

    2023

  75. [78]

    Boström and T

    K. Boström and T. Felbinger, Phys. Rev. Lett.89,187902 (2002)

    2002

  76. [79]

    C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett.69,2881 (1992)

    1992

  77. [80]

    P . J. Coles, M. Berta, M. Tomamichel, and S. Wehner, Rev. Mod. Phys.89,015002(2017). Appendix A: Generalized purification scheme for encoding operation In this section, we are going to prove Eq. (1). Suppose Bob receives the subsystemBof the shared stateρ AB coming from Alice via channelΛ f for the encoding process. Assume that the stateρ AX is the same ...

    2017

  78. [81]

    An equivalent formulation of the BB84protocol can be given in an entanglement-based picture, which is particularly useful for security analysis

    Computing the key rate One can easily check that Alice’s preparation of the signal qubit can be equivalently considered as a measure- ment procedure on a shared maximally entangled Bell state. An equivalent formulation of the BB84protocol can be given in an entanglement-based picture, which is particularly useful for security analysis. This version is clo...