pith. sign in

arxiv: 2605.23547 · v1 · pith:TAKHGZRCnew · submitted 2026-05-22 · 🪐 quant-ph

Non-Maximally Entangled States for Quantum Key Distribution in Underwater Channels: BBM92 Protocol via Kraus Operators

Nour Rizk , Ang\'elique Dr\'emeau , Arnaud Coatanhay This is my paper

Pith reviewed 2026-05-25 04:34 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum key distributionBBM92 protocolunderwater channelsnon-maximally entangled statesKraus operatorsquantum bit error ratesecret key rateamplitude damping
0
0 comments X

The pith

Non-maximally entangled states enable closed-form QBER and SKR expressions for BBM92 in underwater channels modeled by Kraus operators.

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

The paper examines the BBM92 entanglement-based QKD protocol when photon pairs are prepared in non-maximally entangled states and sent through underwater optical channels. It models the propagation medium as a quantum channel that combines amplitude-damping and depolarizing noise, expressed through Kraus operators. Closed-form analytical formulas for the quantum bit error rate and secret key rate are obtained in terms of the entanglement parameter and the channel degradation factors. These expressions are checked against Monte Carlo simulations and then evaluated for concrete cases of clear ocean, coastal, and turbid water under different atmospheric conditions.

Core claim

By preparing photon pairs in non-maximally entangled states and describing the underwater channel with a Kraus-operator map that includes both amplitude damping and depolarization, the BBM92 protocol yields closed-form analytical expressions for QBER and SKR that depend on the degree of entanglement and the channel parameters; the expressions are validated by Monte Carlo simulation and applied to realistic underwater environments with varying water types and atmospheric conditions.

What carries the argument

Kraus operator representation of the combined amplitude-damping and depolarizing channel acting on non-maximally entangled photon pairs within the BBM92 measurement protocol.

If this is right

  • QBER and SKR become explicit functions of entanglement degree and channel loss parameters, allowing direct optimization.
  • Performance predictions can be made for clear ocean, coastal, and turbid water under different atmospheric conditions.
  • Monte Carlo simulations serve as numerical confirmation of the closed-form results.
  • Secret key rates are obtainable for any chosen water type once the damping and depolarization parameters are known.

Where Pith is reading between the lines

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

  • The same Kraus-channel treatment could be applied to other lossy environments such as foggy atmosphere or turbid fiber links.
  • Tuning the entanglement degree away from maximal may reduce error rates in channels dominated by one noise type over the other.
  • The analytic expressions supply a benchmark against which future experimental underwater QKD data can be compared.

Load-bearing premise

The actual underwater optical propagation is fully captured by the specific combination of amplitude-damping and depolarizing Kraus maps without extra effects such as turbulence or memory.

What would settle it

An experimental measurement of QBER in a real underwater BBM92 link using a known entanglement degree that deviates from the derived analytical formula by more than simulation error would show the channel model is incomplete.

read the original abstract

Underwater optical channels pose significant challenges to the security and reliability of quantum communication systems due to absorption and scattering. In this paper, we investigate the BBM92 entanglement-based quantum key distribution (QKD) protocol under realistic underwater channel conditions. Photon pairs are prepared in non-maximally entangled states, and the underwater propagation medium is modeled as a quantum channel incorporating both amplitude-damping and depolarizing effects, described within the Kraus operator formalism. The protocol performance is evaluated in terms of quantum bit error rate (QBER) and secret key rate (SKR), analyzed as functions of the entanglement degree and channel degradation parameters. Closed-form analytical expressions for the QBER and SKR are derived for the proposed channel model and validated through Monte Carlo simulations. The proposed framework is then applied to various realistic underwater scenarios, considering different water types, namely clear ocean, coastal, and turbid water, as well as varying atmospheric conditions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to derive closed-form analytical expressions for the QBER and SKR in the BBM92 entanglement-based QKD protocol using non-maximally entangled states. The underwater channel is modeled via combined amplitude-damping and depolarizing Kraus operators. The expressions are validated by Monte Carlo simulations and then applied to evaluate performance across clear ocean, coastal, and turbid water types under varying atmospheric conditions, with analysis as functions of entanglement degree and channel degradation parameters.

Significance. If the closed-form derivations hold and the Kraus model adequately represents the dominant physics, the work supplies analytical tools for optimizing non-maximal entanglement in underwater QKD, which is relevant for practical secure links in marine environments. The explicit Kraus-operator composition and Monte Carlo validation are strengths that enable reproducible checks of the QBER/SKR formulas.

major comments (2)
  1. [Abstract and channel-model description] Abstract and channel-model description: The central claim applies the derived QBER and SKR expressions to 'realistic underwater scenarios' (clear ocean, coastal, turbid water). However, the model is restricted to amplitude-damping plus depolarizing Kraus maps and omits turbulence-induced phase noise and non-Markovian scattering dynamics that are known to arise in underwater optical propagation. Because the applicability statements rest on this model, the omission is load-bearing and requires either explicit justification that the neglected effects are negligible in the considered regimes or an extended channel description.
  2. [Derivation and validation sections] Derivation and validation sections: The abstract states that closed-form QBER and SKR expressions were obtained by composing the Kraus operators on the non-maximally entangled state and checked by Monte Carlo. Without the explicit intermediate steps (including how the BBM92 measurement projectors enter the error-rate calculation) or details on simulation parameters, trial counts, and error-bar treatment, the support for the analytic formulas cannot be verified at the level required for the central claim.
minor comments (2)
  1. [Introduction] Notation for the entanglement degree parameter should be defined once in the introduction and used consistently in all subsequent equations and figures.
  2. [Results figures] Figure captions should list the exact numerical values of the channel degradation parameters and entanglement degree used in each plot to improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The comments highlight important aspects of model applicability and reproducibility that we address below. We have revised the manuscript to incorporate additional justifications and expanded technical details.

read point-by-point responses

  1. Referee: [Abstract and channel-model description] The central claim applies the derived QBER and SKR expressions to 'realistic underwater scenarios' (clear ocean, coastal, turbid water). However, the model is restricted to amplitude-damping plus depolarizing Kraus maps and omits turbulence-induced phase noise and non-Markovian scattering dynamics that are known to arise in underwater optical propagation. Because the applicability statements rest on this model, the omission is load-bearing and requires either explicit justification that the neglected effects are negligible in the considered regimes or an extended channel description.

    Authors: We agree that the channel model is limited to amplitude-damping and depolarizing Kraus operators, which represent the dominant absorption and scattering processes for the short-range underwater links analyzed. Turbulence-induced phase fluctuations and non-Markovian effects are acknowledged in the broader literature but are secondary for the distances (tens of meters) and water types considered, where attenuation lengths dominate. In the revised manuscript we will add an explicit paragraph in Section II justifying this approximation with references to prior underwater QKD studies, stating that these effects are neglected to first order under the stated conditions. We do not claim the model captures every physical mechanism; the revision will clarify the scope. revision: partial

  2. Referee: [Derivation and validation sections] The abstract states that closed-form QBER and SKR expressions were obtained by composing the Kraus operators on the non-maximally entangled state and checked by Monte Carlo. Without the explicit intermediate steps (including how the BBM92 measurement projectors enter the error-rate calculation) or details on simulation parameters, trial counts, and error-bar treatment, the support for the analytic formulas cannot be verified at the level required for the central claim.

    Authors: We accept that the original manuscript omitted the full intermediate algebra and Monte Carlo specifications. The revised version will include a dedicated subsection detailing the Kraus-operator composition applied to the non-maximally entangled state, the explicit insertion of the BBM92 Bell-state projectors into the error-rate calculation, and the resulting closed-form QBER and SKR expressions. We will also add a simulation-parameters paragraph specifying 10^6 trials per data point, the random-number generation method, and the standard-error treatment used to generate the plotted error bars. These additions will enable direct verification of the analytic results against the numerical checks. revision: yes

Circularity Check

0 steps flagged

No significant circularity; QBER/SKR expressions derived from independent Kraus channel model

full rationale

The paper derives closed-form QBER and SKR by composing amplitude-damping and depolarizing Kraus operators applied to non-maximally entangled states in the BBM92 protocol. These operators and the entanglement parameter are treated as independent inputs; the resulting analytic expressions are then validated by separate Monte Carlo simulations rather than fitted to the target data. No self-citation chain, self-definitional loop, or renaming of known results is indicated in the provided text. The central claims therefore remain self-contained against external benchmarks and do not reduce to their own inputs by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; therefore the ledger is necessarily incomplete. The model treats entanglement degree and channel degradation parameters as free inputs rather than fitted constants. No new entities are introduced.

free parameters (2)
  • entanglement degree
    Treated as a continuous variable that QBER and SKR are expressed in terms of.
  • channel degradation parameters
    Amplitude-damping and depolarizing strengths are independent inputs to the Kraus maps.
axioms (1)
  • standard math Standard quantum mechanics and the Kraus-operator representation of completely positive trace-preserving maps
    Used to define the underwater quantum channel.

pith-pipeline@v0.9.0 · 5701 in / 1417 out tokens · 28608 ms · 2026-05-25T04:34:29.711978+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

59 extracted references · 59 canonical work pages

  1. [1]

    Security of quantum key distribution with entangled photons against individual attacks , author =. Phys. Rev. A , volume =. 2002 , month =. doi:10.1103/PhysRevA.65.052310 , note =

    work page doi:10.1103/physreva.65.052310 2002

  2. [2]

    , booktitle=

    Elkouss, David and Leverrier, Anthony and Alleaume, Romain and Boutros, Joseph J. , booktitle=. Efficient reconciliation protocol for discrete-variable quantum key distribution , year=

    work page

  3. [3]

    C. H. Bennett and G. Brassard , title =. International conference on computers, systems and signal processing , year =

    work page

  4. [4]

    W. K. Wootters and W. H. Zurek , title =. Nature , year =

    work page

  5. [5]

    Physics reports , volume=

    Heisenberg's uncertainty principle , author=. Physics reports , volume=. 2007 , publisher=

    work page 2007

  6. [6]

    Kumar and S

    A. Kumar and S. Garhwal , title =. Archives of Computational Methods in Engineering , year =

    work page

  7. [7]

    Physical review letters , volume=

    Quantum Cryptography Protocols Robust against Photon Number Splitting Attacks for Weak Laser Pulse Implementations , author=. Physical review letters , volume=. 2004 , publisher=

    work page 2004

  8. [8]

    Physical review letters , volume=

    Quantum cryptography without Bell's theorem , author=. Physical review letters , volume=. 1992 , publisher=

    work page 1992

  9. [9]

    Physical review letters , volume=

    Beating the photon-number-splitting attack in practical quantum cryptography , author=. Physical review letters , volume=. 2005 , publisher=

    work page 2005

  10. [10]

    Quantum Engineering , volume=

    A primer on underwater quantum key distribution , author=. Quantum Engineering , volume=. 2023 , publisher=

    work page 2023

  11. [11]

    A. H. F. Raouf , title =. 2021 , note =

    work page 2021

  12. [12]

    P. W. Shor and J. Preskill , journal=. Simple proof of security of the. 2000 , publisher=

    work page 2000

  13. [13]

    Physical Review A , volume=

    Security of two quantum cryptography protocols using the same four qubit states , author=. Physical Review A , volume=. 2005 , publisher=

    work page 2005

  14. [14]

    Physical Review Letters , volume=

    Device-independent security of quantum cryptography against collective attacks , author=. Physical Review Letters , volume=. 2007 , publisher=

    work page 2007

  15. [15]

    Physical Review A , volume=

    Quantum key distribution with entangled photon sources , author=. Physical Review A , volume=. 2007 , publisher=

    work page 2007

  16. [16]

    , journal=

    Biswas, Ayan and al. , journal=. Use of Non-Maximal entangled state for free space. 2023 , note =

    work page 2023

  17. [17]

    Mishra and A

    S. Mishra and A. Biswas and S. Patil and P. Chandravanshi and V. Mongia and T. Sharma and A. Rani and S. Prabhakar and S. Ramachandran and R.P. Singh , journal=. 2022 , publisher=

    work page 2022

  18. [18]

    Meena and S

    Muskan and R. Meena and S. Banerjee , journal=. Performance analysis of satellite-based. 2023 , note =

    work page 2023

  19. [19]

    Zurek , journal=

    W.H. Zurek , journal=. Probabilities from entanglement,. 2005 , publisher=

    work page 2005

  20. [20]

    Physical review letters , volume=

    Nonmaximally entangled states: production, characterization, and utilization , author=. Physical review letters , volume=. 1999 , publisher=

    work page 1999

  21. [21]

    1994 , publisher=

    Light and water: radiative transfer in natural waters , author=. 1994 , publisher=

    work page 1994

  22. [22]

    Rizk and A

    N. Rizk and A. Dr. Performances des protocoles. Proc. of GRETSI, Strasbourg, France , year =

    work page

  23. [23]

    Optics & Laser Technology , volume=

    Underwater turbulence, its effects on optical wireless communication and imaging: A review , author=. Optics & Laser Technology , volume=. 2022 , publisher=

    work page 2022

  24. [24]

    5th IEEE International Conference on Advanced Computing & Communication Technologies [ICACCT-2011] , pages=

    Quantum cryptography & its comparison with classical cryptography: A review , author=. 5th IEEE International Conference on Advanced Computing & Communication Technologies [ICACCT-2011] , pages=. 2011 , note =

    work page 2011

  25. [25]

    Cybernetics and Systems , volume=

    Review of security methods based on classical cryptography and quantum cryptography , author=. Cybernetics and Systems , volume=. 2025 , publisher=

    work page 2025

  26. [26]

    Journal of cryptology , volume=

    Experimental quantum cryptography , author=. Journal of cryptology , volume=. 1992 , publisher=

    work page 1992

  27. [27]

    Physical review letters , volume=

    Practical free-space quantum key distribution over 1 km , author=. Physical review letters , volume=. 1998 , publisher=

    work page 1998

  28. [28]

    Takesue and E

    H. Takesue and E. Diamanti and C. Langrock and M. M. Fejer and Y. Yamamoto , journal=. 10-. 2006 , publisher=

    work page 2006

  29. [29]

    Takesue and S.W

    H. Takesue and S.W. Nam and Q. Zhang and R.H. Hadfield and T. Honjo and K. Tamaki and Y. Yamamoto , journal=. Quantum key distribution over a 40-. 2007 , publisher=

    work page 2007

  30. [30]

    Wang and W

    S. Wang and W. Chen and J. F. Guo and Z. Q. Yin and H. W. Li and Z. Zhou and G. C. Guo and Z. F. Han , journal=. 2. 2012 , publisher=

    work page 2012

  31. [31]

    Quantum Information Processing , volume=

    Controlled bidirectional remote state preparation in noisy environment: a generalized view , author=. Quantum Information Processing , volume=. 2015 , publisher=

    work page 2015

  32. [32]

    IET Quantum Communication , volume=

    Advances in space quantum communications , author=. IET Quantum Communication , volume=. 2021 , publisher=

    work page 2021

  33. [33]

    Ecker and B

    S. Ecker and B. Liu and J. Handsteiner and M. Fink and D. Rauch and F. Steinlechner and T. Scheidl and A. Zeilinger and R. Ursin , journal=. Strategies for achieving high key rates in satellite-based. 2021 , publisher=

    work page 2021

  34. [34]

    New Journal of Physics , volume=

    Analysis of satellite-to-ground quantum key distribution with adaptive optics , author=. New Journal of Physics , volume=. 2024 , publisher=

    work page 2024

  35. [35]

    Telecommunication Systems , volume=

    Underwater communication technologies: A review , author=. Telecommunication Systems , volume=. 2025 , publisher=

    work page 2025

  36. [36]

    IEEE access , volume=

    Underwater optical wireless communication , author=. IEEE access , volume=. 2016 , publisher=

    work page 2016

  37. [37]

    2012 , publisher=

    Underwater communications , author=. 2012 , publisher=

    work page 2012

  38. [38]

    Optics Express , volume=

    Towards quantum communications in free-space seawater , author=. Optics Express , volume=. 2017 , publisher=

    work page 2017

  39. [39]

    Journal of the Optical Society of America A , volume=

    Performance of underwater quantum key distribution with polarization encoding , author=. Journal of the Optical Society of America A , volume=. 2019 , publisher=

    work page 2019

  40. [40]

    IEEE Journal of Oceanic Engineering , year=

    Impact of Natural Turbulent Waters on Quantum Key Distribution: Temperature and Salinity Considerations , author=. IEEE Journal of Oceanic Engineering , year=

    work page

  41. [41]

    Physical Review A , volume=

    Performance of two quantum-key-distribution protocols , author=. Physical Review A , volume=. 2006 , publisher=

    work page 2006

  42. [42]

    IEEE Transactions on Communications , volume=

    Performance characterization of underwater visible light communication , author=. IEEE Transactions on Communications , volume=. 2018 , publisher=

    work page 2018

  43. [43]

    1991 , note =

    Fundamentals of Photonics , author=. 1991 , note =

    work page 1991

  44. [44]

    Rogers and J.C

    D.J. Rogers and J.C. Bienfang and A. Mink and B.J. Hershman and A. Nakassis and X. Tang and L. Ma and D.H. Su and C.J. Williams and C.W. Clark , booktitle=. Free-space quantum cryptography in the. 2006 , organization=

    work page 2006

  45. [45]

    Physical Review A , volume=

    Practical decoy state for quantum key distribution , author=. Physical Review A , volume=. 2005 , publisher=

    work page 2005

  46. [46]

    Ali and S

    S. Ali and S. Mohammed and M.S.H. Chowdhury and A.A. Hasan , journal=. Practical. 2012 , publisher=

    work page 2012

  47. [47]

    Fung and K

    C.H.F. Fung and K. Tamaki and H.K. Lo , journal=. On the performance of two protocols:. 2005 , note =

    work page 2005

  48. [48]

    Physics Letters A , volume=

    Full thermalization of a photonic qubit , author=. Physics Letters A , volume=. 2020 , publisher=

    work page 2020

  49. [49]

    2013 , publisher=

    Quantum information theory , author=. 2013 , publisher=

    work page 2013

  50. [50]

    2007 , publisher=

    Light Absorption in Sea Water , author=. 2007 , publisher=

    work page 2007

  51. [51]

    2013 , publisher=

    Quantum systems, channels, information: a mathematical introduction , author=. 2013 , publisher=

    work page 2013

  52. [52]

    Journal of Computer Engineering , volume=

    Quantum key distribution protocols: a review , author=. Journal of Computer Engineering , volume=. 2014 , note =

    work page 2014

  53. [53]

    Chalupnik and B

    M. Chalupnik and B. Doolittle and S. Seshadri and E.G. Brown and K. Kenemer and D. Winton and D. Sanchez-Rosales and M. Skrzypczyk and C. Alexander and E. Ostby and M. Cubeddu , journal=. Realistic quantum network simulation for experimental. 2025 , note =

    work page 2025

  54. [54]

    Kravtsov , journal=

    K.S. Kravtsov , journal=. Security of entanglement-based. 2023 , publisher=

    work page 2023

  55. [55]

    Physical Review A , volume=

    Quantum decoherence of two qubits , author=. Physical Review A , volume=. 2009 , publisher=

    work page 2009

  56. [56]

    Optics communications , volume=

    Quantum cryptography using partially entangled states , author=. Optics communications , volume=. 2010 , publisher=

    work page 2010

  57. [57]

    Reviews of modern physics , volume=

    Quantum cryptography , author=. Reviews of modern physics , volume=. 2002 , publisher=

    work page 2002

  58. [58]

    Rizk and A

    N. Rizk and A. Dr. 2026 , MONTH = Apr, KEYWORDS =

    work page 2026

  59. [59]

    Rizk and A

    N. Rizk and A. Dremeau and A. Coatanhay , URL =. 2025 , MONTH = Aug, PDF =

    work page 2025