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arxiv: 2606.20346 · v1 · pith:CKFTDV33new · submitted 2026-06-18 · 🪐 quant-ph

Effective discrete-modulated continuous variable QKD under general attacks

Mariana Navarro , Antonio Ac\'in , Carlos Pascual-Garc\'ia This is my paper

Pith reviewed 2026-06-26 17:11 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum key distributioncontinuous variablediscrete modulationfinite-size analysisgeneral attackstrusted detector
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The pith

A finite-size security analysis for discrete-modulated CV-QKD produces positive key rates at block sizes of order 10^8 under general attacks.

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

The paper establishes a security analysis for continuous-variable quantum key distribution that uses discrete modulations and works against general attacks in the finite-size regime. It integrates dimension reduction, marginal-constrained entropy accumulation, and a trusted detector model to handle receiver imperfections without requiring bounded state dimensions or impractically large blocks. This yields positive secret key rates for block sizes around 10^8. Such results matter because they allow information-theoretic security with standard telecom hardware and simplified postprocessing, narrowing the distance to experimental deployment.

Core claim

The authors show that a finite-size security analysis for discrete-modulated continuous variable quantum key distribution protocols, combining the dimension reduction technique with a proof based on marginal-constrained entropy accumulation and a trusted detector model, removes prior restrictions on coherent-state dimensions and achieves positive key rates for relevant block sizes of order 10^8 under general attacks.

What carries the argument

Dimension reduction combined with marginal-constrained entropy accumulation under a trusted detector model for receiver imperfections.

If this is right

  • Positive secret key rates become available for discrete-modulated CV-QKD at block sizes of order 10^8 against general attacks.
  • No bounded-dimension assumption on coherent states is needed to prove security.
  • Receiver imperfections can be incorporated realistically while preserving usable finite-size rates.
  • Standard telecom components and simplified postprocessing suffice for the protocol.

Where Pith is reading between the lines

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

  • The same combination of techniques might extend to other discrete modulation formats or channel models without re-deriving the full proof.
  • If detector trust cannot be maintained, the key-rate claims would require a separate analysis of untrusted components.
  • Experimental implementations at 10^8 blocks could directly test whether the modeled rates match observed performance.

Load-bearing premise

The analysis assumes a trusted detector whose imperfections are accurately modeled and fully accounted for in the security proof.

What would settle it

An experiment or calculation demonstrating that an eavesdropper can extract more information than the trusted-detector model predicts for block sizes near 10^8 would falsify the reported positive key rates.

Figures

Figures reproduced from arXiv: 2606.20346 by Antonio Ac\'in, Carlos Pascual-Garc\'ia, Mariana Navarro.

Figure 1
Figure 1. Figure 1: Phase space representation of the modulations used by Bob for (a) key distillation and (b) parameter [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Dimension reduction penalty ζα with respect to the cutoff size Nc of Bob’s projection and diverse values of the parameter ∆ for (a) n = 1010 with transmittance χ = 8 dB, and for (b) n = 108 with transmittance χ = 4 dB. The parameters α, γ, p K and ∆s were optimized for each data point. The remaining parameters, such as γ, ∆s, and α, were independently optimized for each trans￾mittance point to maximize the… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Dimension reduction penalty ζα with respect to the channel transmittance for diverse values of the block size n. These values lead to the (b) finite-size key rates against the total transmittance χ for different block sizes. We use a dimension reduction with Nc = 20, a modulation of ∆ = 4.7, and reconciliation efficiency β = 95%. The parameters α, γ, p K and ∆s were optimized for each data point. Furth… view at source ↗
read the original abstract

Continuous variable quantum key distribution via discrete modulations ensures information-theoretic security using standard telecom technologies, providing affordable and scalable quantum communications with simplified classical postprocessing. However, existing security proofs against general attacks often rely on restrictive assumptions, such as a bounded dimension for coherent states, or require impractically large block sizes. In this work, we develop a finite-size security analysis that removes these limitations while incorporating realistic experimental features. Our approach combines the dimension reduction technique, a security proof based on the marginal-constrained entropy accumulation, and a trusted detector model accounting for the receiver imperfections. We report positive key rates in the finite-size regime for relevant block sizes of the order of $10^8$. These results contribute to narrowing the gap between theoretical security proofs and practical implementations of discrete-modulated continuous variable quantum key distribution protocols.

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

1 major / 1 minor

Summary. The manuscript develops a finite-size security analysis for discrete-modulated continuous-variable quantum key distribution (CV-QKD) under general attacks. It combines the dimension reduction technique, a security proof based on marginal-constrained entropy accumulation, and a trusted detector model that accounts for receiver imperfections. The central claim is that this approach yields positive secret key rates for block sizes on the order of 10^8, removing prior limitations such as bounded coherent-state dimensions or impractically large blocks while incorporating realistic experimental features.

Significance. If the analysis is sound, the result is significant for the field: it narrows the gap between theoretical security proofs and practical DM-CV-QKD implementations that rely on standard telecom components and simplified post-processing. The integration of dimension reduction with entropy accumulation under a trusted-detector assumption enables finite-size rates at experimentally relevant scales, strengthening the case for scalable, information-theoretically secure CV-QKD.

major comments (1)
  1. The headline claim of positive key rates at block sizes ~10^8 is load-bearing on the trusted detector model fully capturing every receiver imperfection exploitable under general attacks. The abstract states that the model 'accounts for the receiver imperfections,' but without explicit bounds or a section showing that any mismatch between modeled and actual detector behavior leaves the marginal-constrained entropy accumulation bounds intact, the finite-size rates cannot be considered secure.
minor comments (1)
  1. [Abstract] The abstract refers to 'relevant block sizes of the order of 10^8' without specifying the exact modulation cardinality, channel transmittance, or excess noise values at which positive rates are obtained; a short table or sentence in the introduction would clarify the operating regime.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comment on our manuscript. We address the major point below.

read point-by-point responses

  1. Referee: The headline claim of positive key rates at block sizes ~10^8 is load-bearing on the trusted detector model fully capturing every receiver imperfection exploitable under general attacks. The abstract states that the model 'accounts for the receiver imperfections,' but without explicit bounds or a section showing that any mismatch between modeled and actual detector behavior leaves the marginal-constrained entropy accumulation bounds intact, the finite-size rates cannot be considered secure.

    Authors: The security proof is derived under the explicit assumption of a trusted detector model in which all relevant receiver imperfections (efficiency, electronic noise, etc.) are fully characterized and folded into the observed marginal statistics that constrain the entropy accumulation. This modeling choice is standard in CV-QKD analyses; the dimension-reduction and marginal-constrained entropy accumulation steps are performed with respect to these modeled statistics. Any physical mismatch between the assumed model and the actual detector would place the implementation outside the scope of the claimed security guarantee, exactly as occurs with trusted-source or trusted-channel assumptions in other proofs. We will add a clarifying paragraph in the security-analysis section stating this assumption explicitly and noting that the reported rates are conditional on the model fidelity. This revision improves transparency without requiring new bounds or altering the core results. revision: yes

Circularity Check

0 steps flagged

Security analysis combines established techniques (dimension reduction, entropy accumulation, trusted detector) with no reduction of key rates to fitted parameters or self-referential definitions

full rationale

The abstract and description present the finite-size key rate result as arising from a combination of the dimension reduction technique, marginal-constrained entropy accumulation, and a trusted detector model. No equations or steps are shown that define a quantity in terms of itself or rename a fit as a prediction. Any self-citations to prior work on these techniques are not load-bearing in a way that collapses the central claim to a tautology; the reported positive rates at block size ~10^8 rest on the external validity of those methods rather than internal redefinition. This is the normal case of an incremental application of known tools.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; full technical details of parameters, axioms, and entities are unavailable.

axioms (1)
  • domain assumption Trusted detector model accounts for receiver imperfections
    Invoked to incorporate realistic experimental features as stated in the abstract.

pith-pipeline@v0.9.1-grok · 5666 in / 1128 out tokens · 22317 ms · 2026-06-26T17:11:42.561058+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

57 extracted references · 20 canonical work pages

  1. [1]

    C. H. Bennett and G. Brassard, ‘Quantum Cryptography: Public Key Distribution and Coin Tossing’, in Proceedings of the ieee international conference on computers, systems and signal processing (1984), pp. 175–179

    1984

  2. [2]

    C. H. Bennett, G. Brassard and N. D. Mermin, ‘Quantum cryptography without bell’s the- orem’, Phys. Rev. Lett.68, 557–559 (1992),https : / / link . aps . org / doi / 10 . 1103 / PhysRevLett.68.557

    1992

  3. [3]

    A. K. Ekert, ‘Quantum cryptography based on Bell’s theorem’, Physical review letters67, 661–663 (1991)

    1991

  4. [4]

    Renner, ‘Security of quantum key distribution’, PhD thesis (Swiss Federal Institute of Technology, 2006), eprint:ArXiv:0512258[quant-ph]

    R. Renner, ‘Security of quantum key distribution’, PhD thesis (Swiss Federal Institute of Technology, 2006), eprint:ArXiv:0512258[quant-ph]

    2006

  5. [5]

    Portmann and R

    C. Portmann and R. Renner, ‘Security in quantum cryptography’, Rev. Mod. Phys.94, 025008 (2022),https://link.aps.org/doi/10.1103/RevModPhys.94.025008

    work page doi:10.1103/revmodphys.94.025008 2022

  6. [6]

    Diamanti and A

    E. Diamanti and A. Leverrier, ‘Distributing secret keys with quantum continuous variables: principle, security and implementations’, Entropy17, 6072–6092 (2015),https : / / www . mdpi.com/1099-4300/17/9/6072

    2015

  7. [7]

    Pirandola, U

    S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani et al., ‘Advances in quantum cryptography’, Advances in optics and photonics12, 1012–1236 (2020)

    2020

  8. [8]

    Zhang, Y

    Y. Zhang, Y. Bian, Z. Li, S. Yu and H. Guo, ‘Continuous-variable quantum key distribution system: Past, present, and future’, Applied Physics Reviews11, 011318 (2024),https : //doi.org/10.1063/5.0179566

    work page doi:10.1063/5.0179566 2024

  9. [9]

    M. F. Anka, J. A. Mora Rodríguez, D. F. Pinto, L. Q. Galvão, M. A. Dias and A. B. Tacla, ‘An introductory review of the theory of continuous-variable quantum key distribution: fundamentals, protocols, and security’, Brazilian Journal of Physics56,doi: 10.1007/s13538- 025-01975-8 (2026),http://dx.doi.org/10.1007/s13538-025-01975-8

    work page doi:10.1007/s13538- 2026

  10. [10]

    Grosshans and P

    F. Grosshans and P. Grangier, ‘Continuous variable quantum cryptography using coherent states’, Physical Review Letters88, 057902 (2002)

    2002

  11. [11]

    Grosshans, G

    F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf and P. Grangier, ‘Quantum key distribution using gaussian-modulated coherent states’, Nature421, 238–241 (2003), https://www.nature.com/articles/nature01289. 14

    2003

  12. [12]

    Mariani, R

    L. Mariani, R. Yehia, C. Pascual-García, F. Centrone, J. van der Kolk, M. Ángeles Serrano and A. Acín,Quantum key distribution over complex networks, 2025, arXiv:2504 . 02372 [quant-ph],https://arxiv.org/abs/2504.02372

    arXiv 2025

  13. [13]

    Leverrier, ‘Theoretical study of continuous-variable quantum key distribution’, PhD thesis (Telecom ParisTech, 2009)

    A. Leverrier, ‘Theoretical study of continuous-variable quantum key distribution’, PhD thesis (Telecom ParisTech, 2009)

    2009

  14. [14]

    Leverrier and P

    A. Leverrier and P. Grangier, ‘Unconditional security proof of long-distance continuous- variable quantum key distribution with discrete modulation’, Phys. Rev. Lett.102, 180504 (2009),https://link.aps.org/doi/10.1103/PhysRevLett.102.180504

    work page doi:10.1103/physrevlett.102.180504 2009

  15. [15]

    Ghorai, P

    S. Ghorai, P. Grangier, E. Diamanti and A. Leverrier, ‘Asymptotic security of continuous- variable quantum key distribution with a discrete modulation’, Phys. Rev. X9, 021059 (2019),https://link.aps.org/doi/10.1103/PhysRevX.9.021059

    work page doi:10.1103/physrevx.9.021059 2019

  16. [16]

    Roumestan, A

    F. Roumestan, A. Ghazisaeidi, J. Renaudier, L. T. Vidarte, A. Leverrier, E. Diamanti and P. Grangier, ‘Shaped constellation continuous variable quantum key distribution: con- cepts, methods and experimental validation’, Journal of Lightwave Technology42, 5182–5189 (2024),http://dx.doi.org/10.1109/JLT.2024.3391168

    work page doi:10.1109/jlt.2024.3391168 2024

  17. [17]

    S.Bäuml,C.Pascual-García,V.Wright,O.FawziandA.Acín,‘Securityofdiscrete-modulated continuous-variable quantum key distribution’, Quantum8, 1418 (2024),https://doi.org/ 10.22331/q-2024-07-18-1418

    work page doi:10.22331/q-2024-07-18-1418 2024

  18. [18]

    Pascual-García, S

    C. Pascual-García, S. Bäuml, M. Araújo, R. Liss and A. Acín, ‘Improved finite-size key rates fordiscrete-modulatedcontinuous-variablequantumkeydistributionundercoherentattacks’, Phys. Rev. A111, 022610 (2025),https://link.aps.org/doi/10.1103/PhysRevA.111. 022610

    work page doi:10.1103/physreva.111 2025

  19. [19]

    I. W. Primaatmaja, W. Y. Kon and C. Lim,Discrete-modulated continuous-variable quantum key distribution secure against general attacks, 2024, arXiv:2409.02630 [quant-ph],https: //arxiv.org/abs/2409.02630

    arXiv 2024

  20. [20]

    Navarro, A

    M. Navarro, A. G. Lorente, P. V. Parellada, C. Pascual-García and M. Araújo,Finite-size quantum key distribution rates from rényi entropies using conic optimization, 2026, arXiv:2 511.10584 [quant-ph],https://arxiv.org/abs/2511.10584

    Pith/arXiv arXiv 2026

  21. [21]

    J.Lin,T.UpadhyayaandN.Lütkenhaus,‘Asymptoticsecurityanalysisofdiscrete-modulated continuous-variable quantum key distribution’, Phys. Rev. X9, 041064 (2019),https:// link.aps.org/doi/10.1103/PhysRevX.9.041064

    work page doi:10.1103/physrevx.9.041064 2019

  22. [22]

    T.Upadhyaya,T.vanHimbeeck,J.LinandN.Lütkenhaus,‘Dimensionreductioninquantum key distribution for continuous- and discrete-variable protocols’, PRX Quantum2, 020325 (2021),https://link.aps.org/doi/10.1103/PRXQuantum.2.020325

    work page doi:10.1103/prxquantum.2.020325 2021

  23. [23]

    Dupuis, O

    F. Dupuis, O. Fawzi and R. Renner, ‘Entropy accumulation’, Communications in Mathem- atical Physics379, 867–913 (2020)

    2020

  24. [24]

    Dupuis and O

    F. Dupuis and O. Fawzi, ‘Entropy accumulation with improved second-order term’, IEEE Transactions on information theory65, 7596–7612 (2019)

    2019

  25. [25]

    Metger and R

    T. Metger and R. Renner, ‘Security of quantum key distribution from generalised entropy accumulation’, Nature Communications14, 5272 (2023),https://www.nature.com/artic les/s41467-023-40920-8

    2023

  26. [26]

    Arqand and E

    A. Arqand and E. Y. Z. Tan,Marginal-constrained entropy accumulation theorem, 2025, arXiv:2502.02563 [quant-ph]

    arXiv 2025

  27. [27]

    Dupuis, ‘Privacy amplification and decoupling without smoothing’, IEEE Transactions on Information Theory69, 7784–7792 (2023)

    F. Dupuis, ‘Privacy amplification and decoupling without smoothing’, IEEE Transactions on Information Theory69, 7784–7792 (2023)

    2023

  28. [28]

    Kamin, J

    L. Kamin, J. Burniston and E. Y. Z. Tan,Rényi security framework against coherent attacks applied to decoy-state QKD, 2025, arXiv:2504.12248 [quant-ph]

    arXiv 2025

  29. [29]

    Skajaa and Y

    A. Skajaa and Y. Ye, ‘A homogeneous interior-point algorithm for nonsymmetric convex conic optimization’, Mathematical Programming150, 391–422 (2015)

    2015

  30. [30]

    A homogeneous interior-point algorithm for non-symmetric con- vex conic optimization

    D. Papp and S. Yıldız,On “A homogeneous interior-point algorithm for non-symmetric con- vex conic optimization”, 2017, arXiv:1712.00492 [math.OC]. 15

    Pith/arXiv arXiv 2017

  31. [31]

    A. G. Lorente, P. V. Parellada, M. Castillo-Celeita and M. Araújo,Quantum key distribution rates from non-symmetric conic optimization, 2024, arXiv:2407.00152 [quant-ph],https: //arxiv.org/abs/2407.00152

    arXiv 2024

  32. [32]

    Drusvyatskiy and H

    D. Drusvyatskiy and H. Wolkowicz, ‘The many faces of degeneracy in conic optimization’, Foundations and Trends in Optimization3, 77–170 (2017), arXiv:1706.03705 [quant-ph]

    Pith/arXiv arXiv 2017

  33. [33]

    H. Hu, J. Im, J. Lin, N. Lütkenhaus and H. Wolkowicz, ‘Robust interior point method for quantum key distribution rate computation’, Quantum6, 792 (2022). [34]ConicQKD.jl: Implementation of convex cones for quantum key distribution.2025,https: //github.com/araujoms/ConicQKD.jl. [35]Renyi-ConicQKD, 2025,https://github.com/MarianaNvrr/Renyi-ConicQKD

    2022

  34. [34]

    Kanitschar and C

    F. Kanitschar and C. Pacher, ‘Optimizing continuous-variable quantum key distribution with phase-shift keying modulation and postselection’, Phys. Rev. Appl.18, 034073 (2022),http s://link.aps.org/doi/10.1103/PhysRevApplied.18.034073

    work page doi:10.1103/physrevapplied.18.034073 2022

  35. [35]

    Lin and N

    J. Lin and N. Lütkenhaus, ‘Trusted detector noise analysis for discrete modulation schemes of continuous-variable quantum key distribution’, Phys. Rev. Appl.14, 064030 (2020),https: //link.aps.org/doi/10.1103/PhysRevApplied.14.064030

    work page doi:10.1103/physrevapplied.14.064030 2020

  36. [36]

    Kanitschar, I

    F. Kanitschar, I. George, J. Lin, T. Upadhyaya and N. Lütkenhaus, ‘Finite-size security for discrete-modulated continuous-variable quantum key distribution protocols’, PRX Quantum 4, 040306 (2023),https://link.aps.org/doi/10.1103/PRXQuantum.4.040306

    work page doi:10.1103/prxquantum.4.040306 2023

  37. [37]

    Ferradini, M

    C. Ferradini, M. Sandfuchs, R. Wolf and R. Renner,Defining security in quantum key dis- tribution, 2025, arXiv:2509.13405 [quant-ph],https://arxiv.org/abs/2509.13405

    arXiv 2025

  38. [38]

    W. Hoeffding, ‘Probability inequalities for sums of bounded random variables’, Journal of the American Statistical Association58, 13–30 (1963),https://www.tandfonline.com/ doi/abs/10.1080/01621459.1963.10500830

    work page doi:10.1080/01621459.1963.10500830 1963

  39. [39]

    Staffieri, G

    G. Staffieri, G. Scala and C. Lupo,Finite-size secret-key rates of discrete modulation cv qkd under passive attacks, 2025, eprint:2509 . 14345(quant-ph),https : / / arxiv . org / abs / 2509.14345

    arXiv 2025

  40. [40]

    Marwah and F

    A. Marwah and F. Dupuis, ‘Uniform continuity bound for sandwiched rényi conditional entropy’, Journal of Mathematical Physics63,doi: 10.1063/5.0088507 (2022),http://dx. doi.org/10.1063/5.0088507

    work page doi:10.1063/5.0088507 2022

  41. [41]

    Bluhm, A

    A. Bluhm, A. Capel, P. Gondolf and T. Möbus, ‘Unified framework for continuity of sand- wiched rényi divergences’, Annales Henri Poincaré,doi: 10.1007/s00023-024-01519-x (2024), http://dx.doi.org/10.1007/s00023-024-01519-x

    work page doi:10.1007/s00023-024-01519-x 2024

  42. [42]

    Slepian and J

    D. Slepian and J. Wolf, ‘Noiseless coding of correlated information sources’, IEEE Transac- tions on Information Theory19, 471–480 (1973)

    1973

  43. [43]

    Boyd and L

    S. Boyd and L. Vandenberghe,Convex optimization(Cambridge University Press, 2004)

    2004

  44. [44]

    Nesterov,Lectures on convex optimization(Springer, 2018)

    Y. Nesterov,Lectures on convex optimization(Springer, 2018)

    2018

  45. [45]

    A. A. E. Hajomer, F. Kanitschar, N. Jain, M. Hentschel, R. Zhang, N. Lütkenhaus, U. L. An- dersen, C. Pacher and T. Gehring, ‘Experimental composable key distribution using discrete- modulated continuous variable quantum cryptography’, Light: Science & Applications14, 255 (2025),https://doi.org/10.1038/s41377-025-01924-9

    work page doi:10.1038/s41377-025-01924-9 2025

  46. [46]

    C. Coey, L. Kapelevich and J. P. Vielma, ‘Performance enhancements for a generic conic in- terior point algorithm’, Mathematical Programming Computation15, 53–101 (2023), arXiv:2 107.04262 [math.OC]

    2023

  47. [47]

    Tupkary, E

    D. Tupkary, E. Y.-Z. Tan and N. Lütkenhaus, ‘Security proof for variable-length quantum key distribution’, Phys. Rev. Res.6, 023002 (2024), arXiv:2311.01600 [quant-ph]

    arXiv 2024

  48. [48]

    Kanitschar and M

    F. Kanitschar and M. Huber, ‘Composable finite-size security of high-dimensional quantum- key-distribution protocols’, Phys. Rev. Appl.24, 054028 (2025),https://link.aps.org/ doi/10.1103/v51y-vkfr. 16

    work page doi:10.1103/v51y-vkfr 2025

  49. [49]

    Tupkary, S

    D. Tupkary, S. Nahar, A. Arqand, E. Y. Z. Tan and N. Lütkenhaus,A rigorous and com- plete security proof of decoy-state bb84 quantum key distribution, 2026, arXiv:2601.18035 [quant-ph],https://arxiv.org/abs/2601.18035

    arXiv 2026

  50. [50]

    Mizutani, T

    A. Mizutani, T. Sasaki and G. Kato,Protocol-level description and self-contained security proof of decoy-state bb84 qkd protocol,2025,arXiv:2504.20417 [quant-ph],https://arxiv. org/abs/2504.20417

    arXiv 2025

  51. [51]

    Tupkary, E

    D. Tupkary, E. Y. Z. Tan, S. Nahar, L. Kamin and N. Lütkenhaus,Qkd security proofs for decoy-state bb84: protocol variations, proof techniques, gaps and limitations, 2025, arXiv:25 02.10340 [quant-ph],https://arxiv.org/abs/2502.10340

    Pith/arXiv arXiv 2025

  52. [52]

    Tomamichel,Quantum information processing with finite resources(Springer Interna- tional Publishing, 2016)

    M. Tomamichel,Quantum information processing with finite resources(Springer Interna- tional Publishing, 2016)

    2016

  53. [53]

    Berta, F

    M. Berta, F. Furrer and V. B. Scholz, ‘The smooth entropy formalism for von neumann algebras’, Journal of Mathematical Physics57(2016)

    2016

  54. [54]

    Winter, ‘Coding theorem and strong converse for quantum channels’, IEEE Transactions on Information Theory45, 2481–2485 (1999)

    A. Winter, ‘Coding theorem and strong converse for quantum channels’, IEEE Transactions on Information Theory45, 2481–2485 (1999)

    1999

  55. [55]

    R.RennerandJ.I.Cirac,‘Definettirepresentationtheoremforinfinite-dimensionalquantum systems and applications to quantum cryptography’, Phys. Rev. Lett.102, 110504 (2009), https://link.aps.org/doi/10.1103/PhysRevLett.102.110504

    work page doi:10.1103/physrevlett.102.110504 2009

  56. [56]

    C. Coey, L. Kapelevich and J. P. Vielma, ‘Solving natural conic formulations with Hypatia.jl’, INFORMS Journal on Computing34, 2686–2699 (2022)

    2022

  57. [57]

    Nahar, D

    S. Nahar, D. Tupkary, Y. Zhao, N. Lütkenhaus and E. Y.-Z. Tan, ‘Postselection technique for optical quantum key distribution with improved de finetti reductions’, PRX Quantum5, 040315 (2024),https://link.aps.org/doi/10.1103/PRXQuantum.5.040315. 17 A Privacy amplification via Rényi leftover hashing As a starting point in our finite-size analysis, we make u...

    work page doi:10.1103/prxquantum.5.040315 2024