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arxiv: 2605.07229 · v1 · submitted 2026-05-08 · 🪐 quant-ph

Hardware-Free Polarization Stabilization for Measurement-Device-Independent Quantum Key Distribution via Correlated Twirling

Papon Pewkhom , Nattee Jeennugool , Norshamsuri Ali , Rosdisham Endut , Syed Alwee Aljunid , Pruet Kalasuwan This is my paper

Pith reviewed 2026-05-11 01:25 UTC · model grok-4.3

classification 🪐 quant-ph
keywords MDI-QKDpolarization stabilizationtwirling supermapquantum key distributionpost-processingPauli depolarizing channelangular misalignmentHong-Ou-Mandel interference
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The pith

A synchronized public twirling supermap converts asymmetric polarization rotations into isotropic depolarizing noise, extending MDI-QKD misalignment tolerance without hardware.

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

The paper establishes that Alice and Bob can apply a synchronized, public twirling supermap during classical post-processing to transform deterministic asymmetric geometric rotations from channel turbulence into an isotropic Pauli depolarizing channel. This approach suppresses intrinsic channel noise by a factor of two-thirds and raises tolerance to Y-bias from 0.68 to 0.84 radians while extending angular misalignment tolerance for the eleven-percent security threshold from 38.7 to 47.9 degrees. A sympathetic reader would care because the method operates entirely as virtual post-processing, eliminating the need for hardware stabilization or frequent calibration in turbulent optical fibers and enabling secure key distribution over longer distances where standard MDI-QKD fails.

Core claim

By applying a synchronized, public twirling supermap based on a unitary 2-design, Alice and Bob mathematically transform deterministic, asymmetric geometric rotations into an isotropic Pauli depolarizing channel. Executed as a virtual post-processing step during classical sifting, this protocol suppresses intrinsic channel noise by a factor of 2/3. Simulations show it extends Y-bias tolerance from 0.68 to 0.84 radians and absolute angular misalignment tolerance for the 11% security threshold from 38.7° to 47.9°, sustaining secure key distillation in highly turbulent regimes.

What carries the argument

The Correlated Twirling protocol, a synchronized public twirling supermap that maps asymmetric rotations to isotropic Pauli noise during post-processing.

If this is right

  • Secure key distillation remains possible over extended fiber distances in highly turbulent regimes where standard MDI-QKD fails.
  • The protocol remains compatible with decoy-state weak coherent pulses.
  • Induced symmetry neutralizes catastrophic axis-dependent failures in Hong-Ou-Mandel interference.

Where Pith is reading between the lines

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

  • Real-world MDI-QKD deployments could reduce calibration downtime by replacing physical stabilization hardware with this post-processing step.
  • The symmetry-inducing approach may extend to other polarization-sensitive quantum protocols facing similar drift issues.
  • Further testing in actual fiber links would confirm whether the simulated tolerance gains hold under combined loss and turbulence.

Load-bearing premise

Channel polarization changes behave as purely deterministic asymmetric geometric rotations that post-processing twirling can fully correct without adding new errors or rate penalties.

What would settle it

An experiment or simulation in which the noise after applying the twirling supermap is not reduced by two-thirds or the angular misalignment tolerance fails to reach 47.9 degrees at the 11% security threshold.

Figures

Figures reproduced from arXiv: 2605.07229 by Nattee Jeennugool, Norshamsuri Ali, Papon Pewkhom, Pruet Kalasuwan, Rosdisham Endut, Syed Alwee Aljunid.

Figure 1
Figure 1. Figure 1: Total guessing probability (Pguess,total) as a function of the relative rotation angle α. The plot compares the invariant response of the protected system (solid blue line) against the volatile performance of the standard unprotected system (gray line), which fluctuates un￾predictably between the theoretical worst-case ”Bad Axis” (red dashed) and best-case ”Good Axis” (green dashed) fiber stress orientatio… view at source ↗
Figure 2
Figure 2. Figure 2: Total Quantum Bit Error Rate (QBER) as a function of the relative polarization [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Simulated QBER response to discrete rotational biases applied along the Polarization [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Maximum secure fiber length achievable as a function of relative rotation noise (stan [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
read the original abstract

Measurement-Device-Independent Quantum Key Distribution (MDI-QKD) provides unconditional security against detector vulnerabilities, but its practical deployment is severely hindered by asymmetric channel turbulence. Fluctuations in optical fibers induce arbitrary polarization drift, degrading Hong-Ou-Mandel interference and forcing extensive calibration downtime. In this work, we propose a hardware-free polarization stabilization technique utilizing a Correlated Twirling protocol based on a unitary 2-design. By applying a synchronized, public twirling supermap, Alice and Bob mathematically transform deterministic, asymmetric geometric rotations into an isotropic Pauli depolarizing channel. Executed entirely as a virtual post-processing step during classical sifting, this protocol mathematically suppresses intrinsic channel noise by a factor of 2/3. We demonstrate through exact quantum state simulations that this induced symmetry neutralizes catastrophic axis-dependent failures, extending the Y-bias tolerance from 0.68 to 0.84 radians. Furthermore, the protocol passively extends the absolute angular misalignment tolerance for the 11% security threshold from $38.7^\circ$ to $47.9^\circ$, sustaining secure key distillation over extended fiber distances in highly turbulent regimes where standard architectures fail. Inherently compatible with decoy-state weak coherent pulses, this algorithmic approach provides a highly scalable, resource-efficient framework for robust long-distance quantum networks.

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 / 1 minor

Summary. The manuscript proposes a hardware-free polarization stabilization technique for MDI-QKD based on a correlated twirling protocol using a unitary 2-design. By applying a synchronized public twirling supermap entirely as virtual post-processing during classical sifting, the authors claim to mathematically convert deterministic asymmetric geometric rotations into an isotropic Pauli depolarizing channel. This is asserted to suppress intrinsic channel noise by a factor of 2/3, extend Y-bias tolerance from 0.68 to 0.84 radians, and increase absolute angular misalignment tolerance from 38.7° to 47.9° for the 11% security threshold, with compatibility to decoy-state weak coherent pulses, as demonstrated via exact quantum state simulations.

Significance. If the virtual twirling protocol achieves the claimed transformation and noise suppression without hidden rate penalties, it would provide a meaningful practical advance for MDI-QKD by offering a software-only approach to mitigate polarization drift in turbulent fibers. This could reduce reliance on active hardware stabilization and calibration downtime, enabling more robust long-distance operation in challenging environments. The approach builds on standard quantum information tools like unitary 2-designs in a targeted way for a real deployment bottleneck.

major comments (2)
  1. Abstract: The claim that the synchronized public twirling supermap 'mathematically transform[s] deterministic, asymmetric geometric rotations into an isotropic Pauli depolarizing channel' and suppresses noise by 2/3 via virtual post-processing is load-bearing but insufficiently justified. Since the physical channel applies a fixed rotation producing fixed outcome probabilities at the Bell-state measurement, post-processing cannot perform the averaging over the 2-design without either physical randomization (contradicting the hardware-free claim) or post-selection that reduces sifting efficiency by a multiplicative factor; this factor is not modeled in the reported tolerance gains or 2/3 suppression.
  2. Simulation results (as referenced in the abstract): The exact quantum state simulations supporting the 2/3 suppression and specific tolerance extensions (Y-bias from 0.68 to 0.84 rad; angular misalignment from 38.7° to 47.9°) do not specify key parameters such as the number of twirl elements from the 2-design, the precise implementation of correlated twirling in the simulation, or inclusion of any sifting rate reduction. This prevents independent verification of the central mapping and undermines the quantified improvements.
minor comments (1)
  1. The abstract introduces terms such as 'Correlated Twirling protocol' and 'twirling supermap' without defining the specific unitary 2-design or the form of the supermap; these should be explicitly defined with equations in the main text for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and have made revisions to improve clarity and provide the requested details.

read point-by-point responses

  1. Referee: Abstract: The claim that the synchronized public twirling supermap 'mathematically transform[s] deterministic, asymmetric geometric rotations into an isotropic Pauli depolarizing channel' and suppresses noise by 2/3 via virtual post-processing is load-bearing but insufficiently justified. Since the physical channel applies a fixed rotation producing fixed outcome probabilities at the Bell-state measurement, post-processing cannot perform the averaging over the 2-design without either physical randomization (contradicting the hardware-free claim) or post-selection that reduces sifting efficiency by a multiplicative factor; this factor is not modeled in the reported tolerance gains or 2/3 suppression.

    Authors: We appreciate the referee's concern and the opportunity to clarify the mechanism. The synchronized public twirling is implemented by Alice and Bob sharing a public random seed to select the same element U from the unitary 2-design for each pulse pair. Virtual Pauli corrections derived from U are then applied to the classical sifted bits and error estimates during post-processing. Because the selection is public and the corrections are classical, the physical channel and Bell-state measurement remain unaltered, preserving the hardware-free property. The effect on the observed statistics is mathematically equivalent to applying the twirling supermap, converting the fixed rotation into the isotropic depolarizing channel with noise suppressed by 2/3. No events are discarded, so the sifting rate is unchanged. We have added a dedicated subsection with explicit equations in the revised manuscript demonstrating this equivalence. revision: yes

  2. Referee: Simulation results (as referenced in the abstract): The exact quantum state simulations supporting the 2/3 suppression and specific tolerance extensions (Y-bias from 0.68 to 0.84 rad; angular misalignment from 38.7° to 47.9°) do not specify key parameters such as the number of twirl elements from the 2-design, the precise implementation of correlated twirling in the simulation, or inclusion of any sifting rate reduction. This prevents independent verification of the central mapping and undermines the quantified improvements.

    Authors: We agree that the simulation parameters require explicit specification for reproducibility. In the revised manuscript we now state that the unitary 2-design is the single-qubit Clifford group (24 elements). For each simulated pulse pair, a random Clifford element U is sampled, the corresponding virtual correction is applied to the Bell-state measurement outcomes during classical post-processing, and the resulting error rates are averaged over 10^5 independent realizations. No sifting rate reduction is applied. These details have been added to the Methods section together with the simulation code structure, allowing independent verification of the reported 2/3 noise suppression and the tolerance values (Y-bias 0.84 rad, angular misalignment 47.9°). revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives its central claims—the 2/3 noise suppression and extended misalignment tolerances—directly from the standard mathematical properties of applying a unitary 2-design twirling supermap to a deterministic asymmetric rotation channel model. This transformation into an isotropic Pauli depolarizing channel follows from the definition and averaging behavior of the 2-design, which is an established quantum-information construction independent of the present work. The exact quantum state simulations then evaluate the resulting channel under the stated assumptions without introducing fitted parameters, self-referential loops, or load-bearing self-citations. No step reduces by construction to the paper's own inputs; the derivation remains self-contained within conventional quantum channel theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the mathematical averaging property of unitary 2-designs and the modeling assumption that fiber-induced polarization changes are deterministic geometric rotations; no new physical entities or fitted constants are introduced.

axioms (2)
  • standard math Unitary 2-designs exist and their twirling supermap converts any fixed unitary rotation into an isotropic Pauli depolarizing channel.
    Invoked when the paper states that the synchronized twirling transforms asymmetric rotations into isotropic noise.
  • domain assumption Polarization drift in optical fibers consists of deterministic asymmetric geometric rotations induced by environmental turbulence.
    The protocol is designed specifically to neutralize this class of channel noise.

pith-pipeline@v0.9.0 · 5563 in / 1557 out tokens · 42656 ms · 2026-05-11T01:25:35.597357+00:00 · methodology

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Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    Security in quantum cryptography.Rev

    Christopher Portmann and Renato Renner. Security in quantum cryptography.Rev. Mod. Phys., 94:025008, Jun 2022

    work page 2022

  2. [2]

    Secure quantum key distribution

    Hoi-Kwong Lo, Marcos Curty, and Kiyoshi Tamaki. Secure quantum key distribution. Nature Photonics, 8(8):595–604, 2014

    work page 2014

  3. [3]

    Controlling passively quenched single photon detectors by bright light

    Vadim Makarov. Controlling passively quenched single photon detectors by bright light. New Journal of Physics, 11(6):065003, jun 2009

    work page 2009

  4. [4]

    Hacking commercial quantum cryptography systems by tailored bright illumination.Nature Photonics, 4(10):686–689, 2010

    Lars Lydersen, Carlos Wiechers, Christoffer Wittmann, Dominique Elser, Johannes Skaar, and Vadim Makarov. Hacking commercial quantum cryptography systems by tailored bright illumination.Nature Photonics, 4(10):686–689, 2010

    work page 2010

  5. [5]

    After-gate attack on a quantum cryptosystem.New Journal of Physics, 13(1):013043, jan 2011

    C Wiechers, L Lydersen, C Wittmann, D Elser, J Skaar, Ch Marquardt, V Makarov, and G Leuchs. After-gate attack on a quantum cryptosystem.New Journal of Physics, 13(1):013043, jan 2011

    work page 2011

  6. [6]

    Controlling an actively-quenched single photon detector with bright light.Opt

    Sebastien Sauge, Lars Lydersen, Andrey Anisimov, Johannes Skaar, and Vadim Makarov. Controlling an actively-quenched single photon detector with bright light.Opt. Express, 19(23):23590–23600, Nov 2011

    work page 2011

  7. [7]

    Measurement-device-independent quantum key distribution.Physical Review Letters, 108(13):130503, 2012

    Hoi-Kwong Lo, Marcos Curty, and Bing Qi. Measurement-device-independent quantum key distribution.Physical Review Letters, 108(13):130503, 2012

    work page 2012

  8. [8]

    Braunstein and Stefano Pirandola

    Samuel L. Braunstein and Stefano Pirandola. Side-channel-free quantum key distribution. Phys. Rev. Lett., 108:130502, Mar 2012

    work page 2012

  9. [9]

    A cost-effective measurement- device-independent quantum key distribution system for quantum networks.Quantum Science and Technology, 2(4):04LT01, sep 2017

    Raju Valivarthi, Qiang Zhou, Caleb John, Francesco Marsili, Varun B Verma, Matthew D Shaw, Sae Woo Nam, Daniel Oblak, and Wolfgang Tittel. A cost-effective measurement- device-independent quantum key distribution system for quantum networks.Quantum Science and Technology, 2(4):04LT01, sep 2017

    work page 2017

  10. [10]

    Large-scale quantum communication networks with integrated photonics.Nature, 651(8104):68–75, 2026

    Yun Zheng, Hanyu Wang, Xinyu Jia, Jiahui Huang, Huihong Yuan, Chonghao Zhai, Junhao Dai, Jingbo Shi, Lei Zhang, Xuguang Zhang, Minxue Zhuang, Jinchang Liu, Jun Mao, Tianxiang Dai, Zhaorong Fu, Yuqing Jiao, Yaocheng Shi, Daoxin Dai, Xingjun Wang, Yan Li, Qihuang Gong, Zhiliang Yuan, Lin Chang, and Jianwei Wang. Large-scale quantum communication networks wi...

    work page 2026

  11. [11]

    A measurement-device-independent quantum key distri- bution network using optical frequency comb.npj Quantum Information, 11(1):97, 2025

    Wenhan Yan, Xiaodong Zheng, Wenjun Wen, Liangliang Lu, Yifeng Du, Yan-Qing Lu, Shining Zhu, and Xiao-Song Ma. A measurement-device-independent quantum key distri- bution network using optical frequency comb.npj Quantum Information, 11(1):97, 2025

    work page 2025

  12. [12]

    Measurement- device-independent quantum key distribution over untrustful metropolitan network.Phys

    Yan-Lin Tang, Hua-Lei Yin, Qi Zhao, Hui Liu, Xiang-Xiang Sun, Ming-Qi Huang, Wei-Jun Zhang, Si-Jing Chen, Lu Zhang, Li-Xing You, Zhen Wang, Yang Liu, Chao-Yang Lu, Xiao Jiang, Xiongfeng Ma, Qiang Zhang, Teng-Yun Chen, and Jian-Wei Pan. Measurement- device-independent quantum key distribution over untrustful metropolitan network.Phys. Rev. X, 6:011024, Mar 2016

    work page 2016

  13. [13]

    Field test of measurement-device-independent quantum key distribution.IEEE Journal of Selected Topics in Quantum Electronics, 21(3):116–122, 2015

    Yan-Lin Tang, Hua-Lei Yin, Si-Jing Chen, Yang Liu, Wei-Jun Zhang, Xiao Jiang, Lu Zhang, Jian Wang, Li-Xing You, Jian-Yu Guan, Dong-Xu Yang, Zhen Wang, Hao Liang, Zhen Zhang, Nan Zhou, Xiongfeng Ma, Teng-Yun Chen, Qiang Zhang, and Jian- Wei Pan. Field test of measurement-device-independent quantum key distribution.IEEE Journal of Selected Topics in Quantum...

    work page 2015

  14. [14]

    Long-distance free-space measurement-device-independent quantum key distribution.Phys

    Yuan Cao, Yu-Huai Li, Kui-Xing Yang, Yang-Fan Jiang, Shuang-Lin Li, Xiao-Long Hu, Maimaiti Abulizi, Cheng-Long Li, Weijun Zhang, Qi-Chao Sun, Wei-Yue Liu, Xiao Jiang, Sheng-Kai Liao, Ji-Gang Ren, Hao Li, Lixing You, Zhen Wang, Juan Yin, Chao-Yang Lu, Xiang-Bin Wang, Qiang Zhang, Cheng-Zhi Peng, and Jian-Wei Pan. Long-distance free-space measurement-device...

    work page 2020

  15. [15]

    R. I. Woodward, Y. S. Lo, M. Pittaluga, M. Minder, T. K. Para¨ ıso, M. Lucamarini, Z. L. Yuan, and A. J. Shields. Gigahertz measurement-device-independent quantum key distri- bution using directly modulated lasers.npj Quantum Information, 7(1):58, 2021

    work page 2021

  16. [16]

    High-rate continuous-variable measurement device-independent quantum key distribution with finite-size security.Quan- tum Science and Technology, 10(2):025032, feb 2025

    Adnan A E Hajomer, Ulrik L Andersen, and Tobias Gehring. High-rate continuous-variable measurement device-independent quantum key distribution with finite-size security.Quan- tum Science and Technology, 10(2):025032, feb 2025

    work page 2025

  17. [17]

    Measurement-device-independent quantum key distribution over 200 km.Phys

    Yan-Lin Tang, Hua-Lei Yin, Si-Jing Chen, Yang Liu, Wei-Jun Zhang, Xiao Jiang, Lu Zhang, Jian Wang, Li-Xing You, Jian-Yu Guan, Dong-Xu Yang, Zhen Wang, Hao Liang, Zhen Zhang, Nan Zhou, Xiongfeng Ma, Teng-Yun Chen, Qiang Zhang, and Jian- Wei Pan. Measurement-device-independent quantum key distribution over 200 km.Phys. Rev. Lett., 113:190501, Nov 2014

    work page 2014

  18. [18]

    Measurement-device-independent quantum key distribution over a 404 km optical fiber.Phys

    Hua-Lei Yin, Teng-Yun Chen, Zong-Wen Yu, Hui Liu, Li-Xing You, Yi-Heng Zhou, Si-Jing Chen, Yingqiu Mao, Ming-Qi Huang, Wei-Jun Zhang, Hao Chen, Ming Jun Li, Daniel Nolan, Fei Zhou, Xiao Jiang, Zhen Wang, Qiang Zhang, Xiang-Bin Wang, and Jian-Wei Pan. Measurement-device-independent quantum key distribution over a 404 km optical fiber.Phys. Rev. Lett., 117:...

    work page 2016

  19. [19]

    Berrevoets, Thomas Middelburg, Raymond F

    Remon C. Berrevoets, Thomas Middelburg, Raymond F. L. Vermeulen, Luca Della Chiesa, Federico Broggi, Stefano Piciaccia, Rene Pluis, Prathwiraj Umesh, Jorge F. Marques, Wolf- gang Tittel, and Joshua A. Slater. Deployed measurement-device independent quantum key distribution and bell-state measurements coexisting with standard internet data and networking e...

    work page 2022

  20. [20]

    Performance analysis of mdi-qkd in thermal-loss and phase noise channels

    Heyang Peng, Seid Koudia, Leonardo Oleynik, and Symeon Chatzinotas. Performance analysis of mdi-qkd in thermal-loss and phase noise channels. In2025 IEEE International Conference on Quantum Computing and Engineering (QCE), volume 01, pages 976–983, 2025

    work page 2025

  21. [21]

    Measurement of subpicosecond time intervals between two photons by interference.Physical Review Letters, 59(18):2044, 1987

    Chung Ki Hong, Zhe-Yu Ou, and Leonard Mandel. Measurement of subpicosecond time intervals between two photons by interference.Physical Review Letters, 59(18):2044, 1987

    work page 2044

  22. [22]

    Active polarization control for quantum communication in long-distance optical fibers with shared telecom traffic.Microwave and Optical Technology Letters, 53(11):2661–2665, 2011

    Guilherme B Xavier, Guilherme V de Faria, Thiago F da Silva, Guilherme P Tempor˜ ao, and Jean P von der Weid. Active polarization control for quantum communication in long-distance optical fibers with shared telecom traffic.Microwave and Optical Technology Letters, 53(11):2661–2665, 2011

    work page 2011

  23. [23]

    Reference-frame- independent quantum key distribution.Physical Review A, 82(1):012304, 2010

    Anthony Laing, Valerio Scarani, John G Rarity, and Jeremy L O’Brien. Reference-frame- independent quantum key distribution.Physical Review A, 82(1):012304, 2010

    work page 2010

  24. [24]

    Phase- reference-free experiment of measurement-device-independent quantum key distribution

    C Wang, X-T Song, Z-Q Yin, S Wang, W Chen, C-M Zhang, G-C Guo, and Z-F Han. Phase- reference-free experiment of measurement-device-independent quantum key distribution. Physical Review Letters, 115(16):160502, 2015

    work page 2015

  25. [25]

    Ko et al

    H. Ko et al. Secure quantum communication with the preservation of optimal measure- ments.IEEE Journal on Selected Areas in Communications, 43(8):2798, 2025. 18

    work page 2025

  26. [26]

    Kropf, and Joonwoo Bae

    Spiros Kechrimparis, Tanmay Singal, Chahan M. Kropf, and Joonwoo Bae. Preserving measurements for optimal state discrimination over quantum channels.Phys. Rev. A, 99:062302, Jun 2019

    work page 2019

  27. [27]

    Quantum cryptog- raphy.Reviews of Modern Physics, 74(1):145, 2002

    Nicolas Gisin, Gr´ egoire Ribordy, Wolfgang Tittel, and Hugo Zbinden. Quantum cryptog- raphy.Reviews of Modern Physics, 74(1):145, 2002

    work page 2002

  28. [28]

    Full polarization control for quantum key distribution.Electronics Letters, 44(4):256–258, 2008

    G B Xavier, G V de Faria, G P Tempor˜ ao, and J P Von Der Weid. Full polarization control for quantum key distribution.Electronics Letters, 44(4):256–258, 2008

    work page 2008

  29. [29]

    Evenly distributed unitaries: A gen- eralization of haar-measure.Journal of Mathematical Physics, 48(5):052104, 2007

    David Gross, Koenraad Audenaert, and Jens Eisert. Evenly distributed unitaries: A gen- eralization of haar-measure.Journal of Mathematical Physics, 48(5):052104, 2007

    work page 2007

  30. [30]

    Exact and approx- imate unitary 2-designs and their application to fidelity estimation.Physical Review A, 80(1):012304, 2009

    Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. Exact and approx- imate unitary 2-designs and their application to fidelity estimation.Physical Review A, 80(1):012304, 2009

    work page 2009

  31. [31]

    Cambridge University Press, 10th anniversary edition, 2010

    Michael A Nielsen and Isaac L Chuang.Quantum Computation and Quantum Information. Cambridge University Press, 10th anniversary edition, 2010

    work page 2010

  32. [32]

    Simple proof of security of the BB84 quantum key distribution protocol.Physical Review Letters, 85(2):441–444, 2000

    Peter W Shor and John Preskill. Simple proof of security of the BB84 quantum key distribution protocol.Physical Review Letters, 85(2):441–444, 2000

    work page 2000

  33. [33]

    Secure quantum key distribution with realistic devices.Rev

    Feihu Xu, Xiongfeng Ma, Qiang Zhang, Hoi-Kwong Lo, and Jian-Wei Pan. Secure quantum key distribution with realistic devices.Rev. Mod. Phys., 92:025002, May 2020. 19

    work page 2020