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arxiv: 2606.29772 · v1 · pith:G2MTPXKSnew · submitted 2026-06-29 · 🪐 quant-ph

Multipartite quantum resource distillation through local measurement programs

Yan Wang , Shao-Qi Lin , Xi-Nuo Tao , Li-Jiong Shen , Yong-Nan Sun , Ze-Yan Hao , Kai Sun , Qi-Ping Su
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Chui-Ping Yang
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Pith reviewed 2026-06-30 06:18 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum resource distillationlocal measurement programmultipartite steeringphotonic systemsquantum networksentanglement
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The pith

Local measurement programs distill multipartite quantum resources from noisy states without extra copies.

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

The paper introduces local measurement programs (LMP) as a way to distill quantum resources such as steering by converting completely positive maps into sequences of programmable local measurements. This replaces conventional needs for multiple resource copies or dedicated filters. Experiments in photonic setups show the method works for both bipartite and tripartite systems, activating and strengthening multipartite steering. The same framework also covers virtual distillation techniques. The result is a hardware-friendly route to improving resources in multi-user quantum networks.

Core claim

Quantum resource distillation is performed by local measurement programs that realize completely positive trace-preserving maps through programmable local measurements, with experimental demonstrations in photonic bipartite and tripartite systems that activate and enhance multipartite steering, and with virtual distillation naturally included in the same framework.

What carries the argument

Local measurement program (LMP) that transfers completely positive maps into programmable measurement processes.

If this is right

  • Distillation proceeds without requiring additional resource copies or physical filtering elements.
  • Multipartite steering can be activated and enhanced in tripartite photonic systems.
  • Virtual resource distillation fits inside the same LMP framework.
  • The method scales to multipartite and higher-dimensional systems for network use.

Where Pith is reading between the lines

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

  • Existing photonic hardware nodes could perform distillation locally with only measurement control.
  • The approach may extend to distilling other resources such as coherence in similar network settings.
  • Larger networks might use LMP sequences to counteract cumulative channel loss across multiple links.

Load-bearing premise

That the local measurement programs can be realized in photonic hardware with sufficient fidelity and without introducing unaccounted noise or loss that would negate the distillation gain.

What would settle it

An experiment in which the output state after the LMP sequence shows lower steering strength or fidelity than the input state due to accumulated experimental errors.

Figures

Figures reproduced from arXiv: 2606.29772 by Chui-ping Yang, Kai Sun, Li-Jiong Shen, Qi-ping Su, Shao-qi Lin, Xi-Nuo Tao, Yan Wang, Yong-nan Sun, Ze-Yan Hao.

Figure 1
Figure 1. Figure 1: Schematic illustration. (a) The local filtering dis￾tillation is constructed through performing local operations Λ(ρ) on the initial state ρ. (b) Local measurement program (LMP) distillation implements the filtering map on the mea￾surement basis in ΛF (M). (c) LMP distillation in multipartite systems with local operations selectively performing on i-th partite Λ i F (M) (i ∈ {a, b, d}). physical local filt… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of experiment. (a) Pairs of entan￾gled photons (|Φ(θ)b⟩) are generated via parametric down￾conversion in a periodically poled potassium titanyl phos￾phate (PPKTP) crystal, with one output sent to beam dis￾placer (BD) to further prepare tripartite states (|Φ(θ)t⟩). The noise module (NM) placed on Alice and Charlie’s sides are de￾tailed in the blue box. Three dotted boxes are the measure￾ment ap… view at source ↗
Figure 4
Figure 4. Figure 4: Experimental results in the tripartite case. The blue circles and rhombuses (red squares and triangles) repre￾sent experimental results before (after) distillation, and the solid and hollow markers respectively represent the entangle￾ment (E) and steering witness (S), along with the theoretical analysis in solid lines. The parameters Sa|bc (Sab|c) in (c) and (d) denote the steering witness from Alice to Bo… view at source ↗
Figure 5
Figure 5. Figure 5: The process of LMP in a practical experiment. Leong-Chuan Kwek, and Alán Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022). [7] Daniel A. Lidar, Review of decoherence-free subspaces, noiseless subsystems, and dynamical decoupling, in Quantum Information and Computation for Chemistry (John Wiley, Sons, Ltd, 2014) pp. 295–354. [8] Ryszard Horodecki, Paweł Horodecki,… view at source ↗
read the original abstract

Distributed quantum resources in practical multi-user quantum networks are inevitably degraded by environmental noise, channel loss, and device-induced imperfections. To address these issues, quantum resource distillation offers a fundamental approach to recovering stronger resources from imperfect states. However, conventional implementations often require additional copies, dedicated physical filtering elements, or restrict to bipartite systems, posing challenges for scalable multipartite networks. Here, we introduce the method of quantum resource distillation based on the local measurement program (LMP), which transfers completely positive maps into programmable measurement processes. We experimentally demonstrate the performance of resource distillation through LMP in both bipartite and tripartite photonic systems, including the activation and enhancement of multipartite steering configurations. To demonstrate the flexibility and extensibility of the LMP framework, we also show that virtual resource distillation can be naturally reformulated within it. Our results establish a programmable and experimentally economical approach for distilling quantum resources in multipartite and higher-dimensional systems, thereby providing a practical route toward scalable 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

0 major / 3 minor

Summary. The manuscript introduces local measurement programs (LMP) as a method to realize completely positive maps via programmable local measurements for distilling quantum resources. It experimentally demonstrates LMP-based distillation in bipartite and tripartite photonic systems, including quantitative improvements in resource quantifiers, activation and enhancement of multipartite steering, and reformulation of virtual resource distillation within the LMP framework.

Significance. If the experimental results hold, the LMP approach provides a flexible, programmable, and copy-efficient route to multipartite resource distillation that avoids dedicated filters or additional state copies, directly addressing scalability challenges in noisy quantum networks. The photonic implementations with explicit optical setups, post-selection, and reported before/after quantifiers with error bars constitute a concrete experimental validation; the natural inclusion of virtual distillation further demonstrates extensibility of the framework.

minor comments (3)
  1. [§3.2, Fig. 4] §3.2 and Fig. 4: the caption for the tripartite steering activation data should explicitly state the number of experimental runs and the precise definition of the steering quantifier used (e.g., whether it is the steering robustness or a normalized witness).
  2. [Eq. (7)] Eq. (7): the normalization factor in the definition of the LMP Kraus operators appears to depend on the input state; clarify whether this is by design or if an alternative state-independent normalization is possible.
  3. [Table 1] Table 1: the reported fidelity values for the distilled states lack a direct comparison column to the theoretical maximum achievable under the same loss model; adding this would strengthen the claim of near-optimal performance.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report correctly identifies the core contributions of the LMP framework for programmable multipartite resource distillation and its experimental demonstration in photonic systems.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript is an experimental demonstration of LMP-based resource distillation in photonic systems for bipartite and tripartite cases. LMP is introduced as a definition that maps CP maps to programmable measurements, with explicit optical setups, post-selection, and before/after quantifiers reported with error bars. No equations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided text; the central claims rest on measured data rather than any derivation that reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no free parameters, axioms, or invented entities can be identified from the provided text.

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Works this paper leans on

58 extracted references · 1 canonical work pages

  1. [1]

    Otfried Gühne, Erkka Haapasalo, Tristan Kraft, Juha- Pekka Pellonpää, and Roope Uola, Colloquium: Incom- patible measurements in quantum information science, Rev. Mod. Phys.95, 011003 (2023)

    2023

  2. [2]

    Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K

    Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters, Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels, Phys. Rev. Lett. 70, 1895–1899 (1993)

    1993

  3. [3]

    Xiao-Min Hu, Yu Guo, Bi-Heng Liu, Chuan-Feng Li, and Guang-Can Guo, Progress in quantum teleportation, Nat. Rev. Phys.5, 339–353 (2023)

    2023

  4. [4]

    C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys.89, 035002 (2017)

    2017

  5. [5]

    Economou, David Elkouss, Paul Hilaire, Liang Jiang, Hoi-Kwong Lo, and Ilan Tzitrin, Quantum repeaters: From quantum networks to the quantum internet, Rev

    Koji Azuma, Sophia E. Economou, David Elkouss, Paul Hilaire, Liang Jiang, Hoi-Kwong Lo, and Ilan Tzitrin, Quantum repeaters: From quantum networks to the quantum internet, Rev. Mod. Phys.95, 045006 (2023)

    2023

  6. [6]

    Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, 7 Fig

    Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, 7 Fig. 5.The process of LMP in a practical experiment. Leong-Chuan Kwek, and Alán Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022)

    2022

  7. [7]

    Lidar, Review of decoherence-free subspaces, noiseless subsystems, and dynamical decoupling, in Quantum Information and Computation for Chemistry (John Wiley, Sons, Ltd, 2014) pp

    Daniel A. Lidar, Review of decoherence-free subspaces, noiseless subsystems, and dynamical decoupling, in Quantum Information and Computation for Chemistry (John Wiley, Sons, Ltd, 2014) pp. 295–354

    2014

  8. [8]

    Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki, Quantum entanglement, Rev. Mod. Phys.81, 865 (2009)

    2009

  9. [9]

    Roope Uola, Ana C. S. Costa, H. Chau Nguyen, and Otfried Gühne, Quantum steering, Rev. Mod. Phys.92, 015001 (2020)

    2020

  10. [10]

    Bennett, Gilles Brassard, Sandu Popescu, Benjamin Schumacher, John A

    Charles H. Bennett, Gilles Brassard, Sandu Popescu, Benjamin Schumacher, John A. Smolin, and William K. Wootters, Purification of noisy entanglement and faith- ful teleportation via noisy channels, Phys. Rev. Lett.76, 722–725 (1996)

    1996

  11. [11]

    David Deutsch, Artur Ekert, Richard Jozsa, Chiara Mac- chiavello, Sandu Popescu, and Anna Sanpera, Quantum privacy amplification and the security of quantum cryp- tography over noisy channels, Phys. Rev. Lett.77, 2818– 2821 (1996)

    1996

  12. [12]

    Daniel Gottesman and Isaac L Chuang, Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations, Nature402, 390–393 (1999)

    1999

  13. [13]

    Jian-Wei Pan, Sara Gasparoni, Rupert Ursin, Gregor Weihs, and Anton Zeilinger, Experimental entanglement purification of arbitrary unknown states, Nature423, 417–422 (2003)

    2003

  14. [14]

    Yue-Yang Fei, Zhenhuan Liu, Rui Zhang, Zhenyu Cai, Xu-FeiYin, YingqiuMao, LiLi, Nai-LeLiu, Yu-AoChen, andJian-WeiPan, Experimentalquantumchannelpurifi- cation, Phys. Rev. Lett.136, 110804 (2026)

    2026

  15. [15]

    Rui Zhang, Yue-Yang Fei, Zhenhuan Liu, Xingjian Zhang, Xu-Fei Yin, Yingqiu Mao, Li Li, Nai-Le Liu, Otfried Gühne, Xiongfeng Ma, et al., Entanglement superactivation in multiphoton distillation networks, arXiv:2510.26290 (2025)

    work page arXiv 2025

  16. [16]

    Photonics3, 687– 695 (2009)

    Jeremy L O’brien, Akira Furusawa, and Jelena Vučković, Photonic quantum technologies, Nat. Photonics3, 687– 695 (2009)

    2009

  17. [17]

    Vatshal Srivastav, Natalia Herrera Valencia, Will Mc- Cutcheon, Saroch Leedumrongwatthanakun, Sébastien Designolle, RoopeUola, NicolasBrunner, andMehulMa- lik, Quick quantum steering: Overcoming loss and noise with qudits, Phys. Rev. X12, 041023 (2022)

    2022

  18. [18]

    G Wendin, Quantum information processing with su- perconducting circuits: a review, Rep. Prog. Phys.80, 106001 (2017)

    2017

  19. [19]

    Nicolas Sangouard, Christoph Simon, Hugues de Ried- matten, and Nicolas Gisin, Quantum repeaters based on atomic ensembles and linear optics, Rev. Mod. Phys.83, 33–80 (2011)

    2011

  20. [20]

    Commun.6, 6787 (2015)

    Koji Azuma, Kiyoshi Tamaki, and Hoi-Kwong Lo, All- photonic quantum repeaters, Nat. Commun.6, 6787 (2015). 8

    2015

  21. [21]

    Stephanie Wehner, David Elkouss, and Ronald Hanson, Quantum internet: A vision for the road ahead, Science 362, eaam9288 (2018a)

  22. [22]

    Dougal Main, Peter Drmota, David P Nadlinger, Ellis M Ainley, Ayush Agrawal, Bethan C Nichol, Raghaven- dra Srinivas, Gabriel Araneda, and David M Lucas, Dis- tributed quantum computing across an optical network link, Nature638, 383–388 (2025)

    2025

  23. [23]

    Eric Chitambar and Gilad Gour, Quantum resource the- ories, Rev. Mod. Phys.91, 025001 (2019)

    2019

  24. [24]

    Xiao-Min Hu, Cen-Xiao Huang, Yu-Bo Sheng, Lan Zhou, Bi-Heng Liu, Yu Guo, Chao Zhang, Wen-Bo Xing, Yun- FengHuang, Chuan-FengLi, andGuang-CanGuo, Long- distance entanglement purification for quantum commu- nication, Phys. Rev. Lett.126, 010503 (2021)

    2021

  25. [25]

    Lan Zhou, Cen-Xiao Huang, Yu-Bo Sheng, Yu Guo, Xiao-Min Hu, Yun-Feng Huang, Chuan-Feng Li, Guang- Can Guo, and Bi-Heng Liu, Observation of residual entanglement in entanglement purification, Phys. Rev. Lett.135, 050801 (2025)

    2025

  26. [26]

    Yuhang Li, Junjing Xing, Dengke Qu, Huixia Gao, Lei Xiao, Jin-Ming Liu, Yunlong Xiao, and Peng Xue, Tem- poral asymmetry in entanglement distillation, Phys. Rev. Lett.135, 170801 (2025)

    2025

  27. [27]

    Bartosz Regula, Probabilistic transformations of quan- tum resources, Phys. Rev. Lett.128, 110505 (2022)

    2022

  28. [28]

    Nicolas Gisin, Hidden quantum nonlocality revealed by local filters, Phys. Lett. A210, 151–156 (1996)

    1996

  29. [29]

    Paul G Kwiat, Salvador Barraza-Lopez, Andre Stefanov, and Nicolas Gisin, Experimental entanglement distilla- tion and hidden non-locality, Nature409, 1014–1017 (2001)

    2001

  30. [30]

    Res.12, 552–562 (2024a)

    Qian-Xi Zhang, Xiao-Xu Fang, and He Lu, Experimental distillation of tripartite quantum steering with an opti- mal local filtering operation, Photon. Res.12, 552–562 (2024a)

  31. [31]

    Tabia, Kai-Siang Chen, Yeong-Cherng Liang, and He Lu, Experimental single- copy distillation of quantumness from higher-dimensional entanglement, Phys

    Xiao-Xu Fang, Gelo Noel M. Tabia, Kai-Siang Chen, Yeong-Cherng Liang, and He Lu, Experimental single- copy distillation of quantumness from higher-dimensional entanglement, Phys. Rev. Lett.134, 150201 (2025)

    2025

  32. [32]

    Tanumoy Pramanik, Young-Wook Cho, Sang-Wook Han, Sang-Yun Lee, Yong-Su Kim, and Sung Moon, Revealing hidden quantum steerability using local filtering opera- tions, Phys. Rev. A99, 030101 (2019)

    2019

  33. [33]

    Dominy, Alireza Shabani, and Daniel A

    Jason M. Dominy, Alireza Shabani, and Daniel A. Lidar, A general framework for complete positivity, Quantum Inf. Process.15, 465–494 (2016)

    2016

  34. [34]

    R. V. Nery, M. M. Taddei, P. Sahium, S. P. Walborn, L. Aolita, and G. H. Aguilar, Distillation of quantum steering, Phys. Rev. Lett.124, 120402 (2020)

    2020

  35. [35]

    Commun.13, 4973 (2022)

    Huan-Yu Ku, Chung-Yun Hsieh, Shin-Liang Chen, Yueh- Nan Chen, and Costantino Budroni, Complete classifica- tion of steerability under local filters and its relation with measurement incompatibility, Nat. Commun.13, 4973 (2022)

    2022

  36. [36]

    Yang Liu, Kaimin Zheng, Haijun Kang, Dongmei Han, Meihong Wang, Lijian Zhang, Xiaolong Su, and Kunchi Peng, Distillation of gaussian einstein-podolsky-rosen steering with noiseless linear amplification, npj Quantum Inf.8, 38 (2022)

    2022

  37. [37]

    Ze-Yan Hao, Yan Wang, Jia-Kun Li, Yu Xiang, Qiong- Yi He, Zheng-Hao Liu, Mu Yang, Kai Sun, Jin-Shi Xu, Chuan-Feng Li, and Guang-Can Guo, Filtering one- way einstein-podolsky-rosen steering, Phys. Rev. A109, 022411 (2024)

    2024

  38. [38]

    Ryuji Takagi, Xiao Yuan, Bartosz Regula, and Mile Gu, Virtual quantum resource distillation: General frame- work and applications, Phys. Rev. A109, 022403 (2024)

    2024

  39. [39]

    Xiao Yuan, Bartosz Regula, Ryuji Takagi, and Mile Gu, Virtual quantum resource distillation, Phys. Rev. Lett. 132, 050203 (2024)

    2024

  40. [40]

    Ting Zhang, Yukun Zhang, Lu Liu, Xiao-Xu Fang, Qian- Xi Zhang, Xiao Yuan, and He Lu, Experimental virtual distillation of entanglement and coherence, Phys. Rev. Lett.132, 180201 (2024b)

  41. [41]

    Neil Dowling, Pavel Kos, and Xhek Turkeshi, Magic re- sources of the heisenberg picture, Phys. Rev. Lett.135, 050401 (2025)

    2025

  42. [42]

    See Supplemental Material for details of local program operation, experimental setup, criterions of quantum en- tanglement and steering in bipartite and tripartite sys- tems

  43. [43]

    Express15, 15377–15386 (2007)

    Alessandro Fedrizzi, Thomas Herbst, Andreas Poppe, Thomas Jennewein, and Anton Zeilinger, A wavelength- tunable fiber-coupled source of narrowband entangled photons, Opt. Express15, 15377–15386 (2007)

    2007

  44. [44]

    Ya Xiao, Xiang-Jun Ye, Kai Sun, Jin-Shi Xu, Chuan- Feng Li, and Guang-Can Guo, Demonstration of multi- setting one-way einstein-podolsky-rosen steering in two- qubit systems, Phys. Rev. Lett.118, 140404 (2017)

    2017

  45. [45]

    Commun.6, 7941 (2015)

    DanielCavalcanti, PaulSkrzypczyk, GHAguilar, Ranieri V Nery, PH Souto Ribeiro, and SP Walborn, Detection of entanglement in asymmetric quantum networks and multipartite quantum steering, Nat. Commun.6, 7941 (2015)

    2015

  46. [46]

    Daniel F. V. James, Paul G. Kwiat, William J. Munro, and Andrew G. White, Measurement of qubits, Phys. Rev. A64, 052312 (2001)

    2001

  47. [47]

    Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys

    William K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett.80, 2245– 2248 (1998)

    1998

  48. [48]

    Yi-Zheng Zhen, Yu-Lin Zheng, Wen-Fei Cao, Li Li, Zeng-Bing Chen, Nai-Le Liu, and Kai Chen, Certifying einstein-podolsky-rosen steering via the local uncertainty principle, Phys. Rev. A93, 012108 (2016)

    2016

  49. [49]

    A. Acín, D. Bruß, M. Lewenstein, and A. Sanpera, Clas- sification of mixed three-qubit states, Phys. Rev. Lett. 87, 040401 (2001)

    2001

  50. [50]

    J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, Quantum state transfer and entanglement distribution among distant nodes in a quantum network, Phys. Rev. Lett.78, 3221–3224 (1997)

    1997

  51. [51]

    L-M Duan, Mikhail D Lukin, J Ignacio Cirac, and Pe- ter Zoller, Long-distance quantum communication with atomic ensembles and linear optics, Nature414, 413–418 (2001)

    2001

  52. [52]

    Stephanie Wehner, David Elkouss, and Ronald Hanson, Quantum internet: A vision for the road ahead, Science 362, eaam9288 (2018b)

  53. [53]

    Filippo Caruso, Vittorio Giovannetti, Cosmo Lupo, and Stefano Mancini, Quantum channels and memory effects, Rev. Mod. Phys.86, 1203–1259 (2014)

    2014

  54. [54]

    Becerra, Jian Fan, and Alan Migdall, Im- plementation of generalized quantum measurements for unambiguous discrimination of multiple non-orthogonal coherent states, Nat

    Fernando E. Becerra, Jian Fan, and Alan Migdall, Im- plementation of generalized quantum measurements for unambiguous discrimination of multiple non-orthogonal coherent states, Nat. Commun.4, 2028 (2013)

    2028

  55. [55]

    Reinhold A. Bertlmann and Nicolai Friis, Quantum chan- nels and quantum operations, in Modern Quantum The- 9 ory: From Quantum Mechanics to Entanglement and Quantum Information (Oxford University Press, Oxford, UK, 2023)

    2023

  56. [56]

    Cerf, Miloslav Dušek, Norbert Lütkenhaus, and Momtchil Peev, The security of practical quantum key distribution, Rev

    Valerio Scarani, Helle Bechmann-Pasquinucci, Nicolas J. Cerf, Miloslav Dušek, Norbert Lütkenhaus, and Momtchil Peev, The security of practical quantum key distribution, Rev. Mod. Phys.81, 1301–1350 (2009)

    2009

  57. [57]

    Neng-Chun Chiu, Elias C Trapp, Jinen Guo, Mohamed H Abobeih, Luke M Stewart, Simon Hollerith, Pavel L Stroganov, Marcin Kalinowski, Alexandra A Geim, Si- mon J Evered, et al., Continuous operation of a coherent 3,000-qubit system, Nature646, 1075–1080 (2025)

    2025

  58. [58]

    Appl.11, 203 (2022)

    Kai Sun, Ze-Yan Hao, Yan Wang, Jia-Kun Li, Xiao- Ye Xu, Jin-Shi Xu, Yong-Jian Han, Chuan-Feng Li, and Guang-Can Guo, Optical demonstration of quantum fault-tolerant threshold, Light Sci. Appl.11, 203 (2022)

    2022