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研究生: 徐則楷
Hsu, Tse-Kai
論文名稱: 通訊波段糾纏光子源之表徵與任務導向評估:多光子雜訊分析與遠端態準備
Characterization and Task-Oriented Evaluation of Telecom-Wavelength Entangled Photon Sources: Multi-Photon Noise Analysis and Remote State Preparation
指導教授: 李哲明
Li, Che-Ming
學位類別: 碩士
Master
系所名稱: 工學院 - 工程科學系
Department of Engineering Science
論文出版年: 2026
畢業學年度: 114
語文別: 英文
論文頁數: 123
中文關鍵詞: 量子網路薩格納克干涉儀多光子噪音分析超導奈米線單光子偵測器通訊 波段糾纏光子源遠端態準備
外文關鍵詞: Quantum networks, Sagnac interferometer, multi-photon emission noise, Superconducting nanowire single-photon detector, telecom-wavelength entangled photon source, remote state preparation
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  • 本研究致力於通訊波段糾纏光子源之建立與表徵,並以遠端態準備作為代表性的量子網路任務,探討其任務導向評估之應用潛力。在理論方面,我們建立考量多光子對發射的雜訊模型,並以光子對發射振幅與二階相關函數量化多對效應,分析其對雙光子干涉對比度與四光子狀態保真度下界預測,此模型可用於描述自發參數下轉換光源在不同操作條件下的多光子雜訊貢獻,並提供評估糾纏資源品質的理論基礎。實驗上以兩組薩格納克結構之第二型自發參數下轉換產生通訊波段偏振糾纏光子對,並透過符合計數率、依據理論模型取得的光子對發射振幅,以及超導奈米線單光子偵測器之偵測效率,表徵光源平台在實際操作條件下的光子對產生特性與偵測效能。此外,本研究引入遠端態準備作為具代表性的量子網路任務,用以建立任務導向的糾纏資源評估觀點。整體而言,本研究結合通訊波段糾纏光子源建置、多光子雜訊分析、探測器效能表徵與任務導向評估觀點,建立一套由理論到實驗的評估架構,為後續可擴展光量子網路實驗與遠端量子資訊任務之研究提供基礎。

    This study focuses on the construction and characterization of a telecom-wavelength entangled photon source and investigates its application potential for task-oriented evaluation, using remote state preparation as a representative quantum networking task. On the theoretical side, we develop a noise model that takes multi-photon pair emissions into account. The multi-pair contribution is quantified using the pair-emission amplitude and the second-order correlation function, allowing us to analyze its influence on the two-photon interference visibility and the predicted fidelity lower bound of four-photon states. This model provides a theoretical basis for describing the multi-photon noise contribution of spontaneous parametric down-conversion sources under different operating conditions and for evaluating the quality of entanglement resources. Experimentally, telecom-wavelength polarization-entangled photon pairs are generated using two type-II spontaneous parametric down-conversion sources based on Sagnac interferometer configurations. The photon-pair generation characteristics and detection performance of the source platform under practical operating conditions are characterized through the coincidence count rate, the pair-emission amplitude obtained from the theoretical model, and the detection efficiencies of superconducting nanowire single-photon detectors. In addition, this study introduces remote state preparation as a representative quantum networking task to establish a task-oriented perspective for evaluating entanglement resources. Overall, this work combines the construction of a telecom-wavelength entangled photon source, multi-photon noise analysis, detector-performance characterization, and task-oriented evaluation. It establishes a theoretical-to-experimental evaluation framework and provides a foundation for future studies on scalable photonic quantum network experiments and remote quantum information tasks.

    摘要 i Abstract ii 誌謝 iv Table of Contents vi List of Tables ix List of Figures x Nomenclature xii Chapter 1. Introduction to a telecom-wavelength entangled photon source and its task-focused verification 1 1.1. Background 1 1.2. Motivation 5 1.3. Purpose 7 1.4. Outline 9 Chapter 2. Theoretical framework for multi-photon noise analysis and entanglement-resource evaluation 11 2.1. Theoretical framework of photon-pair generation 12 2.1.1. Type-II spontaneous parametric down-conversion (SPDC) 12 2.1.2. Type-II SPDC in polarization Sagnac interferometers (PSI) 15 2.1.3. State fidelity 17 2.2. Multi-photon pair emission and second-order correlation function (g(2)(0)) 20 2.2.1. Multi-pair emission state 20 2.2.2. Photon detection model and g(2)(0) derivation 25 2.2.3. Multi-photon pair contribution in SPDC within PSI 32 2.2.4. Experimental measurement of g(2)(0) 36 2.3. Theoretical estimation of the fidelity lower bound for four-photon entangled states under multi-photon emission noise 42 2.3.1. Four-photon state model with experimentally determined γ 43 2.3.2. Prediction of the steering witness under multi-photon emission noise 51 2.4. Characterization of superconducting nanowire single-photon detectors(SNSPD) for telecom-wavelength photon-counting measurements 61 2.4.1. SNSPD system description 61 Chapter 3. Experimental characterization of telecom-wavelength entangled photon sources 70 3.1. Experimental preparation for source characterization 70 3.1.1. Temperature controller model of ppKTP 71 3.2. Experimental configuration of the entangled photon source platform for four-photon extension 72 3.2.1. Experimental setup of the entangled photon source platform for four-photon extension 73 3.2.2. Detection efficiency of the SNSPD system 76 3.3. Pump-power dependence of the pair-emission amplitude 80 Chapter 4. Coherence quantum benefit in remote state preparation with an extension to four-photon networks 82 4.1. Coherence quantum benefits in remote state preparation (RSP) 84 4.1.1. Concept of RSP and resource perspective 85 4.1.2. Coherence quantum benefit 87 4.2. Experimental demonstration using a two-photon entangled source 89 4.2.1. Experimental setup and measurement procedure 89 4.3. Theoretical extension to a four-photon RSP network 94 Chapter 5. Summary and Outlook 98 5.1. Summary 98 5.2. Outlook 99 References 102

    [1] S. Wehner, D. Elkouss, and R. Hanson, “Quantum internet: A vision for the road ahead,”Science, vol. 362, no. 6412, p. eaam9288, 2018.
    [2] M. F. Riedel, D. Binosi, R. Thew, and T. Calarco, “The european quantum technologies flagship programme,” Quantum Science and Technology, vol. 2, no. 3, p. 030501, 2017.
    [3] A. Acín, I. Bloch, H. Buhrman, T. Calarco, C. Eichler, J. Eisert, D. Esteve, N. Gisin, S. J. Glaser, F. Jelezko, S. Kuhr, M. Lewenstein, M. F. Riedel, P. O. Schmidt, R. Thew, A. Wallraff, I. Walmsley, and F. K. Wilhelm, “The quantum technologies roadmap: a european community view,” New Journal of Physics, vol. 20, no. 8, p. 080201, 2018.
    [4] M. Riedel, M. Kovacs, P. Zoller, J. Mlynek, and T. Calarco, “Europe's quantum flagship initiative,” Quantum Science and Technology, vol. 4, no. 2, p. 020501, 2019.
    [5] H. J. Kimble, “The quantum internet,” Nature, vol. 453, no. 7198, pp. 1023–1030, 2008.
    [6] A. Pirker, J. Wallnöfer, and W. Dür, “Modular architectures for quantum networks,”New Journal of Physics, vol. 20, no. 5, p. 053054, 2018.
    [7] H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest,“Measurement-based quantum computation,” Nature Physics, vol. 5, no. 1, pp. 19–26, 2009.
    [8] D. E. Browne, E. Kashefi, M. Mhalla, and S. Perdrix, “Generalized flow and deter-minism in measurement-based quantum computation,” New Journal of Physics, vol. 9, no. 8, p. 250, 2007.
    [9] A. Broadbent, J. Fitzsimons, and E. Kashefi, “Universal blind quantum computation,”in 2009 50th annual IEEE symposium on foundations of computer science, pp. 517–526, IEEE, 2009.
    [10] A. K. Pati, “Minimum cbits for remote preparation and measurement of a qubit,” Phys. Rev. A, vol. 63, no. 1, p. 014302, 2000.
    [11] C. H. Bennett, D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, and W. K. Wootters, “Remote state preparation,” Phys. Rev. Lett., vol. 87, p. 077902, 2001.
    [12] I. Marcikic, H. De Riedmatten, W. Tittel, H. Zbinden, and N. Gisin, “Long-distance teleportation of qubits at telecommunication wavelengths,” Nature, vol. 421, no. 6922, pp. 509–513, 2003.
    [13] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels,” Physical review letters, vol. 70, no. 13, p. 1895, 1993.
    [14] Z. Zhao, Y.-A. Chen, A.-N. Zhang, T. Yang, H. J. Briegel, and J.-W. Pan, “Experimental demonstration of five-photon entanglement and open-destination teleportation,”Nature, vol. 430, no. 6995, pp. 54–58, 2004.
    [15] Y.-H. Luo, H.-S. Zhong, M. Erhard, X.-L. Wang, L.-C. Peng, M. Krenn, X. Jiang, L. Li, N.-L. Liu, C.-Y. Lu, et al., “Quantum teleportation in high dimensions,” Physical review letters, vol. 123, no. 7, p. 070505, 2019.
    [16] X.-M. Hu, C. Zhang, B.-H. Liu, Y. Cai, X.-J. Ye, Y. Guo, W.-B. Xing, C.-X. Huang, Y.-F. Huang, C.-F. Li, et al., “Experimental high-dimensional quantum teleportation,”Physical Review Letters, vol. 125, no. 23, p. 230501, 2020.
    [17] R.-B. Jin, M. Takeoka, U. Takagi, R. Shimizu, and M. Sasaki, “Highly efficient entanglement swapping and teleportation at telecom wavelength,” Scientific reports, vol. 5, no. 1, p. 9333, 2015.
    [18] X. Dong, S. Liu, Y.-L. Mao, B. Guo, S. Xu, H. Chen, Y. Guo, Z.-D. Li, and J. Fan “Experimental quantum teleportation of a toffoli gate across three spatially distributed parties in a photonic quantum network,” Optics Express, vol. 32, no. 22, pp. 3967539684, 2024.
    [19] X.-H. Bao, X.-F. Xu, C.-M. Li, and J.-W. Pan, “Quantum teleportation between remote atomic-ensemble quantum memories,” Proceedings of the National Academy of Sciences, vol. 109, pp. 20347–20351, Nov. 2012.
    [20] M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Physical Review A, vol. 59, no. 3, p. 1829, 1999.
    [21] D. Markham and B. C. Sanders, “Erratum: Graph states for quantum secret sharing [phys. rev. a 78, 042309 (2008)],” Physical Review A Atomic, Molecular, and Optical Physics, vol. 83, no. 1, p. 019901, 2011.
    [22] B. Bell, D. Markham, D. Herrera-Martí, A. Marin, W. Wadsworth, J. Rarity, and M. Tame, “Experimental demonstration of graph-state quantum secret sharing,” Nature communications, vol. 5, no. 1, pp. 1–12, 2014.
    [23] Y.-A. Chen, A.-N. Zhang, Z. Zhao, X.-Q. Zhou, C.-Y. Lu, C.-Z. Peng, T. Yang, and J. W. Pan, “Experimental quantum secret sharing and third-man quantum cryptography,”Physical review letters, vol. 95, no. 20, p. 200502, 2005.
    [24] S.-H. Chen, C. Hsu, Y.-C. Kao, B.-Y. Lee, Y.-S. Liu, Y.-N. Chen, and C.-M. Li “Preparing remote states for genuine quantum networks,” Communications Physics, vol. 7, no. 1, p. 144, 2024.
    [25] B. Dakic, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, Č. Brukner, and P. Walther, “Quantum discord as resource for remote state preparation,” Nature Physics, vol. 8, no. 9, pp. 666–670, 2012.
    [26] R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Physical Review Letters, vol. 86, pp. 5188–5191, May 2001.
    [27] V. Danos and E. Kashefi, “Determinism in the one-way model,” Physical Review A, vol. 74, p. 052310, Nov. 2006.
    [28] S. Barz, E. Kashefi, A. Broadbent, J. F. Fitzsimons, A. Zeilinger, and P. Walther “Demonstration of blind quantum computing,” Science, vol. 335, pp. 303–308, Jan. 2012.
    [29] T. Morimae and K. Fujii, “Blind quantum computation protocol in which alice only makes measurements,” Physical Review A, vol. 87, p. 050301(R), May 2013.
    [30] C. Greganti, M.-C. Roehsner, S. Barz, T. Morimae, and P. Walther, “Demonstration of measurement-only blind quantum computing,” New Journal of Physics, vol. 18, p. 013020, Jan. 2016.
    [31] C. Couteau, “Spontaneous parametric down-conversion,” Contemporary Physics, vol. 59, no. 3, pp. 291–304, 2018.
    [32] R. W. Boyd, A. L. Gaeta, and E. Giese, “Nonlinear optics,” in Springer Handbook of Atomic, Molecular, and Optical Physics, pp. 1097–1110, Springer, 2008.
    [33] P. Lambropoulos and D. Petrosyan, Fundamentals of Quantum Information. Springer, 2007.
    [34] P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Physical Review Letters, vol. 75, no. 24, p. 4337, 1995.
    [35] C. Kurtsiefer, M. Oberparleiter, and H. Weinfurter, “Generation of correlated photon pairs in type-ii parametric down conversion—revisited,” Journal of Modern Optics, vol. 48, no. 13, pp. 1997–2007, 2001.
    [36] Y. Shih, “Entangled biphoton source-property and preparation,” Reports on Progress in Physics, vol. 66, no. 6, p. 1009, 2003.
    [37] M. H. Rubin, D. N. Klyshko, Y. Shih, and A. Sergienko, “Theory of two-photon entanglement in type-ii optical parametric down-conversion,” Physical Review A, vol. 50, no. 6, p. 5122, 1994.
    [38] T. B. Pittman, Y. Shih, D. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Physical Review A, vol. 52, no. 5, p. R3429, 1995.
    [39] P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Physical Review A, vol. 60, no. 2, p. R773,1999.
    [40] T. Yang, Q. Zhang, J. Zhang, J. Yin, Z. Zhao, M. Żukowski, Z.-B. Chen, and J.-W. Pan, “All-versus-nothing violation of local realism by two-photon, four dimensional entanglement,” Physical Review Letters, vol. 95, no. 24, p. 240406, 2005.
    [41] Z.-D. Li, X. Yuan, X.-F. Yin, L.-Z. Liu, R. Zhang, Y.-Y. Fei, L. Li, N.-L. Liu, X. Ma, and H. Lu, “Experimental random-party entanglement distillation via weak measurement,”Physical Review Research, vol. 2, no. 2, p. 023047, 2020.
    [42] X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, and Z.-D. Li, “Experimental ten-photon entanglement,” Physical review letters, vol. 117, no. 21, p. 210502, 2016.
    [43] Y.-A. Chen, R. Zhang, Y.-Y. Fei, Z. Liu, X. Zhang, X.-F. Yin, Y. Mao, N.-L. Li, N.-L. Liu, and X. Ma, “Entanglement activation in multiphoton distillation networks,” 2024. keywords: multipartite entanglement.
    [44] C. Zhang, Y.-F. Huang, C.-J. Zhang, J. Wang, B.-H. Liu, C.-F. Li, and G.-C. Guo “Generation and applications of an ultrahigh-fidelity four-photon greenberger-horne-zeilinger state,” Optics Express, vol. 24, no. 24, pp. 27059–27069, 2016.
    [45] Z.-D. Li, X.-F. Yin, Z. Wang, L.-Z. Liu, R. Zhang, Y.-Z. Zhang, X. Jiang, J. Zhang, L. Li, and N.-L. Liu, “Photonic realization of quantum resetting,” Optica, vol. 7, no. 7, pp. 766–770, 2020.
    [46] L.-Z. Liu, Y.-Z. Zhang, Z.-D. Li, R. Zhang, X.-F. Yin, Y.-Y. Fei, L. Li, N.-L. Liu, F. Xu, and Y.-A. Chen, “Distributed quantum phase estimation with entangled photons,” Nature Photonics, vol. 15, no. 2, pp. 137–142, 2021.
    [47] L.-Z. Liu, Y.-Y. Fei, Y. Mao, Y. Hu, R. Zhang, X.-F. Yin, X. Jiang, L. Li, N.-L. Liu, and F. Xu, “Full-period quantum phase estimation,” Physical Review Letters, vol. 130, no. 12, p. 120802, 2023.
    [48] B. Liu, K.-X. Yang, Y.-L. Mao, L. Feng, B. Guo, S. Xu, H. Chen, Z.-D. Li, and J. Fan “Experimental adaptive bayesian estimation for a linear function of distributed phases in photonic quantum networks,” Optica, vol. 11, no. 10, pp. 1419–1424, 2024.
    [49] R.-B. Jin, M. Fujiwara, T. Yamashita, S. Miki, H. Terai, Z. Wang, K. Wakui, R. Shimizu, and M. Sasaki, “Efficient detection of an ultra-bright single-photon source using superconducting nanowire single-photon detectors,” Optics Communications, vol. 336, pp. 47–54, 2015.
    [50] L. D. S. Martins, N. Laurent-Puig, P. Lefebvre, S. Neves, and E. Diamanti “Realizing a compact, high-fidelity, telecom-wavelength source of multipartite entangled photons,”arXiv preprint arXiv:2407.00802, 2024.
    [51] D. Wu, Q. Zhao, X.-M. Gu, H.-S. Zhong, Y. Zhou, L.-C. Peng, J. Qin, Y.-H. Luo, K. Chen, and L. Li, “Robust self-testing of multiparticle entanglement,” Physical Review Letters, vol. 127, no. 23, p. 230503, 2021.
    [52] C. Zhang, Y. Li, X.-M. Hu, Y. Xiang, C.-F. Li, G.-C. Guo, J. Tura, Q. Gong, Q. He, and B.-H. Liu, “Randomness versus nonlocality in multi-input and multi-output quantum scenario,” arXiv preprint arXiv:2408.04566, 2024.
    [53] Y.-L. Mao, H. Chen, B. Guo, S. Liu, Z.-D. Li, M.-X. Luo, and J. Fan, “Certifying network topologies and nonlocalities of triangle quantum networks,” Physical Review Letters, vol. 132, no. 24, p. 240801, 2024.
    [54] D. Wu, Y.-F. Jiang, X.-M. Gu, L. Huang, B. Bai, Q.-C. Sun, X. Zhang, Q. Gong, Y. Mao, and H.-S. Zhong, “Experimental refutation of real-valued quantum mechanics under strict locality conditions,” Physical Review Letters, vol. 129, no. 14, p. 140401, 2022.
    [55] D. Wu, Q. Zhao, C. Wang, L. Huang, Y.-F. Jiang, B. Bai, Y. Zhou, X.-M. Gu, F.-M. Liu, and Y.-Q. Mao, “Closing the locality and detection loopholes in multiparticle entanglement self-testing,” Physical Review Letters, vol. 128, no. 25, p. 250401, 2022.
    [56] Z.-D. Li, Y.-L. Mao, M. Weilenmann, A. Tavakoli, H. Chen, L. Feng, S.-J. Yang, M. O. Renou, D. Trillo, and T. P. Le, “Testing real quantum theory in an optical quantum network,” Physical Review Letters, vol. 128, no. 4, p. 040402, 2022.
    [57] Y.-L. Mao, Z.-D. Li, S. Yu, and J. Fan, “Test of genuine multipartite nonlocality,” Physical Review Letters, vol. 129, no. 15, p. 150401, 2022.
    [58] S. Chen, L.-C. Peng, Y.-P. Guo, X.-M. Gu, X. Ding, R.-Z. Liu, J.-Y. Zhao, X. You, J. Qin, and Y.-F. Wang, “Heralded three-photon entanglement from a single-photon source on a photonic chip,” Physical Review Letters, vol. 132, no. 13, p. 130603, 2024.
    [59] W. McCutcheon, A. Pappa, B. A. Bell, A. McMillan, A. Chailloux, T. Lawson, M. Mafu, D. Markham, E. Diamanti, and I. Kerenidis, “Experimental verification of multipartite entanglement in quantum networks,” Nature communications, vol. 7, no. 1, p. 13251, 2016.
    [60] T. Faleo, E. Brunner, J. W. Webb, A. Pickston, J. Ho, G. Weihs, A. Buchleitner, C. Dittel, G. Dufour, and A. Fedrizzi, “Entanglement-induced collective many-body interference,” Science Advances, vol. 10, no. 35, p. eadp9030, 2024.
    [61] J. W. Webb, J. Ho, F. Grasselli, G. Murta, A. Pickston, A. Ulibarrena, and A. Fedrizzi,“Experimental anonymous quantum conferencing,” Optica, vol. 11, no. 6, pp. 872–875, 2024.
    [62] H.-P. Lo, T. Ikuta, K. Azuma, T. Honjo, W. J. Munro, and H. Takesue, “Generation of a time-bin greenberger–horne–zeilinger state with an optical switch,” Quantum Science and Technology, vol. 8, no. 3, p. 035003, 2023.
    [63] T. Kim, M. Fiorentino, and F. N. Wong, “Phase-stable source of polarization-entangled photons using a polarization sagnac interferometer,” Physical Review A Atomic, Molecular, and Optical Physics, vol. 73, no. 1, p. 012316, 2006.
    [64] A. Motazedifard, S. A. Madani, J. J. Dashkasan, and N. S. Vayaghan, “Nonlocal realism tests and quantum state tomography in sagnac-based type-ii polarization-entanglement spdc-source,” Heliyon, vol. 7, no. 6, 2021.
    [65] M. Zwerger, W. Dür, and H. J. Briegel, “Experimental verification of multipartite entanglement in quantum networks,” Nature Communications, vol. 7, p. 13251, 2016.
    [66] K. Wei, M.-C. Li, Z.-H. Hu, et al., “Entanglement-induced collective many-body interference,” Science Advances, vol. 8, no. 19, p. eadp9030, 2022.
    [67] J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Reviews of Modern Physics, vol. 84, no. 2, pp. 777–838, 2012.
    [68] J.-W. Pan, C. Simon, Č. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature, vol. 410, no. 6832, pp. 1067–1070, 2001.
    [69] R. W. Boyd, Nonlinear Optics. San Diego: Academic Press, 3 ed., 2008.
    [70] M. M. Fejer, G. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE Journal of quantum electronics, vol. 28, no. 11, pp. 2631–2654, 1992.
    [71] C. Lung, “Experimental realization of telecommunication wavelength four-photon entanglement for quantum state tomography in untrusted quantum networks,” master’s thesis, National Cheng Kung University, Tainan, Taiwan, Jan. 2025.
    [72] C.-C. Chu, “Experimental realization of a telecommunication wavelength four-photon entanglement source for untrusted quantum networking gate-set tomography,” master’s thesis, National Cheng Kung University, Tainan, Taiwan, Jan. 2025.
    [73] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information. Cambridge university press, 2010.
    [74] S. Sempere-Llagostera, G. Thekkadath, R. Patel, W. Kolthammer, and I. Walmsley,“Reducing g (2)(0) of a parametric down-conversion source via photon-number resolution with superconducting nanowire detectors,” Optics Express, vol. 30, no. 2, pp. 3138–3147, 2022.
    [75] A. M. Branczyk, T. C. Ralph, W. Helwig, and C. Silberhorn, “Optimised generation of heralded fock states using parametric down-conversion,” arXiv preprint, 2009.
    [76] A. M. Bránczyk, A. Fedrizzi, T. M. Stace, T. C. Ralph, and A. G. White,“Engineered optical nonlinearity for quantum light sources,” Optics Express, vol. 19, no. 1, pp. 55–65, 2011.
    [77] D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” American Journal of Physics, vol. 58, no. 12, pp. 1131–1143, 1990.
    [78] D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, “Observation of three-photon greenberger-horne-zeilinger entanglement,” Physical Review Letters, vol. 82, no. 7, p. 1345, 1999.
    [79] C.-Y. Lu, X.-Q. Zhou, O. Gühne, W.-B. Gao, J. Zhang, Z.-S. Yuan, A. Goebel, T. Yang, and J.-W. Pan, “Experimental entanglement of six photons in graph states,” Nature Physics, vol. 3, pp. 91–95, 2007.
    [80] C.-M. Li, “Benchmarking quantum entanglement networks with system-size independent measurements,” 2025. Manuscript in preparation.
    [81] I. Quantique, “Idqube,” https://www.idquantique.com/quantum-detection systems/products/id-qube-nir-gated/.
    [82] C. M. Natarajan, M. G. Tanner, and R. H. Hadfield, “Superconducting nanowire single-photon detectors: physics and applications,” Superconductor Science and Technology, vol. 25, no. 6, p. 063001, 2012.
    [83] ID Quantique, ID281 Cryogenic System User Manual. ID Quantique SA, 2022. Accessed from manufacturer documentation.
    [84] Single Quantum, Superconducting Nanowire Single Photon Detectors: Operation Principle. Single Quantum B.V., 2022. Accessed from product documentation.
    [85] I. Quantique, “Id1000,” https://www.idquantique.com/quantum-detection systems/products/id1000-time-controller/.
    [86] A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, “A wavelength tunable fiber-coupled source of narrowband entangled photons,” Optics Express, vol. 15, no. 23, pp. 15377–15386, 2007.
    [87] C. Hsu, “Photonic dynamics and entangled photon source:from identification of experimental non-markovian dynamics to generation of polarization entangled photonsource based on sagnac interferometer,” master’s thesis, National Cheng Kung University, Tainan, Taiwan, Aug. 2022.
    [88] B.-Y. Lee, “Experimental photonic quantum tomography using untrusted devices,” National Cheng Kung University, 2022.
    [89] Thorlabs, “Mpc320,” https://www.thorlabs.com/thorproduct.cfm?partnumber=MPC320.
    [90] F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nature Photonics, vol. 7, pp. 210–214, 2013.
    [91] R. H. Hadfield, “Single-photon detectors for optical quantum information applications,”Nature photonics, vol. 3, no. 12, pp. 696–705, 2009.
    [92] S. Wehner, D. Elkouss, and R. Hanson, “Quantum internet: A vision for the road ahead,”Science, vol. 362, no. 6412, p. eaam9288, 2018.
    [93] Y.-S. Liu, “Verification of necessary and sufficient quantum beneficial condition of quantum remote state preparation in entangled photon sagnac interferometer,” master’s thesis, National Cheng Kung University, Tainan, Taiwan, 2023.
    [94] Y.-S. Liu, S.-H. Chen, B.-Y. Lee, C. Hsu, G.-Y. Chen, Y.-N. Chen, and C.-M. Li, “Efficient detection of preparing quantum remote states using coherence quantum benefits,”arXiv preprint arXiv:2401.03674, 2024.
    [95] C.-M. Li, N. Lambert, Y.-N. Chen, G.-Y. Chen, and F. Nori, “Witnessing quantum coherence: from solid-state to biological systems,” Scientific Reports, vol. 2, p. 885, 2012.
    [96] R. F. Werner, “Quantum states with einstein-podolsky-rosen correlations admitting a hidden-variable model,” Physical Review A, vol. 40, no. 8, pp. 4277–4281, 1989.
    [97] H.-P. Lo, K. Asaoka, and H. Takesue, “Improving quantum interference visibility between independent sources by enhancing the purity of correlated photon pairs,”Japanese Journal of Applied Physics, vol. 65, no. 2, p. 022004, 2026.

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