與裝置無關量子密鑰分發和單邊與裝置無關量子密鑰分發是兩個基於量子物理定律的使用糾纏態之量子密鑰分發方案，分別依靠違反貝爾不等式和愛因斯坦—波多爾斯基—羅森操控性不等式來保障其安全性。前者不須對協定中所用的裝置做任何假設，即兩位參與者的裝置皆為不可信任的，而在後者中只有其中一位的不可信任。在此研究中，我們提出兩個量子密鑰分發協定，高效率與裝置無關量子密鑰分發協定和高效率單邊與裝置無關量子密鑰分發協定，透過由其中一位參與者用糾纏態準備特定量子態並透過量子通道傳送至另一方做測量，並使用過程斷層掃描來特徵化所使用之傳送過程以保障安全性。相同情境下，在保證集體攻擊下之安全性上，我們提出的前者協定比一般的協定更有效率，節省一百倍以上所需的最少糾纏態與測量的資源，且能容忍安全密鑰的量子位元錯誤率高達6.5%。而當以往的單邊與裝置無關量子密鑰分發協定無用以測量之基底的錯誤率低於平均三個測量基底的錯誤率，我們提出的後者協定比其更有效率。再者，透過改用最小錯誤率的測量基底以得到密鑰，後者比以往的更有效率。這些優勢利於參與者節省在密鑰分發上所需的糾纏態和測量的資源。此外，以目前最先進的光子實驗所能達到的最高偵測效率來看，我們提出的協定能在非零的密鑰產生率下被安全地實現，且後者所需的偵測效率比前者的更低。用過程斷層掃描的安全性分析方法使我們提出的協定高效率，並能應用於兩種情境下識別一般準備並測量量子態的量子資訊任務。

The device-independent quantum key distribution (DIQKD) and the one-sided device-independent quantum key distribution (1SDIQKD) are two entanglement-based key distribution schemes based on the laws of quantum physics, whose security relies on a Bell-inequality violation and an Einstein-Podolsky-Rosen-steering inequality violation, respectively. DIQKD does not require any assumptions about the devices used in the protocol, namely both participants' devices are untrusted in DIQKD, while only one of participants' devices is untrusted in 1SDIQKD. Here, we propose two quantum key distribution (QKD) protocols, named an efficient device-independent QKD (EDIQKD) protocol and an efficient one-sided device-independent QKD (E1SDIQKD) protocol, in which one participant prepares states and transmits them through a quantum channel to the other participant to measure, and the process between participants is characterized according to the process tomography (PT) for security. Comparing the minimum number of rounds to guarantee security against collective attacks, the efficiency of EDIQKD protocol is two orders of magnitude more than that of DIQKD protocol for the reliable key of which quantum bit error rate is allowed up to 6.5%. E1SDIQKD protocol can be more efficient when the error rate of the measurement setting which is not used in the usual 1SDIQKD is lower than the average error rate of three measurement settings. Furthermore, E1SDIQKD is more efficient than the usual 1SDIQKD through changing key basis to the measurement basis of which the error rate is minimum. These advantages help participants to conserve demanded resources of the entangled pairs and the measurement. Besides, considering the highest detection efficiency which can be achieved in the recent most advanced photonic experiment, we found that our protocols can be realized with non-zero key rate and that E1SDIQKD demands lower detection efficiency than EDIQKD. Security analyses by PT make our protocols efficient, and these can be applied to identifying the usual prepare-and-measure quantum information tasks with device-independent scenario and one-sided device-independent scenario, respectively.

[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information. Cambridge Univ. Press, 2000.
[2] C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, pp. 175–179, 1984.
[3] A. K. Ekert, “Quantum cryptography based on Bell's theorem,” Phys. Rev. Lett., vol. 67, pp. 661–663, 1991.
[4] A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Deviceindependent security of quantum cryptography against collective attacks,” Phys. Rev. Lett., vol. 98, p. 230501, 2007.
[5] S. Pironio, A. Acín, N. Brunner, N. Gisin, S. Massar, and V. Scarani, “Deviceindependent quantum key distribution secure against collective attacks,” New J. Phys., vol. 11, p. 045021, 2009.
[6] F. Xu, X. Ma, Q. Zhang, H.-K. Lo, and J.-W. Pan, “Secure quantum key distribution with realistic devices,” Rev. Mod. Phys., vol. 92, p. 025002, 2020.
[7] W.-Z. Liu, Y.-Z. Zhang, Y.-Z. Zhen, M.-H. Li, Y. Liu, J. Fan, F. Xu, Q. Zhang, and J.-W. Pan, “Toward a photonic demonstration of device-independent quantum key distribution,” Phys. Rev. Lett., vol. 129, p. 050502, 2022.
[8] C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “Onesided device-independent quantum key distribution: Security, feasibility, and the connection with steering,” Phys. Rev. A, vol. 85, p. 010301, 2012.
[9] T. Moroder, O. Gühne, N. Beaudry, M. Piani, and N. Lütkenhaus, “Entanglement verification with realistic measurement devices via squashing operations,” Phys. Rev. A, vol. 81, p. 052342, 2010.
[10] I. W. Primaatmaja, K. T. Goh, E. Y.-Z. Tan, J. T.-F. Khoo, S. Ghorai, and C. C.-W. Lim, “Security of device-independent quantum key distribution protocols: a review,” Quantum, vol. 7, p. 932, 2023.
[11] V. Zapatero, T. van Leent, R. Arnon-Friedman, W.-Z. Liu, Q. Zhang, H. Weinfurter, and M. Curty, “Advances in device-independent quantum key distribution,” npj Quantum Inf., vol. 9, p. 10, 2023.
[12] S.-H. Chen, H. Lu, Q.-C. Sun, Q. Zhang, Y.-A. Chen, and C.-M. Li, “Discriminating quantum correlations with networking quantum teleportation,” Phys. Rev. Res., vol. 2, p. 013043, 2020.
[13] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems. Oxford Univ. Press, 2002.
[14] J.-H. Hsieh, S.-H. Chen, and C.-M. Li, “Quantifying quantum-mechanical processes,” Sci. Rep., vol. 7, p. 13588, 2017.
[15] H. Kragh, Quantum generations: A history of physics in the twentieth century. Princeton Univ. Press, 2002.
[16] K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A, vol. 40, p. 2847, 1989.
[17] U. Leonhardt, Measuring the quantum state of light. Cambridge Univ. Press, 1997.
[18] K. Jones, “Fundamental limits upon the measurement of state vectors,” Phys. Rev. A, vol. 50, p. 3682, 1994.
[19] I. L. Chuang and M. A. Nielsen, “Prescription for experimental determination of the dynamics of a quantum black box,” J. Mod. Opt., vol. 44, pp. 2455–2467, 1997.
[20] J. Poyatos, J. I. Cirac, and P. Zoller, “Complete characterization of a quantum process: the two-bit quantum gate,” Phys. Rev. Lett., vol. 78, p. 390, 1997.
[21] A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Phys. Rev., vol. 47, p. 777, 1935.
[22] N. D. Mermin, “Is the moon there when nobody looks? Reality and the quantum theory,” Phys. Today, vol. 38, pp. 38–47, 1985.
[23] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature, vol. 464, pp. 45–53, 2010.
[24] N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics, vol. 1, pp. 165–171, 2007.
[25] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys., vol. 86, pp. 419–478, 2014.
[26] K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A, vol. 40, p. 2847(R), 1989.
[27] J. Lofberg, “YALMIP: A toolbox for modeling and optimization in MATLAB,” in 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508), pp. 284–289, 2004.
[28] K. C. Toh, M. J. Todd, and R. H. Tütüncü, “SDPT3 – A MATLAB software package for semidefinite programming,” Optim. Methods Softw., vol. 11, pp. 545–581, 1999.
[29] A. Pais, “Einstein and the quantum theory,” Rev. Mod. Phys., vol. 51, pp. 863–914, 1979.
[30] C. Wagenknecht, C.-M. Li, A. Reingruber, X.-H. Bao, A. Goebel, Y.-A. Chen, Q. Zhang, K. Chen, and J.-W. Pan, “Experimental demonstration of a heralded entanglement source,” Nat. Photon, vol. 4, pp. 549–552, 2010.
[31] I. L. Chuang and M. A. Nielsen, “Prescription for experimental determination of the dynamics of a quantum black box,” J. Mod. Opt., vol. 44, pp. 2455–2467, 1997.
[32] G. Murta, S. B. van Dam, J. Ribeiro, R. Hanson, and S. Wehner, “Towards a realization of device-independent quantum key distribution,” Quantum Sci. Technol., vol. 4, p. 035011, 2019.
[33] M. Iqbal, L. Velasco, M. Ruiz, A. Napoli, J. Pedro, and N. Costa, “Quantum bit retransmission using universal quantum copying machine,” in 2022 International Conference on Optical Network Design and Modeling (ONDM), pp. 1–3, 2022.
[34] V. Scarani, S. Iblisdir, N. Gisin, and A. Acín, “Quantum cloning,” Rev. Mod. Phys., vol. 77, pp. 1225–1256, 2005.
[35] A. Ferenczi and N. Lütkenhaus, “Symmetries in quantum key distribution and the connection between optimal attacks and optimal cloning,” Phys. Rev. A, vol. 85, p. 052310, 2012.
[36] C.-Y. Chiu, N. Lambert, T.-L. Liao, F. Nori, and C.-M. Li, “No-cloning of quantum steering,” npj Quantum Inf., vol. 2, p. 16020, 2016.
[37] I. Devetak and A.Winter, “Distillation of secret key and entanglement from quantum states,” Proc. R. Soc. A, vol. 461, p. 207, 2005.
[38] L. Sheridan and V. Scarani, “Security proof for quantum key distribution using qudit systems,” Phys. Rev. A, vol. 82, p. 030301, 2010.
[39] A. Acín, S. Massar, and S. Pironio, “Efficient quantum key distribution secure against no-signalling eavesdroppers,” New J. Phys., vol. 8, p. 126, 2006.
[40] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett., vol. 23, pp. 880–884, 1969.
[41] R. Arnon-Friedman, F. Dupuis, O. Fawzi, R. Renner, and T. Vidick, “Practical deviceindependent quantum cryptography via entropy accumulation,” Nat. Commun., vol. 9, p. 459, 2018.
[42] M. Tomamichel and A. Leverrier, “A largely self-contained and complete security proof for quantum key distribution,” Quantum, vol. 1, p. 14, 2017.
[43] If the experimental data can not be used to build a completely positive process matrix, we use maximum-likelihood estimation (MLE) technique [62] to estimate the process matrix, and the threshold of detection efficiency will rise about 0.2% ∼ 0.4%. For the results shown in Fig. 3.5 with MLE, the threshold of detection efficiency for key generation in our protocol becomes 88.9% with the entangled state fidelity 99.52% (plotted by blue curve), and the detection efficiency threshold drops to 88.5% for the entangled state fidelity 100% (plotted by orange curve). With random post-selection, noisy preprocessing and practical imperfections, the threshold of detection efficiency becomes 88.8% (plotted by purple curve).
[44] Practical optical elements include half-wave plates (HWPs), dual-wavelength HWPs, quarter-wave plates, dichoric mirrors, spherical lenses, aspherical lenses, periodically poled potassium titanyl phosphate crystals, polarizing beam splitters (PBSs), dualwavelength PBSs, fiber couplers, and superconducting nanowire single-photon detectors.
[45] N.-N. Huang, J.-C. Xu, and C.-M. Li, “Device-independent quantum computing,” in OSA Quantum 2.0 Conference, p. QTu8B.11, Optica Publishing Group, 2020.
[46] S. Wehner, D. Elkouss, and R. Hanson, “Quantum internet: A vision for the road ahead,” Science, vol. 362, p. eaam9288, 2018.
[47] S. Bäuml, S. Das, and M. M. Wilde, “Fundamental limits on the capacities of bipartite quantum interactions,” Phys. Rev. Lett., vol. 121, p. 250504, 2018.
[48] W.-H. Huang, S.-H. Chen, C.-H. Chang, T.-L. Hsu, K.-J. Wang, and C.-M. Li, “Quantum correlation generation capability of experimental processes,” Adv. Quantum Technol., vol. 2300113, 2023.
[49] Y. Wang, W.-S. Bao, H.-W. Li, C. Zhou, and Y. Li, “Finite-key analysis for one-sided device-independent quantum key distribution,” Phys. Rev. A, vol. 88, p. 052322, 2013.
[50] E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the einstein-podolsky-rosen paradox,” Phys. Rev. A, vol. 80, p. 032112, 2009.
[51] C. Jebarathinam, A. Khan, S. Kanjilal, and D. Home, “Revealing the quantitative relation between simultaneous correlations in complementary bases and quantum steering for two-qubit bell diagonal states,” Phys. Rev. A, vol. 98, p. 042306, 2018.
[52] M. Tomamichel and R. Renner, “Uncertainty relation for smooth entropies,” Phys. Rev. Lett., vol. 106, p. 110506, 2011.
[53] J.-S. Xu, X.-Y. Xu, C.-F. Li, C.-J. Zhang, X.-B. Zou, and G.-C. Guo, “Experimental investigation of classical and quantum correlations under decoherence,” Nat. Commun., vol. 1, p. 7, 2010.
[54] A. Perez-Leija, D. Guzmán-Silva, R. de J. León-Montiel, M. Gräfe, M. Heinrich, H. Moya-Cessa, K. Busch, and A. Szameit, “Endurance of quantum coherence due to particle indistinguishability in noisy quantum networks,” npj Quantum Inf., vol. 4, p. 45, 2018.
[55] T. Gehring, V. Händchen, J. Duhme, F. Furrer, T. Franz, C. Pacher, R. F. Werner, and R. Schnabel, “Implementation of continuous-variable quantum key distribution with composable and one-sided-device-independent security against coherent attacks,” Nat. Commun., vol. 6, p. 8795, 2015.
[56] N. Walk, S. Hosseini, J. Geng, O. Thearle, J. Y. Haw, S. Armstrong, S. M. Assad, J. Janousek, T. C. Ralph, T. Symul, H. M. Wiseman, and P. K. Lam, “Experimental demonstration of gaussian protocols for one-sided device-independent quantum key distribution,” Optica, vol. 3, pp. 634–642, 2016.
[57] S. Goswami, B. Bhattacharya, D. Das, S. Sasmal, C. Jebaratnam, and A. S. Majumdar, “One-sided device-independent self-testing of any pure two-qubit entangled state,” Phys. Rev. A, vol. 98, p. 022311, 2018.
[58] A. Gheorghiu, P. Wallden, and E. Kashefi, “Rigidity of quantum steering and one-sided device-independent verifiable quantum computation,” New J. Phys., vol. 19, p. 023043, 2017.
[59] H.-K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” Phys. Rev. Lett., vol. 108, p. 130503, 2012.
[60] T.-T. Song, Q.-Y. Wen, F.-Z. Guo, and X.-Q. Tan, “Finite-key analysis for measurementdevice-independent quantum key distribution,” Phys. Rev. A, vol. 86, p. 022332, 2012.
[61] B. A. Bell, M. S. Tame, A. S. Clark, R. W. Nock, W. J. Wadsworth, and J. G. Rarity, “Experimental characterization of universal one-way quantum computing,” New J. Phys., vol. 15, p. 053030, 2013.
[62] J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K. Langford, T. C. Ralph, and A. G. White, “Quantum process tomography of a controlled-not gate,” Phys. Rev. Lett., vol. 93, p. 080502, 2004.
[63] M.-D. Choi, “Completely positive linear maps on complex matrices,” Linear Algebra Appl., vol. 10, pp. 285–290, 1975.
[64] R. Bhatia, Matrix Analysis. Springer International Publishing, 1997.
[65] M. Tomamichel, Quantum Information Processing with Finite Resources. Springer International Publishing, 2016.
[66] Y. Guo and S. Wu, “Quantum correlation exists in any non-product state,” Sci. Rep., vol. 4, p. 7179, 2014.