Optimized Entanglement Purification

Stefan Krastanov; Victor V. Albert; Liang Jiang

doi:10.22331/q-2019-02-18-123

Optimized Entanglement Purification

Stefan Krastanov^1,2, Victor V. Albert^1,2,3, and Liang Jiang^1,2

¹Departments of Applied Physics and Physics, Yale University, New Haven, CT 06511, USA
²Yale Quantum Institute, Yale University, New Haven, CT 06520, USA
³Walter Burke Institute for Theoretical Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA

Published:	2019-02-18, volume 3, page 123
Eprint:	arXiv:1712.09762v3
Doi:	https://doi.org/10.22331/q-2019-02-18-123
Citation:	Quantum 3, 123 (2019).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We investigate novel protocols for entanglement purification of qubit Bell pairs. Employing genetic algorithms for the design of the purification circuit, we obtain shorter circuits achieving higher success rates and better final fidelities than what is currently available in the literature. We provide a software tool for analytical and numerical study of the generated purification circuits, under customizable error models. These new purification protocols pave the way to practical implementations of modular quantum computers and quantum repeaters. Our approach is particularly attentive to the effects of finite resources and imperfect local operations - phenomena neglected in the usual asymptotic approach to the problem. The choice of the building blocks permitted in the construction of the circuits is based on a thorough enumeration of the local Clifford operations that act as permutations on the basis of Bell states.

Featured image: A short optimized entanglement purification circuit for a 4-qubit register designed by a genetic algorithm.

► BibTeX data

@article{Krastanov2019optimized,
  doi = {10.22331/q-2019-02-18-123},
  url = {https://doi.org/10.22331/q-2019-02-18-123},
  title = {Optimized {E}ntanglement {P}urification},
  author = {Krastanov, Stefan and Albert, Victor V. and Jiang, Liang},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {123},
  month = feb,
  year = {2019}
}

► References

[1] Julia Cramer, Norbert Kalb, M Adriaan Rol, Bas Hensen, Machiel S Blok, Matthew Markham, Daniel J Twitchen, Ronald Hanson, and Tim H Taminiau. Repeated quantum error correction on a continuously encoded qubit by real-time feedback. Nature communications, 7, 2016. 10.1038/ncomms11526.
https://doi.org/10.1038/ncomms11526

[2] Nissim Ofek, Andrei Petrenko, Reinier Heeres, Philip Reinhold, Zaki Leghtas, Brian Vlastakis, Yehan Liu, Luigi Frunzio, SM Girvin, L Jiang, et al. Extending the lifetime of a quantum bit with error correction in superconducting circuits. Nature, 536 (7617): 441–445, 2016. 10.1038/nature18949.
https://doi.org/10.1038/nature18949

[3] Daniel Gottesman and Isaac L Chuang. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature, 402 (6760): 390–393, 1999. 10.1038/46503.
https://doi.org/10.1038/46503

[4] C Monroe, R Raussendorf, A Ruthven, KR Brown, P Maunz, L-M Duan, and J Kim. Large-scale modular quantum-computer architecture with atomic memory and photonic interconnects. Physical Review A, 89 (2): 022317, 2014. 10.1103/physreva.89.022317.
https://doi.org/10.1103/physreva.89.022317

[5] A Narla, S Shankar, M Hatridge, Z Leghtas, KM Sliwa, E Zalys-Geller, SO Mundhada, W Pfaff, L Frunzio, RJ Schoelkopf, et al. Robust concurrent remote entanglement between two superconducting qubits. Physical Review X, 6 (3): 031036, 2016. 10.1103/physrevx.6.031036.
https://doi.org/10.1103/physrevx.6.031036

[6] D Hucul, IV Inlek, G Vittorini, C Crocker, S Debnath, SM Clark, and Cl Monroe. Modular entanglement of atomic qubits using photons and phonons. Nature Physics, 11 (1): 37–42, 2015. 10.1038/nphys3150.
https://doi.org/10.1038/nphys3150

[7] Ramil Nigmatullin, Christopher J Ballance, Niel de Beaudrap, and Simon C Benjamin. Minimally complex ion traps as modules for quantum communication and computing. New Journal of Physics, 18 (10): 103028, 2016. 10.1088/1367-2630/18/10/103028.
https://doi.org/10.1088/1367-2630/18/10/103028

[8] Andreas Reiserer, Norbert Kalb, Machiel S Blok, Koen JM van Bemmelen, Tim H Taminiau, Ronald Hanson, Daniel J Twitchen, and Matthew Markham. Robust quantum-network memory using decoherence-protected subspaces of nuclear spins. Physical Review X, 6 (2): 021040, 2016. 10.1103/physrevx.6.021040.
https://doi.org/10.1103/physrevx.6.021040

[9] Naomi H Nickerson, Ying Li, and Simon C Benjamin. Topological quantum computing with a very noisy network and local error rates approaching one percent. Nature communications, 4: 1756, 2013. 10.1038/ncomms2773.
https://doi.org/10.1038/ncomms2773

[10] W Dür, H-J Briegel, JI Cirac, and P Zoller. Quantum repeaters based on entanglement purification. Physical Review A, 59 (1): 169, 1999. 10.1103/physreva.59.169.
https://doi.org/10.1103/physreva.59.169

[11] Jian-Wei Pan, Christoph Simon, Časlav Brukner, and Anton Zeilinger. Entanglement purification for quantum communication. Nature, 410 (6832): 1067–1070, 2001. 10.1038/35074041.
https://doi.org/10.1038/35074041

[12] L Childress, JM Taylor, Anders Søndberg Sørensen, and Mikhail D Lukin. Fault-tolerant quantum repeaters with minimal physical resources and implementations based on single-photon emitters. Physical Review A, 72 (5): 052330, 2005. 10.1103/physreva.72.052330.
https://doi.org/10.1103/physreva.72.052330

[13] L Jiang, JM Taylor, and MD Lukin. Fast and robust approach to long-distance quantum communication with atomic ensembles. Physical Review A, 76 (1): 012301, 2007. 10.1103/physreva.76.012301.
https://doi.org/10.1103/physreva.76.012301

[14] Naomi H Nickerson, Joseph F Fitzsimons, and Simon C Benjamin. Freely scalable quantum technologies using cells of 5-to-50 qubits with very lossy and noisy photonic links. Physical Review X, 4 (4): 041041, 2014. 10.1103/physrevx.4.041041.
https://doi.org/10.1103/physrevx.4.041041

[15] DL Moehring, P Maunz, S Olmschenk, KC Younge, DN Matsukevich, L-M Duan, and C Monroe. Entanglement of single-atom quantum bits at a distance. Nature, 449 (7158): 68–71, 2007. 10.1038/nature06118.
https://doi.org/10.1038/nature06118

[16] Wolfgang Pfaff, Tim H Taminiau, Lucio Robledo, Hannes Bernien, Matthew Markham, Daniel J Twitchen, and Ronald Hanson. Demonstration of entanglement-by-measurement of solid-state qubits. Nature Physics, 9 (1): 29–33, 2013. 10.1038/nphys2444.
https://doi.org/10.1038/nphys2444

[17] Bas Hensen, Hannes Bernien, Anaïs E Dréau, Andreas Reiserer, Norbert Kalb, Machiel S Blok, Just Ruitenberg, Raymond FL Vermeulen, Raymond N Schouten, Carlos Abellán, et al. Loophole-free bell inequality violation using electron spins separated by 1.3 kilometres. Nature, 526 (7575): 682–686, 2015. 10.1038/nature15759.
https://doi.org/10.1038/nature15759

[18] Stephan Ritter, Christian Nölleke, Carolin Hahn, Andreas Reiserer, Andreas Neuzner, Manuel Uphoff, Martin Mücke, Eden Figueroa, Joerg Bochmann, and Gerhard Rempe. An elementary quantum network of single atoms in optical cavities. Nature, 484 (7393): 195–200, 2012. 10.1364/icqi.2011.qwb2.
https://doi.org/10.1364/icqi.2011.qwb2

[19] David Deutsch, Artur Ekert, Richard Jozsa, Chiara Macchiavello, Sandu Popescu, and Anna Sanpera. Quantum privacy amplification and the security of quantum cryptography over noisy channels. Physical review letters, 77 (13): 2818, 1996. 10.1103/physrevlett.77.2818.
https://doi.org/10.1103/physrevlett.77.2818

[20] Charles H Bennett, Gilles Brassard, Sandu Popescu, Benjamin Schumacher, John A Smolin, and William K Wootters. Purification of noisy entanglement and faithful teleportation via noisy channels. Physical review letters, 76 (5): 722, 1996a. 10.1103/physrevlett.76.722.
https://doi.org/10.1103/physrevlett.76.722

[21] Wolfgang Dür and Hans J Briegel. Entanglement purification and quantum error correction. Reports on Progress in Physics, 70 (8): 1381, 2007. 10.1088/0034-4885/70/8/r03.
https://doi.org/10.1088/0034-4885/70/8/r03

[22] Keisuke Fujii and Katsuji Yamamoto. Entanglement purification with double selection. Physical Review A, 80 (4): 042308, 2009. 10.1103/physreva.80.042308.
https://doi.org/10.1103/physreva.80.042308

[23] Hans Aschauer. Quantum communication in noisy environments. PhD thesis, lmu, 2005.

[24] John Preskill. Quantum computing in the nisq era and beyond. arXiv preprint arXiv:1801.00862, 2018. 10.22331/q-2018-08-06-79.
https://doi.org/10.22331/q-2018-08-06-79
arXiv:1801.00862

[25] Charles H Bennett, David P DiVincenzo, John A Smolin, and William K Wootters. Mixed-state entanglement and quantum error correction. Physical Review A, 54 (5): 3824, 1996b. 10.1103/physreva.54.3824.
https://doi.org/10.1103/physreva.54.3824

[26] Jeroen Dehaene, Maarten Van den Nest, Bart De Moor, and Frank Verstraete. Local permutations of products of bell states and entanglement distillation. Physical Review A, 67 (2): 022310, 2003. 10.1103/physreva.67.022310.
https://doi.org/10.1103/physreva.67.022310

[27] H Bombin and MA Martin-Delgado. Entanglement distillation protocols and number theory. Physical Review A, 72 (3): 032313, 2005. 10.1103/physreva.72.032313.
https://doi.org/10.1103/physreva.72.032313

[28] Maris Ozols. Clifford group. 2008. 10.1117/12.266854.
https://doi.org/10.1117/12.266854

[29] A Robert Calderbank, Eric M Rains, PM Shor, and Neil JA Sloane. Quantum error correction via codes over gf (4). IEEE Transactions on Information Theory, 44 (4): 1369–1387, 1998. 10.1109/isit.1997.613213.
https://doi.org/10.1109/isit.1997.613213

[30] Thomas Geijtenbeek, Michiel van de Panne, and A Frank van der Stappen. Flexible muscle-based locomotion for bipedal creatures. ACM Transactions on Graphics (TOG), 32 (6): 206, 2013. 10.1145/2508363.2508399.
https://doi.org/10.1145/2508363.2508399

[31] CA Weidner, Hoon Yu, Ronnie Kosloff, and Dana Z Anderson. Atom interferometry using a shaken optical lattice. Physical Review A, 95 (4): 043624, 2017. 10.1103/physreva.95.043624.
https://doi.org/10.1103/physreva.95.043624

[32] PA Knott. A search algorithm for quantum state engineering and metrology. New Journal of Physics, 18 (7): 073033, 2016. 10.1088/1367-2630/18/7/073033.
https://doi.org/10.1088/1367-2630/18/7/073033

[33] Robin Harper, Robert J Chapman, Christopher Ferrie, Christopher Granade, Richard Kueng, Daniel Naoumenko, Steven T Flammia, and Alberto Peruzzo. Explaining quantum correlations through evolution of causal models. Physical Review A, 95 (4): 042120, 2017. 10.1103/physreva.95.042120.
https://doi.org/10.1103/physreva.95.042120

[34] Rami Barends, Julian Kelly, Anthony Megrant, Andrzej Veitia, Daniel Sank, Evan Jeffrey, Ted C White, Josh Mutus, Austin G Fowler, Brooks Campbell, et al. Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature, 508 (7497): 500–503, 2014. 10.1038/nature13171.
https://doi.org/10.1038/nature13171

[35] CJ Ballance, TP Harty, NM Linke, MA Sepiol, and DM Lucas. High-fidelity quantum logic gates using trapped-ion hyperfine qubits. Physical review letters, 117 (6): 060504, 2016. 10.1103/physrevlett.117.060504.
https://doi.org/10.1103/physrevlett.117.060504

[36] Joseph M Renes, David Sutter, Frédéric Dupuis, and Renato Renner. Efficient quantum polar codes requiring no preshared entanglement. IEEE Transactions on Information Theory, 61 (11): 6395–6414, 2015. 10.1109/isit.2013.6620247.
https://doi.org/10.1109/isit.2013.6620247

[37] Lan Zhou and Yu-Bo Sheng. Purification of logic-qubit entanglement. Scientific reports, 6: 28813, 2016. 10.1038/srep28813.
https://doi.org/10.1038/srep28813

[38] Lan Zhou and Yu-Bo Sheng. Polarization entanglement purification for concatenated greenberger–horne–zeilinger state. Annals of Physics, 385: 10–35, 2017. 10.1016/j.aop.2017.07.012.
https://doi.org/10.1016/j.aop.2017.07.012

[39] M Murao, MB Plenio, Sandu Popescu, V Vedral, and PL Knight. Multiparticle entanglement purification protocols. Physical Review A, 57 (6): R4075, 1998. 10.1103/physreva.57.r4075.
https://doi.org/10.1103/physreva.57.r4075

[40] F Fröwis and W Dür. Stable macroscopic quantum superpositions. Physical review letters, 106 (11): 110402, 2011. 10.1103/physrevlett.106.110402.
https://doi.org/10.1103/physrevlett.106.110402

[41] Florian Fröwis and Wolfgang Dür. Stability of encoded macroscopic quantum superpositions. Physical Review A, 85 (5): 052329, 2012. 10.1103/physreva.85.052329.
https://doi.org/10.1103/physreva.85.052329

Cited by

[1] Francesco Preti, Tommaso Calarco, Juan Mauricio Torres, and József Zsolt Bernád, "Optimal two-qubit gates in recurrence protocols of entanglement purification", Physical Review A 106 2, 022422 (2022).

[2] Pei-Shun Yan, Lan Zhou, Wei Zhong, and Yu-Bo Sheng, "Feasible time-bin entanglement purification based on sum-frequency generation", Optics Express 29 2, 571 (2021).

[3] Cheng-Chen Luo, Lan Zhou, Wei Zhong, and Yu-Bo Sheng, "Purification for hybrid logical qubit entanglement", Quantum Information Processing 21 8, 300 (2022).

[4] Guo Xian Yau, Thirupathaiah Vasantam, and Gayane Vardoyan, 2025 International Conference on Quantum Communications, Networking, and Computing (QCNC) 1 (2025) ISBN:979-8-3315-3159-1.

[5] Naphan Benchasattabuse, Michal Hajdušek, and Rodney Van Meter, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 885 (2025) ISBN:979-8-3315-5736-2.

[6] Vaishnavi L. Addala, Shu Ge, and Stefan Krastanov, "Faster-than-Clifford simulations of entanglement purification circuits and their full-stack optimization", npj Quantum Information 11 1, 12 (2025).

[7] Hoon Yu and Sangkyung Lee, "Miniaturizing coil system for uniform magnetic field in atom spin gyroscope by using natural selection algorithm", AIP Advances 15 4, 045104 (2025).

[8] Nikhil Bansal, Wai-Keong Mok, Kishor Bharti, Dax Enshan Koh, and Tobias Haug, "Pseudorandom Density Matrices", PRX Quantum 6 2, 020322 (2025).

[9] Áron Rozgonyi, Gábor Széchenyi, Orsolya Kálmán, and Tamás Kiss, "Training iterated protocols for distillation of GHZ states with variational quantum algorithms", Physics Letters A 499, 129349 (2024).

[10] Vivek Kumar, Nitish K. Chandra, Kaushik P. Seshadreesan, Alan Scheller-Wolf, and Sridhar Tayur, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 1198 (2023) ISBN:979-8-3503-4323-6.

[11] Vikesh Siddhu, Erick Winston, David C. McKay, and Ali Javadi-Abhari, "Basic entanglement distillation with realistic noise", Physical Review Research 7 4, 043069 (2025).

[12] Orsolya Kálmán, Aurél Gábris, Igor Jex, and Tamás Kiss, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 579 (2024) ISBN:979-8-3315-4137-8.

[13] Xuanqiang Zhao, Benchi Zhao, Zihe Wang, Zhixin Song, and Xin Wang, "Practical distributed quantum information processing with LOCCNet", npj Quantum Information 7 1, 159 (2021).

[14] Shinichi Sunami, Shiro Tamiya, Ryotaro Inoue, Hayata Yamasaki, and Akihisa Goban, "Scalable Networking of Neutral-Atom Qubits: Nanofiber-Based Approach for Multiprocessor Fault-Tolerant Quantum Computers", PRX Quantum 6 1, 010101 (2025).

[15] Francisco Ferreira da Silva, Ariana Torres-Knoop, Tim Coopmans, David Maier, and Stephanie Wehner, "Optimizing entanglement generation and distribution using genetic algorithms", Quantum Science and Technology 6 3, 035007 (2021).

[16] Hemant Sharma, Aleksandr Mokeev, Jonas Helsen, and Johannes Borregaard, "Catalytic entanglement transformations with noisy hardware", Quantum 10, 2117 (2026).

[17] Francesco Preti and József Zsolt Bernád, "Statistical evaluation and optimization of entanglement purification protocols", Physical Review A 110 2, 022619 (2024).

[18] Fang-Fang Du, Gang Fan, Yi-Ming Wu, and Bao-Cang Ren, "Faithful and efficient hyperentanglement purification for spatial-polarization-time-bin photon system", Chinese Physics B 32 6, 060304 (2023).

[19] Li Ding and Lee Spector, "Multi-Objective Evolutionary Architecture Search for Parameterized Quantum Circuits", Entropy 25 1, 93 (2023).

[20] Stav Haldar, Pratik J. Barge, Sumeet Khatri, and Hwang Lee, "Fast and reliable entanglement distribution with quantum repeaters: Principles for improving protocols using reinforcement learning", Physical Review Applied 21 2, 024041 (2024).

[21] Kenneth Goodenough, Sébastian de Bone, Vaishnavi Addala, Stefan Krastanov, Sarah Jansen, Dion Gijswijt, and David Elkouss, "Near-Term n to k Distillation Protocols Using Graph Codes", IEEE Journal on Selected Areas in Communications 42 7, 1830 (2024).

[22] Koray Aydoğan, Anthony W. Schlimgen, and Kade Head-Marsden, "Stabilizing steady-state properties of open quantum systems with parameter engineering", Physical Review Research 7 2, 023057 (2025).

[23] Sebastian de Bone, Runsheng Ouyang, Kenneth Goodenough, and David Elkouss, "Protocols for Creating and Distilling Multipartite GHZ States With Bell Pairs", IEEE Transactions on Quantum Engineering 1, 1 (2020).

[24] Josiah Sinclair, Joshua Ramette, Brandon Grinkemeyer, Dolev Bluvstein, Mikhail D. Lukin, and Vladan Vuletić, "Fault-tolerant optical interconnects for neutral-atom arrays", Physical Review Research 7 1, 013313 (2025).

[25] Stefan Krastanov, Hamza Raniwala, Jeffrey Holzgrafe, Kurt Jacobs, Marko Lončar, Matthew J. Reagor, and Dirk R. Englund, "Optically Heralded Entanglement of Superconducting Systems in Quantum Networks", Physical Review Letters 127 4, 040503 (2021).

[26] Lan Zhou and Yu-Bo Sheng, "One-step device-independent quantum secure direct communication", Science China Physics, Mechanics & Astronomy 65 5, 250311 (2022).

[27] Joshua Carlo A. Casapao, Ananda G. Maity, Naphan Benchasattabuse, Michal Hajdušek, Akihito Soeda, Rodney Van Meter, and David Elkouss, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 1089 (2025) ISBN:979-8-3315-5736-2.

[28] Gui-Long Jiang, Wen-Qiang Liu, and Hai-Rui Wei, "Heralded and high-efficient entanglement concentrations based on linear optics assisted by time-delay degree of freedom", Optics Express 30 26, 47836 (2022).

[29] Juan Mauricio Torres, József Zsolt Bernád, and Rocío Gómez-Rosas, "Performance of entanglement purification including maximally entangled mixed states", Journal of Physics A: Mathematical and Theoretical 57 31, 315302 (2024).

[30] J. Pablo Bonilla Ataides, Hengyun Zhou, Qian Xu, Gefen Baranes, Bikun Li, Mikhail D. Lukin, and Liang Jiang, "Constant-Overhead Fault-Tolerant Bell-Pair Distillation Using High-Rate Codes", Physical Review Letters 135 13, 130804 (2025).

[31] Sajede Harraz and Shuang Cong, "Efficient entanglement enhancement of partially entangled pairs", Physica Scripta 99 8, 085110 (2024).

[32] Hudson Leone, Thinh Le, S. Srikara, and Simon Devitt, "Resource overheads and attainable rates for trapped-ion lattice surgery", Physical Review Research 7 2, 023088 (2025).

[33] Sébastian de Bone, Paul Möller, Conor E. Bradley, Tim H. Taminiau, and David Elkouss, "Thresholds for the distributed surface code in the presence of memory decoherence", AVS Quantum Science 6 3, 033801 (2024).

[34] Guus Avis and Stefan Krastanov, "Optimization of quantum-repeater networks using stochastic automatic differentiation", Physical Review Research 7 3, 033111 (2025).

[35] Benjamin Desef and Martin B. Plenio, "Optimizing quantum codes with an application to the loss channel with partial erasure information", Quantum 6, 667 (2022).

[36] Michelle Victora, Spyros Tserkis, Stefan Krastanov, Alexander Sanchez de la Cerda, Steven Willis, and Prineha Narang, "Entanglement purification on quantum networks", Physical Review Research 5 3, 033171 (2023).

[37] Narayanan Rengaswamy, Nithin Raveendran, Ankur Raina, and Bane Vasić, "Entanglement Purification with Quantum LDPC Codes and Iterative Decoding", Quantum 8, 1233 (2024).

[38] Sajede Harraz and Shuang Cong, "Universal Entanglement Protection for Quantum Teleportation Over Amplitude Damping Channels With Various Entangled States", Advanced Quantum Technologies 8 11, e00382 (2025).

[39] F. Riera-Sàbat, P. Sekatski, A. Pirker, and W. Dür, "Entanglement-Assisted Entanglement Purification", Physical Review Letters 127 4, 040502 (2021).

[40] Michele Amoretti, Mattia Pizzoni, and Stefano Carretta, "Enhancing distributed functional monitoring with quantum protocols", Quantum Information Processing 18 12, 371 (2019).

[41] Andi Gu, Lorenzo Leone, Kenneth Goodenough, and Sumeet Khatri, "Constant Overhead Entanglement Distillation via Scrambling", Physical Review Letters 136 11, 110805 (2026).

[42] Sumeet Khatri, Corey T. Matyas, Aliza U. Siddiqui, and Jonathan P. Dowling, "Practical figures of merit and thresholds for entanglement distribution in quantum networks", Physical Review Research 1 2, 023032 (2019).

[43] Yu Shi, Ashlesha Patil, and Saikat Guha, "Stabilizer Entanglement Distillation and Efficient Fault-Tolerant Encoders", PRX Quantum 6 1, 010339 (2025).

[44] Bernard Aondofa Atsuwe, Akpenongun Roy Atser, Newton Felix Gesa, and Iorkyaa Ahemen, "The Role of Quantum Optics in Next-Generation Secure Telecommunications", Nigerian Journal of Physics 35 1, 228 (2026).

[45] André Roque, Diogo Cruz, Francisco A. Monteiro, and Bruno C. Coutinho, "Efficient entanglement purification based on noise guessing decoding", Quantum 8, 1476 (2024).

[46] Álvaro G. Iñesta, Bethany Davies, Sounak Kar, and Stephanie Wehner, "Entanglement buffering with multiple quantum memories", npj Quantum Information 12 1, 64 (2026).

[47] Peng-Liang Guo, Cheng-Yan Gao, and Bao-Cang Ren, "Polarization-transverse-spatial logical qubit entanglement purification using linear optics", Optics & Laser Technology 185, 112566 (2025).

[48] Guus Avis, Robert Knegjens, Anders S. Sørensen, and Stephanie Wehner, "Asymmetric node placement in fiber-based quantum networks", Physical Review A 109 5, 052627 (2024).

[49] Lan Zhou, Wei Zhong, and Yu-Bo Sheng, "Purification of the residual entanglement", Optics Express 28 2, 2291 (2020).

[50] Pei-Shun Yan, Lan Zhou, Wei Zhong, and Yu-Bo Sheng, "Feasible measurement-based entanglement purification in linear optics", Optics Express 29 6, 9363 (2021).

[51] Lin Chen and Ziyue Jia, "On Optimum Entanglement Purification Scheduling in Quantum Networks", IEEE Journal on Selected Areas in Communications 42 7, 1779 (2024).

[52] Stefan Krastanov, Alexander Sanchez de la Cerda, and Prineha Narang, "Heterogeneous multipartite entanglement purification for size-constrained quantum devices", Physical Review Research 3 3, 033164 (2021).

[53] Cheng-Chen Luo, Lan Zhou, Wei Zhong, and Yu-Bo Sheng, "Multipartite entanglement purification using time-bin entanglement ", Laser Physics Letters 18 6, 065205 (2021).

[54] Hugo Jacinto, Élie Gouzien, and Nicolas Sangouard, "Network requirements for distributed quantum computation", Physical Review Research 8 1, 013205 (2026).

[55] 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", Reviews of Modern Physics 95 4, 045006 (2023).

[56] F. Riera-Sàbat, P. Sekatski, A. Pirker, and W. Dür, "Entanglement purification by counting and locating errors with entangling measurements", Physical Review A 104 1, 012419 (2021).

[57] Xiaoyu Liu, Niv Bharos, Liubov Markovich, and Johannes Borregaard, "Error correlations in photonic qudit-mediated entanglement generation", Physical Review Research 6 2, 023075 (2024).

[58] Kun Fang and Zi-Wen Liu, "No-Go Theorems for Quantum Resource Purification", Physical Review Letters 125 6, 060405 (2020).

[59] Sajede Harraz, Jiao-Yang Zhang, and Shuang Cong, "High-fidelity quantum teleportation through noisy channels via weak measurement and environment-assisted measurement", Results in Physics 55, 107164 (2023).

[60] Si-Yi Chen, Gang Xu, Xiu-Bo Chen, Tao Shang, and Yi-Xian Yang, "Quantum Cooperative Multicast in a Quantum Hybrid Topology Network", Frontiers in Physics 10, 842035 (2022).

[61] Gui‐Long Jiang, Hai‐Rui Wei, Guo‐Zhu Song, and Ming Hua, "Synthesis and Upper Bound of Schmidt Rank of Bipartite Controlled‐Unitary Gates", Annalen der Physik 534 11, 2200317 (2022).

[62] Prateek Mantri, Kenneth Goodenough, and Don Towsley, "Comparing one- and two-way quantum repeater architectures", Communications Physics 8 1, 300 (2025).

[63] Derek S. Wang, Tomáš Neuman, and Prineha Narang, "Dipole-coupled emitters as deterministic entangled photon-pair sources", Physical Review Research 2 4, 043328 (2020).

[64] Jude Alnas, Muneer Alshowkan, Nageswara S. V. Rao, Nicholas A. Peters, and Joseph M. Lukens, "Optimal resource allocation for flexible-grid entanglement distribution networks", Optics Express 30 14, 24375 (2022).

[65] Pei-Shun Yan, Lan Zhou, Wei Zhong, and Yu-Bo Sheng, "Advances in quantum entanglement purification", Science China Physics, Mechanics & Astronomy 66 5, 250301 (2023).

[66] Gongyu Ni, Holger Claussen, and Lester Ho, 2026 International Conference on Quantum Communications, Networking, and Computing (QCNC) 259 (2026) ISBN:979-8-3315-6110-9.

[67] Lu‐Cong Lu, Bao‐Cang Ren, Xin Wang, Mei Zhang, and Fu‐Guo Deng, "General Quantum Entanglement Purification Protocol using a Controlled‐Phase‐Flip Gate", Annalen der Physik 532 4, 2000011 (2020).

[68] Lorenzo Valentini, René Bødker Christensen, Petar Popovski, and Marco Chiani, "Reliable Quantum Communications Based on Asymmetry in Distillation and Coding", IEEE Transactions on Quantum Engineering 5, 1 (2024).

[69] Haoxiong Yan, Youpeng Zhong, Hung-Shen Chang, Audrey Bienfait, Ming-Han Chou, Christopher R. Conner, Étienne Dumur, Joel Grebel, Rhys G. Povey, and Andrew N. Cleland, "Entanglement Purification and Protection in a Superconducting Quantum Network", Physical Review Letters 128 8, 080504 (2022).

[70] Francisco Ferreira da Silva, Guus Avis, Joshua A Slater, and Stephanie Wehner, "Requirements for upgrading trusted nodes to a repeater chain over 900 km of optical fiber", Quantum Science and Technology 9 4, 045041 (2024).

[71] Xuanxuan Xin, Shiwen He, Yongxing Li, and Chong Li, "Deterministic joint remote state preparation via a non-maximally entangled channel", Physica Scripta 98 6, 065103 (2023).

[72] Pei-Shun Yan, Lan Zhou, and Yu-Bo Sheng, "Single-copy entanglement purification for Greenberger–Horne–Zeilinger states", Journal of the Optical Society of America B 40 8, 2050 (2023).

[73] Lan Zhou, Ze-Kai Liu, Zi-Xuan Xu, Yi-Lun Cui, Hai-Jiang Ran, and Yu-Bo Sheng, "Economical multi-photon polarization entanglement purification with Bell state", Quantum Information Processing 20 8, 257 (2021).

[74] Karoki A. Mũgambi and Geoffrey O. Okeng’o, "Design and Implementation of Quantum Repeaters: Insights on Quantum Entanglement Purification", Journal of Quantum Computing 5, 25 (2023).

[75] Adam Kinos, Andreas Walther, Stefan Kröll, and Lars Rippe, "Complete analysis of a realistic fiber-based quantum repeater scheme", Physical Review Research 7 3, 033199 (2025).

[76] Diego Forlivesi, Lorenzo Valentini, Andrea Talarico, and Marco Chiani, 2026 International Conference on Quantum Communications, Networking, and Computing (QCNC) 522 (2026) ISBN:979-8-3315-6110-9.

[77] Ashlesha Patil, Michele Pacenti, Bane Vasić, Saikat Guha, and Narayanan Rengaswamy, "Quantum Repeater Protocol Using Quantum Error Correction for Distillation", IEEE Internet Computing 30 1, 38 (2026).

[78] Allen Zang, Xinan Chen, Eric Chitambar, Martin Suchara, and Tian Zhong, "No-Go Theorems for Universal Entanglement Purification", Physical Review Letters 134 19, 190803 (2025).

[79] V Domínguez Tubío, M C Dijksman, and J Borregaard, "Satellite-assisted entanglement distribution with high-dimensional photonic encoding", Quantum Science and Technology 11 3, 035029 (2026).

[80] Harraz Sajede, Wang Yueyan, and Shuang Cong, "Unilateral protection scheme for N-qubit GHZ states against decoherence: a resource-efficient approach", The Journal of Supercomputing 80 12, 17004 (2024).

[81] Orsolya Kálmán, Aurél Gábris, Igor Jex, and Tamás Kiss, "Universal, Unambiguous Concentration and Distillation of Bell pairs", Physical Review Letters 135 26, 260202 (2025).

[82] András Mihály and László Bacsárdi, 2025 IEEE 19th International Symposium on Applied Computational Intelligence and Informatics (SACI) 000261 (2025) ISBN:979-8-3315-1547-8.

[83] Sajede Harraz and Shuang Cong, "Single-copy entanglement purification: a robust approach for diverse noise sources", EPJ Quantum Technology 12 1, 41 (2025).

[84] Hamza Jnane, Brennan Undseth, Zhenyu Cai, Simon C. Benjamin, and Bálint Koczor, "Multicore Quantum Computing", Physical Review Applied 18 4, 044064 (2022).

[85] Joshua Carlo A. Casapao, Ananda G. Maity, Naphan Benchasattabuse, Michal Hajdušek, Rodney Van Meter, and David Elkouss, "Disti-Mator, an entanglement distillation-based state estimator", Communications Physics 8 1, 461 (2025).

[86] Joshua Ramette, Josiah Sinclair, Nikolas P. Breuckmann, and Vladan Vuletić, "Fault-tolerant connection of error-corrected qubits with noisy links", npj Quantum Information 10 1, 58 (2024).

[87] Cen-Xiao Huang, Xiao-Min Hu, Bi-Heng Liu, Lan Zhou, Yu-Bo Sheng, Chuan-Feng Li, and Guang-Can Guo, "Experimental one-step deterministic polarization entanglement purification", Science Bulletin 67 6, 593 (2022).

[88] Jorge Miguel-Ramiro, Alexander Pirker, and Wolfgang Dür, "Improving entanglement purification through coherent superposition of roles", Quantum 9, 1702 (2025).

[89] Wolfgang Tittel, Mikael Afzelius, Adam Kinos, Lars Rippe, and Andreas Walther, "Quantum networks using rare-earth ions", Quantum Science and Technology 10 3, 033002 (2025).

[90] Hao Zhang, Xusheng Xu, Chen Zhang, Man-Hong Yung, Tao Huang, and Yunjie Liu, "Variational quantum circuit learning of entanglement purification in multiple degrees of freedom", Physical Review A 108 4, 042611 (2023).

[91] Alejandro Contreras Reynoso and Thomas Gorin, "Normal quantum channels and Markovian correlated two-qubit quantum errors", Journal of Physics A: Mathematical and Theoretical 57 22, 225301 (2024).

[92] James Ang, Gabriella Carini, Yanzhu Chen, Isaac Chuang, Michael Demarco, Sophia Economou, Alec Eickbusch, Andrei Faraon, Kai-Mei Fu, Steven Girvin, Michael Hatridge, Andrew Houck, Paul Hilaire, Kevin Krsulich, Ang Li, Chenxu Liu, Yuan Liu, Margaret Martonosi, David McKay, Jim Misewich, Mark Ritter, Robert Schoelkopf, Samuel Stein, Sara Sussman, Hong Tang, Wei Tang, Teague Tomesh, Norm Tubman, Chen Wang, Nathan Wiebe, Yongxin Yao, Dillon Yost, and Yiyu Zhou, "ARQUIN: Architectures for Multinode Superconducting Quantum Computers", ACM Transactions on Quantum Computing 5 3, 1 (2024).

[93] Noah Berthusen, Dhruv Devulapalli, Eddie Schoute, Andrew M. Childs, Michael J. Gullans, Alexey V. Gorshkov, and Daniel Gottesman, "Toward a 2D Local Implementation of Quantum Low-Density Parity-Check Codes", PRX Quantum 6 1, 010306 (2025).

[94] Sarah Jansen, Kenneth Goodenough, Sébastian de Bone, Dion Gijswijt, and David Elkouss, "Enumerating all bilocal Clifford distillation protocols through symmetry reduction", Quantum 6, 715 (2022).

[95] Mohammad Mobayenjarihani, Gayane Vardoyan, and Don Towsley, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 1143 (2023) ISBN:979-8-3503-4323-6.

[96] Sajede Harraz and Shuang Cong, "Efficient Multi‐Stage Entanglement Distillation for W States", Advanced Quantum Technologies 9 2, e00952 (2026).

[97] Narayanan Rengaswamy, Ankur Raina, Nithin Raveendran, and Bane Vasić, "Distilling GHZ States using Stabilizer Codes", arXiv:2109.06248, (2021).

[98] Naphan Benchasattabuse, Michal Hajdušek, and Rodney Van Meter, "Integrating Entanglement Purification into All-Photonic Quantum Repeaters", arXiv:2504.18121, (2025).

[99] Yonghe Yu, Mujtaba Zahidy, Siyan Zhou, Caterina Viligar, Karsten Rottwitt, Leif Katsuo Oxenløwe, and Yunhong Ding, "Chip-to-chip hyperentanglement distribution and entanglement purification using silicon integrated photonics", arXiv:2510.18562, (2025).

[100] Naphan Benchasattabuse, "Resource Management in Heterogeneous Quantum Repeater Networks", arXiv:2605.25132, (2026).

[101] Hudson Leone, Thinh Le, S. Srikara, and Simon Devitt, "Resource overheads and attainable rates for trapped-ion lattice surgery", arXiv:2406.18764, (2024).

[102] Eneet Kaur and Saikat Guha, "Entanglement distribution in two-dimensional square grid network", arXiv:2306.03319, (2023).

[103] Sumeet Khatri, "Towards a General Framework for Practical Quantum Network Protocols", arXiv:2412.20472, (2024).

[104] Mingyuan Wang, Guus Avis, and Stefan Krastanov, "GHZ-Preserving Gates and Optimized Distillation Circuits", arXiv:2510.25854, (2025).

[105] Jibing Yuan, Shiqing Tang, Xinwen Wang, and Dengyu Zhang, "One-step distillation of local-unitary-equivalent GHZ-type states", Quantum Information Processing 17 10, 259 (2018).

[106] Benjamin Desef, "Optimization Techniques in Quantum Information", arXiv:2512.15831, (2025).

The above citations are from Crossref's cited-by service (last updated successfully 2026-07-15 12:38:36) and SAO/NASA ADS (last updated successfully 2026-07-15 12:38:37). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.