Enumerating all bilocal Clifford distillation protocols through symmetry reduction

Sarah Jansen; Kenneth Goodenough; Sébastian de Bone; Dion Gijswijt; David Elkouss

doi:10.22331/q-2022-05-19-715

Enumerating all bilocal Clifford distillation protocols through symmetry reduction

Sarah Jansen^1,2, Kenneth Goodenough³, Sébastian de Bone^3,4, Dion Gijswijt¹, and David Elkouss^3,5

¹Delft Institute of Applied Mathematics, Delft University of Technology, The Netherlands.
²Korteweg-de Vries Institute for Mathematics, University of Amsterdam, The Netherlands
³QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
⁴QuSoft, CWI, Science Park 123, 1098 XG Amsterdam, The Netherlands
⁵Networked Quantum Devices Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa, Japan

Published:	2022-05-19, volume 6, page 715
Eprint:	arXiv:2103.03669v5
Doi:	https://doi.org/10.22331/q-2022-05-19-715
Citation:	Quantum 6, 715 (2022).

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

Abstract

Entanglement distillation is an essential building block in quantum communication protocols. Here, we study the class of near-term implementable distillation protocols that use bilocal Clifford operations followed by a single round of communication. We introduce tools to enumerate and optimise over all protocols for up to $n=5$ (not necessarily equal) Bell-diagonal states using a commodity desktop computer. Furthermore, by exploiting the symmetries of the input states, we find all protocols for up to $n=8$ copies of a Werner state. For the latter case, we present circuits that achieve the highest fidelity with perfect operations and no decoherence. These circuits have modest depth and number of two-qubit gates. Our results are based on a correspondence between distillation protocols and double cosets of the symplectic group, and improve on previously known protocols.

Featured image: Probabilities that describe the state of a system consisting of 3 qubit pairs. The light blue cylinders highlight the probabilities that correspond to the 'pillars'. The sum of these probabilities yields the success probability of a distillation protocol. The darker circles highlight the probabilities that correspond to the 'base'. These probabilities correspond to the fidelity of the output state of the protocol. Note that we have only labelled the coefficients that are on the front, right and top face.

Popular summary

Entanglement forms a key component for quantum networks, but is usually left imperfect by noise in experimental setups. By performing distillation on the noisy entanglement, one can compensate for this. Distillation trades off quantity with quality — it transforms noisy entanglement into a smaller amount of entanglement, but which is less noisy. Understanding how to perform such distillation is crucial for future networks. We use group theoretic tools to exponentially reduce the number of protocols, allowing us to find new and improved such protocols, that can be implemented in the near-term.

► BibTeX data

@article{Jansen2022enumeratingall,
  doi = {10.22331/q-2022-05-19-715},
  url = {https://doi.org/10.22331/q-2022-05-19-715},
  title = {Enumerating all bilocal {C}lifford distillation protocols through symmetry reduction},
  author = {Jansen, Sarah and Goodenough, Kenneth and de Bone, S{\'{e}}bastian and Gijswijt, Dion and Elkouss, David},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {715},
  month = may,
  year = {2022}
}

► References

[1] Charles H. Bennett and Gilles Brassard. ``Quantum cryptography: Public key distribution and coin tossing''. Theoretical Computer Science 560, 7–11 (2014).
https://doi.org/10.1016/j.tcs.2014.05.025

[2] Kai Chen and Hoi-Kwong Lo. ``Conference key agreement and quantum sharing of classical secrets with noisy ghz states''. In Proceedings. International Symposium on Information Theory, 2005. ISIT 2005. Pages 1607–1611. IEEE (2005).
https://doi.org/10.1109/ISIT.2005.1523616

[3] Jérémy Ribeiro, Gláucia Murta, and Stephanie Wehner. ``Fully device-independent conference key agreement''. Physical Review A 97, 022307 (2018).
https://doi.org/10.1103/PhysRevA.97.022307

[4] Richard Jozsa, Daniel S. Abrams, Jonathan P. Dowling, and Colin P. Williams. ``Quantum clock synchronization based on shared prior entanglement''. Physical Review Letters 85, 2010 (2000).
https://doi.org/10.1103/PhysRevLett.85.2010

[5] Lov K. Grover. ``Quantum telecomputation'' (1997). arXiv:quant-ph/9704012.
arXiv:quant-ph/9704012

[6] J. Ignacio Cirac, Artur Ekert, Susana F. Huelga, and Chiara Macchiavello. ``Distributed quantum computation over noisy channels''. Physical Review A 59, 4249 (1999).
https://doi.org/10.1103/PhysRevA.59.4249

[7] 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, 1–5 (2013).
https://doi.org/10.1038/ncomms2773

[8] 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, 722 (1996).
https://doi.org/10.1103/PhysRevLett.76.722

[9] Charles H. Bennett, David P. Divincenzo, John A. Smolin, and William K. Wootters. ``Mixed-state entanglement and quantum error correction''. Physical Review A 54, 3824–3851 (1996).
https://doi.org/10.1103/physreva.54.3824

[10] 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, 2818–2821 (1996).
https://doi.org/10.1103/physrevlett.77.2818

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

[12] Igor Devetak and Andreas Winter. ``Distillation of secret key and entanglement from quantum states''. Proceedings of the Royal Society A: Mathematical, Physical and engineering sciences 461, 207–235 (2005).
https://doi.org/10.1098/rspa.2004.1372

[13] Francesco Buscemi and Nilanjana Datta. ``Distilling entanglement from arbitrary resources''. Journal of Mathematical Physics 51, 102201 (2010).
https://doi.org/10.1063/1.3483717

[14] Felix Leditzky, Nilanjana Datta, and Graeme Smith. ``Useful states and entanglement distillation''. IEEE Transactions on Information Theory 64, 4689–4708 (2017).
https://doi.org/10.1109/tit.2017.2776907

[15] Vlatko Vedral. ``On bound entanglement assisted distillation''. Physics Letters A 262, 121–124 (1999).
https://doi.org/10.1016/s0375-9601(99)00686-6

[16] Satoshi Ishizaka. ``Bound entanglement provides convertibility of pure entangled states''. Physical Review Letters 93, 190501 (2004).
https://doi.org/10.1103/physrevlett.93.190501

[17] Ferran Riera-Sàbat, Pavel Sekatski, Alexander Pirker, and Wolfgang Dür. ``Entanglement purification by counting and locating errors with entangling measurements''. Physical Review A 104, 012419 (2021).
https://doi.org/10.1103/PhysRevA.104.012419

[18] Ferran Riera-Sàbat, Pavel Sekatski, Alexander Pirker, and Wolfgang Dür. ``Entanglement-assisted entanglement purification''. Physical Review Letters 127, 040502 (2021).
https://doi.org/10.1103/PhysRevLett.127.040502

[19] Nicolas Gisin. ``Hidden quantum nonlocality revealed by local filters''. Physics Letters A 210, 151–156 (1996).
https://doi.org/10.1016/s0375-9601(96)80001-6

[20] Frank Verstraete, Jeroen Dehaene, and Bart De Moor. ``Local filtering operations on two qubits''. Physical Review A 64, 010101 (2001).
https://doi.org/10.1103/physreva.64.010101

[21] Francesco Buscemi and Nilanjana Datta. ``General theory of environment-assisted entanglement distillation''. IEEE transactions on information theory 59, 1940–1954 (2012).
https://doi.org/10.1109/tit.2012.2227673

[22] Miguel A. Martín-Delgado and Miguel Navascués. ``Single-step distillation protocol with generalized beam splitters''. Physical Review A 68, 012322 (2003).
https://doi.org/10.1103/physreva.68.012322

[23] Hector Bombin and Miguel A. Martín-Delgado. ``Entanglement distillation protocols and number theory''. Physical Review A 72, 032313 (2005).
https://doi.org/10.1103/physreva.72.032313

[24] Fernando G. S. L. Brandao and Nilanjana Datta. ``One-shot rates for entanglement manipulation under non-entangling maps''. IEEE Transactions on Information Theory 57, 1754–1760 (2011).
https://doi.org/10.1109/TIT.2011.2104531

[25] Bartosz Regula, Kun Fang, Xin Wang, and Mile Gu. ``One-shot entanglement distillation beyond local operations and classical communication''. New Journal of Physics 21, 103017 (2019).
https://doi.org/10.1088/1367-2630/ab4732

[26] Paul G. Kwiat, Salvador Barraza-Lopez, Andre Stefanov, and Nicolas Gisin. ``Experimental entanglement distillation and ‘hidden’ non-locality''. Nature 409, 1014–1017 (2001).
https://doi.org/10.1038/35059017

[27] Jian-Wei Pan, Sara Gasparoni, Rupert Ursin, Gregor Weihs, and Anton Zeilinger. ``Experimental entanglement purification of arbitrary unknown states''. Nature 423, 417–422 (2003).
https://doi.org/10.1038/nature01623

[28] Takashi Yamamoto, Masato Koashi, Şahin Kaya Özdemir, and Nobuyuki Imoto. ``Experimental extraction of an entangled photon pair from two identically decohered pairs''. Nature 421, 343–346 (2003).
https://doi.org/10.1038/nature01358

[29] Philip Walther, Kevin J. Resch, Časlav Brukner, Aephraim M. Steinberg, Jian-Wei Pan, and Anton Zeilinger. ``Quantum nonlocality obtained from local states by entanglement purification''. Physical Review Letters 94, 040504 (2005).
https://doi.org/10.1103/physrevlett.94.040504

[30] Luo-Kan Chen, Hai-Lin Yong, Ping Xu, Xing-Can Yao, Tong Xiang, Zheng-Da Li, Chang Liu, He Lu, Nai-Le Liu, Li Li, et al. ``Experimental nested purification for a linear optical quantum repeater''. Nature Photonics 11, 695–699 (2017).
https://doi.org/10.1038/s41566-017-0010-6

[31] Sebastian Ecker, Philipp Sohr, Lukas Bulla, Marcus Huber, Martin Bohmann, and Rupert Ursin. ``Experimental single-copy entanglement distillation''. Physical Review Letters 127, 040506 (2021).
https://doi.org/10.1103/PhysRevLett.127.040506

[32] Rainer Reichle, Dietrich Leibfried, Emanuel Knill, Joseph W. Britton, R. Bradford Blakestad, John D. Jost, Christopher Langer, Roee Ozeri, Signe Seidelin, and David J. Wineland. ``Experimental purification of two-atom entanglement''. Nature 443, 838–841 (2006).
https://doi.org/10.1038/nature05146

[33] Norbert Kalb, Andreas A. Reiserer, Peter C. Humphreys, Jacob J. W. Bakermans, Sten J. Kamerling, Naomi H. Nickerson, Simon C. Benjamin, Daniel J. Twitchen, Matthew Markham, and Ronald Hanson. ``Entanglement distillation between solid-state quantum network nodes''. Science 356, 928–932 (2017).
https://doi.org/10.1126/science.aan0070

[34] Filip Rozpędek, Thomas Schiet, Le Phuc Thinh, David Elkouss, Andrew C. Doherty, and Stephanie Wehner. ``Optimizing practical entanglement distillation''. Physical Review A 97, 062333 (2018).
https://doi.org/10.1103/physreva.97.062333

[35] Kun Fang, Xin Wang, Marco Tomamichel, and Runyao Duan. ``Non-asymptotic entanglement distillation''. IEEE Transactions on Information Theory 65, 6454–6465 (2019).
https://doi.org/10.1109/tit.2019.2914688

[36] Stefan Krastanov, Victor V. Albert, and Liang Jiang. ``Optimized entanglement purification''. Quantum 3, 123 (2019).
https://doi.org/10.22331/q-2019-02-18-123

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

[38] Hans J. Briegel, Wolfgang Dür, J. Ignacio Cirac, and Peter Zoller. ``Quantum repeaters: the role of imperfect local operations in quantum communication''. Physical Review Letters 81, 5932 (1998).
https://doi.org/10.1103/physrevlett.81.5932

[39] Wolfgang Dür and Hans J. Briegel. ``Entanglement purification for quantum computation''. Physical Review Letters 90, 067901 (2003).
https://doi.org/10.1103/PhysRevLett.90.067901

[40] Wolfgang Dür, Hans J. Briegel, J. Ignacio Cirac, and Peter Zoller. ``Quantum repeaters based on entanglement purification''. Physical Review A 59, 169 (1999).
https://doi.org/10.1103/physreva.59.169

[41] Liangzhong Ruan, Wenhan Dai, and Moe Z. Win. ``Adaptive recurrence quantum entanglement distillation for two-kraus-operator channels''. Physical Review A 97, 052332 (2018).
https://doi.org/10.1103/PhysRevA.97.052332

[42] Karl Gerd H. Vollbrecht and Frank Verstraete. ``Interpolation of recurrence and hashing entanglement distillation protocols''. Physical Review A 71, 062325 (2005).
https://doi.org/10.1103/PhysRevA.71.062325

[43] Daniel Gottesman. ``Theory of fault-tolerant quantum computation''. Physical Review A 57, 127 (1998).
https://doi.org/10.1103/physreva.57.127

[44] Emil Artin. ``Geometric algebra''. Chapter 6, pages 143–147. Interscience Publishers New York. (1957). 1 edition.
https://doi.org/10.1002/9781118164518

[45] 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, 022310 (2003).
https://doi.org/10.1103/physreva.67.022310

[46] Mark M. Wilde. ``Quantum information theory''. Cambridge University Press. (2013). arXiv:1106.1445.
https://doi.org/10.1017/CBO9781139525343
arXiv:1106.1445

[47] Jeroen Dehaene and Bart De Moor. ``Clifford group, stabilizer states, and linear and quadratic operations over gf (2)''. Physical Review A 68, 042318 (2003).
https://doi.org/10.1103/physreva.68.042318

[48] ``Enumerating distillation protocols code''. DOI: 10.4121/15082515. Accessed: 2021-02-04.
https://doi.org/10.4121/15082515

[49] Xuanqiang Zhao, Benchi Zhao, Zihe Wang, Zhixin Song, and Xin Wang. ``Practical distributed quantum information processing with LOCCNet''. npj Quantum Information 7, 159 (2021).
https://doi.org/10.1038/s41534-021-00496-x

[50] Ernest Y.-Z. Tan, Pavel Sekatski, Jean-Daniel Bancal, René Schwonnek, Renato Renner, Nicolas Sangouard, and Charles C.-W. Lim. ``Improved DIQKD protocols with finite-size analysis'' (2020). arXiv:2012.08714.
arXiv:2012.08714

[51] Vlatko Vedral, Martin B. Plenio, Michael A. Rippin, and Peter L. Knight. ``Quantifying entanglement''. Physical Review Letters 78, 2275 (1997).
https://doi.org/10.1103/physrevlett.78.2275

[52] Vlatko Vedral and Martin B. Plenio. ``Entanglement measures and purification procedures''. Physical Review A 57, 1619 (1998).
https://doi.org/10.1103/physreva.57.1619

[53] Vlatko Vedral. ``The role of relative entropy in quantum information theory''. Reviews of Modern Physics 74, 197 (2002).
https://doi.org/10.1103/revmodphys.74.197

[54] Takaaki Matsuo, Clément Durand, and Rodney Van Meter. ``Quantum link bootstrapping using a RuleSet-based communication protocol''. Physical Review A 100, 052320 (2019).
https://doi.org/10.1103/physreva.100.052320

[55] Michelle Victora, Stefan Krastanov, Alexander Sanchez de la Cerda, Steven Willis, and Prineha Narang. ``Purification and entanglement routing on quantum networks'' (2020). arXiv:2011.11644.
arXiv:2011.11644

[56] Dmitri Maslov and Martin Roetteler. ``Shorter stabilizer circuits via bruhat decomposition and quantum circuit transformations''. IEEE Transactions on Information Theory 64, 4729–4738 (2018).
https://doi.org/10.1109/tit.2018.2825602

[57] Sergey Bravyi and Dmitri Maslov. ``Hadamard-free circuits expose the structure of the Clifford group''. IEEE Transactions on Information Theory 67, 4546–4563 (2021).
https://doi.org/10.1109/TIT.2021.3081415

Cited by

[1] 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).

[2] Tanmay Singal, Che Chiang, Eugene Hsu, Eunsang Kim, Hsi-Sheng Goan, and Min-Hsiu Hsieh, "Counting stabiliser codes for arbitrary dimension", Quantum 7, 1048 (2023).

[3] 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).

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

[5] 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).

The above citations are from Crossref's cited-by service (last updated successfully 2023-12-06 21:22:22) and SAO/NASA ADS (last updated successfully 2023-12-06 21:22:23). 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.