Minimal criteria for continuous-variable genuine multipartite entanglement

Olga Leskovjanová; Ladislav Mišta Jr.

doi:10.22331/q-2025-08-27-1837

Minimal criteria for continuous-variable genuine multipartite entanglement

Olga Leskovjanová and Ladislav Mišta Jr.

Department of Optics, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic

Published:	2025-08-27, volume 9, page 1837
Editor:	Remigiusz Augusiak
Eprint:	arXiv:2401.04376v3
Doi:	https://doi.org/10.22331/q-2025-08-27-1837
Citation:	Quantum 9, 1837 (2025).

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Abstract

We derive a set of genuine multi-mode entanglement criteria for second moments of the quadrature operators. The criteria have a common form of the uncertainty relation between sums of variances of position and momentum quadrature combinations. A unique feature of the criteria is that the sums contain the least possible number of variances of at most two-mode combinations. The number of second moments we need to know to apply the criteria thus scales only linearly with the number of modes, as opposed to the quadratic scaling of the already existing criteria. Each criterion is associated with a tree graph, which allowed us to develop a direct method of construction of the criteria based solely on the structure of the underlying tree. The practicality of the proposed criteria is demonstrated by finding a number of examples of Gaussian states of up to six modes, whose genuine multi-mode entanglement is detected by them. The designed criteria are particularly suitable for verification of genuine multipartite entanglement in large multi-mode states or when only a set of two-mode nearest-neighbour marginal covariance matrices of the investigated state is available.

► BibTeX data

@article{Leskovjanova2025minimalcriteria,
  doi = {10.22331/q-2025-08-27-1837},
  url = {https://doi.org/10.22331/q-2025-08-27-1837},
  title = {Minimal criteria for continuous-variable genuine multipartite entanglement},
  author = {Leskovjanov{\'{a}}, Olga and Mi{\v{s}}ta Jr., Ladislav},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1837},
  month = aug,
  year = {2025}
}

► References

[1] Reinhard F. Werner. ``Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model''. Phys. Rev. A 40, 4277–4281 (1989).
https://doi.org/10.1103/PhysRevA.40.4277

[2] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. ``Quantum entanglement''. Rev. Mod. Phys. 81, 865–942 (2009).
https://doi.org/10.1103/RevModPhys.81.865

[3] Howard M. Wiseman, Steven J. Jones, and Andrew C. Doherty. ``Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox''. Phys. Rev. Lett. 98, 140402 (2007).
https://doi.org/10.1103/PhysRevLett.98.140402

[4] Roope Uola, Ana C. S. Costa, H. Chau Nguyen, and Otfried Gühne. ``Quantum steering''. Rev. Mod. Phys. 92, 015001 (2020).
https://doi.org/10.1103/RevModPhys.92.015001

[5] John S. Bell. ``On the Einstein Podolsky Rosen paradox''. Phys. Phys. Fiz. 1, 195–200 (1964).
https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195

[6] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. ``Bell nonlocality''. Rev. Mod. Phys. 86, 419–478 (2014).
https://doi.org/10.1103/RevModPhys.86.419

[7] Michael N. Nielsen and Isaac L. Chuang. ``Quantum computation and quantum information''. Cambridge University Press. (2000).

[8] Jean-Daniel Bancal, Nicolas Gisin, Yeong-Cherng Liang, and Stefano Pironio. ``Device-independent witnesses of genuine multipartite entanglement''. Phys. Rev. Lett. 106, 250404 (2011).
https://doi.org/10.1103/PhysRevLett.106.250404

[9] Daniel M. Greenberger, Michael A. Horne, and Anton Zeilinger. ``Bell's Theorem, Quantum Theory, and Conceptions of the Universe''. Page 69. Kluwer, Dordrecht. (1989).

[10] Jian-Wei Pan, Dik Bouwmeester, Matthew Daniell, Harald Weinfurter, and Anton Zeilinger. ``Experimental test of quantum nonlocality in three-photon Greenberger–Horne–Zeilinger entanglement''. Nature 403 (2000).
https://doi.org/10.1038/35000514

[11] Thomas Monz, Philipp Schindler, Julio T. Barreiro, Michael Chwalla, Daniel Nigg, William A. Coish, Maximilian Harlander, Wolfgang Hänsel, Markus Hennrich, and Rainer Blatt. ``14-qubit entanglement: Creation and coherence''. Phys. Rev. Lett. 106, 130506 (2011).
https://doi.org/10.1103/PhysRevLett.106.130506

[12] Mark Hillery, Vladimír Bužek, and André Berthiaume. ``Quantum secret sharing''. Phys. Rev. A 59, 1829–1834 (1999).
https://doi.org/10.1103/PhysRevA.59.1829

[13] Anders Karlsson and Mohamed Bourennane. ``Quantum teleportation using three-particle entanglement''. Phys. Rev. A 58, 4394–4400 (1998).
https://doi.org/10.1103/PhysRevA.58.4394

[14] Hidehiro Yonezawa, Takao Aoki, and Akira Furusawa. ``Demonstration of a quantum teleportation network for continuous variables''. Nature 431 (2004).
https://doi.org/10.1038/nature02858

[15] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. ``Quantum-enhanced measurements: Beating the standard quantum limit''. Science 306, 1330–1336 (2004).
https://doi.org/10.1126/science.1104149

[16] Géza Tóth. ``Multipartite entanglement and high-precision metrology''. Phys. Rev. A 85, 022322 (2012).
https://doi.org/10.1103/PhysRevA.85.022322

[17] Robert Raussendorf and Hans J. Briegel. ``A one-way quantum computer''. Phys. Rev. Lett. 86, 5188–5191 (2001).
https://doi.org/10.1103/PhysRevLett.86.5188

[18] Philip Walther, Kevin J. Resch, Terry Rudolph, Emmanuel Schenck, Harald Weinfurter, Vlatko Vedral, Markus Aspelmeyer, and Anton Zeilinger. ``Experimental one-way quantum computing''. Nature 434, 169–176 (2005).
https://doi.org/10.1038/nature03347

[19] Michael Epping, Hermann Kampermann, Chiara Macchiavello, and Dagmar Bruß. ``Multi-partite entanglement can speed up quantum key distribution in networks''. New J. Phys. 19, 093012 (2017).
https://doi.org/10.1088/1367-2630/aa8487

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

[21] Nicolai Friis, Oliver Marty, Christine Maier, Cornelius Hempel, Milan Holzäpfel, Petar Jurcevic, Martin B. Plenio, Marcus Huber, Christian Roos, Rainer Blatt, and Ben Lanyon. ``Observation of entangled states of a fully controlled 20-qubit system''. Phys. Rev. X 8, 021012 (2018).
https://doi.org/10.1103/PhysRevX.8.021012

[22] Ming Gong, Ming-Cheng Chen, Yarui Zheng, Shiyu Wang, Chen Zha, Hui Deng, Zhiguang Yan, Hao Rong, Yulin Wu, Shaowei Li, Fusheng Chen, Youwei Zhao, Futian Liang, Jin Lin, Yu Xu, Cheng Guo, Lihua Sun, Juno Clark, Haohua Wang, Chengzhi Peng, Chao-Yang Lu, Xiaobo Zhu, and Jian-Wei Pan. ``Genuine 12-qubit entanglement on a superconducting quantum processor''. Phys. Rev. Lett. 122, 110501 (2019).
https://doi.org/10.1103/PhysRevLett.122.110501

[23] Xi-Lin Wang, Yi-Han Luo, He-Liang Huang, Ming-Cheng Chen, Zu-En Su, Chang Liu, Chao Chen, Wei Li, Yu-Qiang Fang, Xiao Jiang, Jun Zhang, Li Li, Nai-Le Liu, Chao-Yang Lu, and Jian-Wei Pan. ``18-qubit entanglement with six photons' three degrees of freedom''. Phys. Rev. Lett. 120, 260502 (2018).
https://doi.org/10.1103/PhysRevLett.120.260502

[24] Sirui Cao, Bujiao Wu, Fusheng Chen, Ming Gong, Yulin Wu, Yangsen Ye, Chen Zha, Haoran Qian, Chong Ying, Shaojun Guo, et al. ``Generation of genuine entanglement up to 51 superconducting qubits''. Nature 619, 738–742 (2023).
https://doi.org/10.1038/s41586-023-06195-1

[25] Christian Weedbrook, Stefano Pirandola, Raúl García-Patrón, Nicolas J. Cerf, Timothy C. Ralph, Jeffrey H. Shapiro, and Seth Lloyd. ``Gaussian quantum information''. Rev. Mod. Phys. 84, 621–669 (2012).
https://doi.org/10.1103/RevModPhys.84.621

[26] Lynden K. Shalm, Deny R. Hamel, Zhizhong Yan, Christoph Simon, Kevin J. Resch, and Thomas Jennewein. ``Three-photon energy–time entanglement''. Nat. Phys. 9, 19–22 (2013).
https://doi.org/10.1038/nphys2492

[27] Seiji Armstrong, Meng Wang, Run Yan Teh, Qihuang Gong, Qiongyi He, Jiri Janousek, Hans-Albert Bachor, Margaret D Reid, and Ping Koy Lam. ``Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks''. Nat. Phys. 11, 167–172 (2015).
https://doi.org/10.1038/nphys3202

[28] Stefan Gerke, Jan Sperling, Werner Vogel, Yin Cai, Jonathan Roslund, Nicolas Treps, and Claude Fabre. ``Full multipartite entanglement of frequency-comb Gaussian states''. Phys. Rev. Lett. 114, 050501 (2015).
https://doi.org/10.1103/PhysRevLett.114.050501

[29] Evgeny Shchukin and Peter van Loock. ``Generalized conditions for genuine multipartite continuous-variable entanglement''. Phys. Rev. A 92, 042328 (2015).
https://doi.org/10.1103/PhysRevA.92.042328

[30] Moran Chen, Nicolas C. Menicucci, and Olivier Pfister. ``Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb''. Phys. Rev. Lett. 112, 120505 (2014).
https://doi.org/10.1103/PhysRevLett.112.120505

[31] Shota Yokoyama, Ryuji Ukai, Seiji C. Armstrong, Chanond Sornphiphatphong, Toshiyuki Kaji, Shigenari Suzuki, Jun-ichi Yoshikawa, Hidehiro Yonezawa, Nicolas C. Menicucci, and Akira Furusawa. ``Ultra-large-scale continuous-variable cluster states multiplexed in the time domain''. Nat. Photonics 7, 982–986 (2013).
https://doi.org/10.1038/nphoton.2013.287

[32] Jun-ichi Yoshikawa, Shota Yokoyama, Toshiyuki Kaji, Chanond Sornphiphatphong, Yu Shiozawa, Kenzo Makino, and Akira Furusawa. ``Invited article: Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing''. APL Photonics 1 (2016).
https://doi.org/10.1063/1.4962732

[33] Philipp Hyllus and Jens Eisert. ``Optimal entanglement witnesses for continuous-variable systems''. New J. Phys. 8, 51 (2006).
https://doi.org/10.1088/1367-2630/8/4/051

[34] Mauro Paternostro, David Vitali, Sylvain Gigan, M. S. Kim, Caslav Brukner, Jens Eisert, and Markus Aspelmeyer. ``Creating and probing multipartite macroscopic entanglement with light''. Phys. Rev. Lett. 99, 250401 (2007).
https://doi.org/10.1103/PhysRevLett.99.250401

[35] Shan W. Jolin, Gustav Andersson, J. C. Rivera Hernández, Ingrid Strandberg, Fernando Quijandría, José Aumentado, Riccardo Borgani, Mats O. Tholén, and David B. Haviland. ``Multipartite entanglement in a microwave frequency comb''. Phys. Rev. Lett. 130, 120601 (2023).
https://doi.org/10.1103/PhysRevLett.130.120601

[36] Run Y. Teh and Margaret D. Reid. ``Criteria for genuine $N$-partite continuous-variable entanglement and Einstein-Podolsky-Rosen steering''. Phys. Rev. A 90, 062337 (2014).
https://doi.org/10.1103/PhysRevA.90.062337

[37] Fabricio Toscano, A. Saboia, Ardiley T. Avelar, and Stephen P. Walborn. ``Systematic construction of genuine-multipartite-entanglement criteria in continuous-variable systems using uncertainty relations''. Phys. Rev. A 92, 052316 (2015).
https://doi.org/10.1103/PhysRevA.92.052316

[38] Peter van Loock and Akira Furusawa. ``Detecting genuine multipartite continuous-variable entanglement''. Phys. Rev. A 67, 052315 (2003).
https://doi.org/10.1103/PhysRevA.67.052315

[39] Lu-Ming Duan, Géza Giedke, J. Ignacio Cirac, and Peter Zoller. ``Inseparability criterion for continuous variable systems''. Phys. Rev. Lett. 84, 2722–2725 (2000).
https://doi.org/10.1103/PhysRevLett.84.2722

[40] Vittorio Giovannetti, Stefano Mancini, David Vitali, and Paolo Tombesi. ``Characterizing the entanglement of bipartite quantum systems''. Phys. Rev. A 67, 022320 (2003).
https://doi.org/10.1103/PhysRevA.67.022320

[41] Marius Paraschiv, Nikolai Miklin, Tobias Moroder, and Otfried Gühne. ``Proving genuine multiparticle entanglement from separable nearest-neighbor marginals''. Phys. Rev. A 98, 062102 (2018).
https://doi.org/10.1103/PhysRevA.98.062102

[42] Michal Mičuda, Robert Stárek, Jan Provazník, Olga Leskovjanová, and Ladislav Mišta Jr. ``Verifying genuine multipartite entanglement of the whole from its separable parts''. Optica 6, 896–901 (2019).
https://doi.org/10.1364/OPTICA.6.000896

[43] Viktor Nordgren, Olga Leskovjanová, Jan Provazník, Adam Johnston, Natalia Korolkova, and Ladislav Mišta. ``Certifying emergent genuine multipartite entanglement with a partially blind witness''. Phys. Rev. A 106, 062410 (2022).
https://doi.org/10.1103/PhysRevA.106.062410

[44] Oleg Gittsovich, Philipp Hyllus, and Otfried Gühne. ``Multiparticle covariance matrices and the impossibility of detecting graph-state entanglement with two-particle correlations''. Phys. Rev. A 82, 032306 (2010).
https://doi.org/10.1103/PhysRevA.82.032306

[45] Atsushi Higuchi, Anthony Sudbery, and Jason Szulc. ``One-qubit reduced states of a pure many-qubit state: Polygon inequalities''. Phys. Rev. Lett. 90, 107902 (2003).
https://doi.org/10.1103/PhysRevLett.90.107902

[46] Sergey Bravyi. ``Requirements for compatibility between local and multipartite quantum states'' (2003). arXiv:quant-ph/0301014.
arXiv:quant-ph/0301014

[47] Lars Erik Würflinger, Jean-Daniel Bancal, Antonio Acín, Nicolas Gisin, and Tamás Vértesi. ``Nonlocal multipartite correlations from local marginal probabilities''. Phys. Rev. A 86, 032117 (2012).
https://doi.org/10.1103/PhysRevA.86.032117

[48] Wolfgang Dür, J. Ignacio Cirac, and Rolf Tarrach. ``Separability and distillability of multiparticle quantum systems''. Phys. Rev. Lett. 83, 3562–3565 (1999).
https://doi.org/10.1103/PhysRevLett.83.3562

[49] Géza Giedke, Barbara Kraus, Maciej Lewenstein, and J. Ignacio Cirac. ``Separability properties of three-mode gaussian states''. Phys. Rev. A 64, 052303 (2001).
https://doi.org/10.1103/PhysRevA.64.052303

[50] Marco Piani and Caterina E. Mora. ``Class of positive-partial-transpose bound entangled states associated with almost any set of pure entangled states''. Phys. Rev. A 75, 012305 (2007).
https://doi.org/10.1103/PhysRevA.75.012305

[51] Otfried Gühne and Géza Tóth. ``Entanglement detection''. Phys. Rep. 474, 1–75 (2009).
https://doi.org/10.1016/j.physrep.2009.02.004

[52] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. ``Separability of mixed states: necessary and sufficient conditions''. Phys. Lett. A 223, 1–8 (1996).
https://doi.org/10.1016/S0375-9601(96)00706-2

[53] Barbara M. Terhal. ``Bell inequalities and the separability criterion''. Phys. Lett. A 271, 319–326 (2000).
https://doi.org/10.1016/S0375-9601(00)00401-1

[54] Rajiah Simon. ``Peres-Horodecki separability criterion for continuous variable systems''. Phys. Rev. Lett. 84, 2726–2729 (2000).
https://doi.org/10.1103/PhysRevLett.84.2726

[55] Peter van Loock and Samuel L. Braunstein. ``Multipartite entanglement for continuous variables: A quantum teleportation network''. Phys. Rev. Lett. 84, 3482–3485 (2000).
https://doi.org/10.1103/PhysRevLett.84.3482

[56] Janet Anders. ``Estimating the degree of entanglement of unknown Gaussian states'' (2006). arXiv:quant-ph/0610263.
arXiv:quant-ph/0610263

[57] Douglas Brent West et al. ``Introduction to graph theory''. Volume 2. Prentice hall Upper Saddle River. (2001).

[58] Peter Steinbach. ``Field guide to simple graphs''. Volume 3. Design Lab, Albuquerque. (1999).

[59] Richard Otter. ``The number of trees''. Ann. Math. 49, 583–599 (1948).
https://doi.org/10.2307/1969046

[60] ``The on-line encyclopedia of integer sequences''. url: oeis.org. Sequence A000055.
https://oeis.org

[61] Paul R. Halmos. ``Finite dimensional vector spaces''. Annals of Mathematics Studies. Princeton University Press. (1948).

[62] Roger A. Horn and Charles R. Johnson. ``Matrix analysis''. Cambridge university press. (1985).

[63] Gene H. Golub and Charles F. Van Loan. ``Matrix computations''. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press. (2013). 3 edition.

[64] Jack J. Dongarra, James R. Bunch, Cleve B. Moler, and G. W. Stewart. ``LINPACK Users' Guide''. Other Titles in Applied Mathematics. Society for Industrial and Applied Mathematics. (1979).

[65] S. Mitchell Hedetniemi, E. J. Cockayne, and S. T. Hedetniemi. ``Linear algorithms for finding the Jordan center and path center of a tree''. Transp. Sci. 15, 98–114 (1981).
https://doi.org/10.1287/trsc.15.2.98

[66] Holger F. Hofmann and Shigeki Takeuchi. ``Violation of local uncertainty relations as a signature of entanglement''. Phys. Rev. A 68, 032103 (2003).
https://doi.org/10.1103/PhysRevA.68.032103

[67] Wolfram Research, Inc. ``Mathematica, Version 12.3''. Champaign, IL, 2021.

[68] Run Yan Teh, Manuel Gessner, Margaret D. Reid, and Matteo Fadel. ``Full multipartite steering inseparability, genuine multipartite steering, and monogamy for continuous-variable systems''. Phys. Rev. A 105, 012202 (2022).
https://doi.org/10.1103/PhysRevA.105.012202

[69] John Williamson. ``On the algebraic problem concerning the normal forms of linear dynamical systems''. Am. J. Math. 58, 141–163 (1936).
https://doi.org/10.2307/2371062

[70] Evgeny Shchukin and Peter van Loock. ``Tripartite separability conditions exponentially violated by Gaussian states''. Phys. Rev. A 90, 012334 (2014).
https://doi.org/10.1103/PhysRevA.90.012334

[71] Da Zhang, David Barral, Yanpeng Zhang, Min Xiao, and Kamel Bencheikh. ``Genuine tripartite non-Gaussian entanglement''. Phys. Rev. Lett. 130, 093602 (2023).
https://doi.org/10.1103/PhysRevLett.130.093602

[72] Qiongyi Y. He and Margaret D. Reid. ``Genuine multipartite Einstein-Podolsky-Rosen steering''. Phys. Rev. Lett. 111, 250403 (2013).
https://doi.org/10.1103/PhysRevLett.111.250403

[73] Jaromír Fiurášek and Ladislav Mišta. ``Gaussian localizable entanglement''. Phys. Rev. A 75, 060302 (2007).
https://doi.org/10.1103/PhysRevA.75.060302

[74] Guifre Vidal and Reinhard F. Werner. ``Computable measure of entanglement''. Phys. Rev. A 65, 032314 (2002).
https://doi.org/10.1103/PhysRevA.65.032314

[75] Jason L. Pereira, Leonardo Banchi, and Stefano Pirandola. ``Symplectic decomposition from submatrix determinants''. Proc. R. Soc. A 477, 20210513 (2021).
https://doi.org/10.1098/rspa.2021.0513

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[2] Lin Htoo Zaw, Jiajie Guo, Qiongyi He, Matteo Fadel, and Shuheng Liu, "Sufficient Wigner Negativity Implies Genuine Multipartite Entanglement", Physical Review Letters 137 4, 040202 (2026).

[3] E. Shchukin and P. van Loock, "Revisiting Gaussian genuine entanglement witnesses with modern software", Physical Review Research 8 2, 023124 (2026).

[4] E. Shchukin and P. van Loock, "Revisiting Gaussian genuine entanglement witnesses with modern software", arXiv:2412.09757, (2024).

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