Entanglement of Local Hidden States

Matteo Fadel; Manuel Gessner

doi:10.22331/q-2022-02-15-651

Entanglement of Local Hidden States

Matteo Fadel^1,2 and Manuel Gessner^3,4

¹Department of Physics, ETH Zürich, 8093 Zürich, Switzerland
²Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
³ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860, Castelldefels (Barcelona), Spain
⁴Laboratoire Kastler Brossel, ENS-Université PSL, CNRS, Sorbonne Université, Collège de France, 24 Rue Lhomond, 75005, Paris, France

Published:	2022-02-15, volume 6, page 651
Eprint:	arXiv:2107.12944v2
Doi:	https://doi.org/10.22331/q-2022-02-15-651
Citation:	Quantum 6, 651 (2022).

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Abstract

Steering criteria are conditions whose violation excludes the possibility of describing the observed measurement statistics with local hidden state (LHS) models. When the available data do not allow to exclude arbitrary LHS models, it may still be possible to exclude LHS models with a specific separability structure. Here, we derive experimentally feasible criteria that put quantitative bounds on the multipartite entanglement of LHS. Our results reveal that separable states may contain hidden entanglement that can be unlocked by measurements on another system, even if no steering between the two systems is possible.

Featured image: Hierarchy of Local-Hidden-State (LHS) separability models

Popular summary

The Einstein-Podolsky-Rosen paradox is a counterintuitive phenomenon enabled by a particular form of quantum correlations, known as steering. An observation of this paradox implies that no quantum state assigned locally to a system, together with classical correlations, is able to explain the observed measurement results. In many cases, excluding all possible local quantum states is extremely challenging due to demanding requirements on experimental noise. We show that even if the data cannot reveal steering, it may still be possible to exclude certain classes of local quantum states, in particular those that are separable. Our work introduces families of experimentally testable criteria that reveal correlations that can only be reproduced with entangled local states. Moreover, the entanglement that is hidden in these states can be unlocked by remote measurements and classical communication.

► BibTeX data

@article{Fadel2022entanglementoflocal,
  doi = {10.22331/q-2022-02-15-651},
  url = {https://doi.org/10.22331/q-2022-02-15-651},
  title = {Entanglement of {L}ocal {H}idden {S}tates},
  author = {Fadel, Matteo and Gessner, Manuel},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {651},
  month = feb,
  year = {2022}
}

► References

[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, New York, NY, 2000).

[2] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
https://doi.org/10.1103/RevModPhys.81.865

[3] M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications, Rev. Mod. Phys. 81, 1727 (2009).
https://doi.org/10.1103/RevModPhys.81.1727

[4] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell Nonlocality, Rev. Mod. Phys. 86, 419 (2014).
https://doi.org/10.1103/RevModPhys.86.419

[5] D. Cavalcanti and P. Skrzypczyk, Quantum steering: a review with focus on semidefinite programming, Rep. Prog. Phys. 80, 024001 (2017).
https://doi.org/10.1088/1361-6633/80/2/024001

[6] N. Friis, G. Vitagliano, M. Malik, and M. Huber, Entanglement certification from theory to experiment, Nat. Rev. Phys. 1, 72 (2019).
https://doi.org/10.1038/s42254-018-0003-5

[7] R. Uola, A. C. S. Costa, H. C. Nguyen, and O. Gühne, Quantum steering, Rev. Mod. Phys. 92, 015001 (2020).
https://doi.org/10.1103/RevModPhys.92.015001

[8] W. Dür, G. Vidal, and J. I. Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A 62, 062314 (2000).
https://doi.org/10.1103/PhysRevA.62.062314

[9] O. Gühne and G. Tóth, Entanglement Detection, Phys. Rep. 474, 1 (2009).
https://doi.org/10.1016/j.physrep.2009.02.004

[10] F. Levi and F. Mintert, Hierarchies of Multipartite Entanglement, Phys. Rev. Lett. 110, 150402 (2013).
https://doi.org/10.1103/PhysRevLett.110.150402

[11] M. Walter, D. Gross, J. Eisert, Multi-partite entanglement, arXiv:1612.02437.
arXiv:1612.02437

[12] S. Szalay, k-stretchability of entanglement, and the duality of k-separability and k-producibility, Quantum 3, 204 (2019).
https://doi.org/10.22331/q-2019-12-02-204

[13] J.-S. Bell, Physics (Long Island City, N.Y.) 1, 195 (1964).

[14] H. M. Wiseman, S. J. Jones, and A. 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

[15] R. Gallego and L. Aolita, Resource Theory of Steering, Phys. Rev. X 5, 041008 (2015).
https://doi.org/10.1103/PhysRevX.5.041008

[16] A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?, Phys. Rev. 47, 777 (1935).
https://doi.org/10.1103/PhysRev.47.777

[17] M. D. Reid, Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification, Phys. Rev. A 40, 913 (1989).
https://doi.org/10.1103/PhysRevA.40.913

[18] B. Yadin, M. Fadel, and M. Gessner, Metrological complementarity reveals the Einstein-Podolsky-Rosen paradox, Nat. Commun. 12, 2401 (2021).
https://doi.org/10.1038/s41467-021-22353-3

[19] J. Guo, F.-X. Sun, D. Zhu, M. Gessner, Q. He, M. Fadel, Detecting Einstein-Podolsky-Rosen steering in non-Gaussian spin states from conditional spin-squeezing parameters, arXiv:2106.13106.
arXiv:2106.13106

[20] S. P. Walborn, A. Salles, R. M. Gomes, F. Toscano, and P. H. Souto Ribeiro, Revealing Hidden Einstein-Podolsky-Rosen Nonlocality, Phys. Rev. Lett. 106, 130402 (2013).
https://doi.org/10.1103/PhysRevLett.106.130402

[21] J. Schneeloch, C. J. Broadbent, S. P. Walborn, E. G. Cavalcanti and J. C. Howell, Einstein-Podolsky-Rosen steering inequalities from entropic uncertainty relations, Phys. Rev. A 87, 062103 (2013).
https://doi.org/10.1103/PhysRevA.87.062103

[22] A. C. S. Costa, R. Uola, O. Gühne, Entropic Steering Criteria: Applications to Bipartite and Tripartite Systems, Entropy 20, 763 (2018).
https://doi.org/10.3390/e20100763

[23] A. Riccardi, C. Macchiavello, and L. Maccone, Multipartite steering inequalities based on entropic uncertainty relations, Phys. Rev. A 97, 052307 (2018).
https://doi.org/10.1103/PhysRevA.97.052307

[24] Q. Y. He and M. D. Reid, Genuine Multipartite Einstein-Podolsky-Rosen Steering, Phys. Rev. Lett. 111, 250403 (2013).
https://doi.org/10.1103/PhysRevLett.111.250403

[25] M. D. Reid, Monogamy inequalities for the Einstein-Podolsky-Rosen paradox and quantum steering, Phys. Rev. A 88, 062108 (2013).
https://doi.org/10.1103/PhysRevA.88.062108

[26] R. Y. Teh and M. 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

[27] S. Armstrong, M. Wang, R. Y. Teh, Q. Gong, Q. He, J. Janousek, H.-A. Bachor, M. D. Reid, P. K. Lam, Multipartite Einstein–Podolsky–Rosen steering and genuine tripartite entanglement with optical networks, Nature Phys. 11, 167 (2015).
https://doi.org/10.1038/nphys3202

[28] D. Cavalcanti, P. Skrzypczyk, G. H. Aguilar, R. V. Nery, P.H. Souto Ribeiro, S. P. Walborn, Detection of entanglement in asymmetric quantum networks and multipartite quantum steering, Nat. Commun. 6, 7941 (2015).
https://doi.org/10.1038/ncomms8941

[29] Y. Xiang, X. Su, L. Mišta, Jr., G. Adesso, and Q. He, Multipartite Einstein-Podolsky-Rosen steering sharing with separable states, Phys. Rev. A 99, 010104(R) (2019).
https://doi.org/10.1103/PhysRevA.99.010104

[30] S. L. Braunstein and C. M. Caves, Statistical Distance and the Geometry of Quantum States, Phys. Rev. Lett. 72, 3439 (1994).
https://doi.org/10.1103/PhysRevLett.72.3439

[31] V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nature Phot. 5, 222 (2011).
https://doi.org/10.1038/nphoton.2011.35

[32] G. Tóth and I. Apellaniz, Quantum metrology from a quantum information science perspective, J. Phys. A 47, 424006 (2014).
https://doi.org/10.1088/1751-8113/47/42/424006

[33] L. Pezzè and A. Smerzi, Quantum theory of phase estimation, in Atom Interferometry, Proceedings of the International School of Physics ``Enrico Fermi'', Course 188, Varenna, edited by G. M. Tino and M. A. Kasevich (IOS Press, Amsterdam) p. 691 (2014).
arXiv:1411.5164

[34] See the Supplemental Material for a proof of Eq. (3).

[35] M. Gessner, L. Pezzè, A. Smerzi, Efficient entanglement criteria for discrete, continuous, and hybrid variables, Phys. Rev. A 94, 020101(R) (2016).
https://doi.org/10.1103/PhysRevA.94.020101

[36] O. Cohen, Unlocking Hidden Entanglement with Classical Information, Phys. Rev. Lett. 80, 2493 (1998).
https://doi.org/10.1103/PhysRevLett.80.2493

[37] O. Cohen, Unlocking Hidden Entanglement with Classical Information, Phys. Rev. A 60, 80 (1999).
https://doi.org/10.1103/PhysRevA.60.80

[38] D. M. Greenberger, M. A. Horne, and A. Zeilinger, Going Beyond Bell's Theorem, in ``Bell's Theorem, Quantum Theory, and Conceptions of the Universe'', M. Kafatos (Ed.), Kluwer, Dordrecht, pp. 69 (1989).
https://doi.org/10.1007/978-94-017-0849-4_10

[39] H. F. Hofmann and S. Takeuchi, Violation of local uncertainty relations as a signature of entanglement, Phys. Rev. A 68, 6 (2003).
https://doi.org/10.1103/PhysRevA.68.032103

[40] M. Fadel and M. Gessner, Relating spin squeezing to multipartite entanglement criteria for particles and modes, Phys. Rev. A 102, 012412 (2020).
https://doi.org/10.1103/PhysRevA.102.012412

[41] M. Gessner, L. Pezzè, and A. Smerzi, Entanglement and squeezing in continuous-variable systems, Quantum 1, 17 (2017).
https://doi.org/10.22331/q-2017-07-14-17

[42] O. Gühne, G. Tóth anf H. Briegel, Multipartite entanglement in spin chains, New J. Phys. 7, 229 (2005).
https://doi.org/10.1088/1367-2630/7/1/229

[43] Z. Ren, W. Li, A. Smerzi, and M. Gessner, Metrological Detection of Multipartite Entanglement from Young Diagrams, Phys. Rev. Lett. 126, 080502 (2021).
https://doi.org/10.1103/PhysRevLett.126.080502

[44] P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, L. Pezzè, and A. Smerzi, Fisher information and multiparticle entanglement, Phys. Rev. A 85, 022321 (2012).
https://doi.org/10.1103/PhysRevA.85.022321

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

[46] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum metrology, Phys. Rev. Lett. 96, 010401 (2006).
https://doi.org/10.1103/PhysRevLett.96.010401

[47] M. Gessner and A. Smerzi, Encrypted quantum correlations: Delayed choice of quantum statistics and other applications, EPJ Quantum Technol. 6, 4 (2019).
https://doi.org/10.1140/epjqt/s40507-019-0074-y

[48] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).
https://doi.org/10.1103/PhysRevLett.70.1895

[49] A. K. Ekert, Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett. 67, 661 (1991).
https://doi.org/10.1103/PhysRevLett.67.661

[50] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Quantifying Entanglement, Phys. Rev. Lett. 78, 2275 (1997).
https://doi.org/10.1103/PhysRevLett.78.2275

[51] A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, Device-Independent Security of Quantum Cryptography against Collective Attacks, Phys. Rev. Lett. 98, 230501 (2007).
https://doi.org/10.1103/PhysRevLett.98.230501

[52] B. Morris, B. Yadin, M. Fadel, T. Zibold, P. Treutlein and G. Adesso, Entanglement between identical particles is a useful and consistent resource, Phys. Rev. X 10, 041012 (2020).
https://doi.org/10.1103/PhysRevX.10.041012

[53] M. Fadel, A. Usui, M. Huber, N. Friis and G. Vitagliano, Entanglement Quantification in Atomic Ensembles, Phys. Rev. Lett. 127, 010401 (2021).
https://doi.org/10.1103/PhysRevLett.127.010401

[54] H. Strobel, W. Muessel, D. Linnemann, T. Zibold, D. B. Hume, L. Pezzè, A. Smerzi, and M. K. Oberthaler, Fisher information and entanglement of non-Gaussian spin states, Science 345, 424 (2014).
https://doi.org/10.1126/science.1250147

[55] M. Fadel, T. Zibold, B. Décamps and P. Treutlein, Spatial entanglement patterns and Einstein-Podolsky-Rosen steering in Bose-Einstein condensates, Science 360, 409 (2018).
https://doi.org/10.1126/science.aao1850

[56] P. Kunkel, M. Prüfer, H. Strobel, D. Linnemann, A. Frölian, T. Gasenzer, M. Gärttner and M. K. Oberthaler, Spatially distributed multipartite entanglement enables EPR steering of atomic clouds, Science 360, 413 (2018).
https://doi.org/10.1126/science.aao2254

[57] K. Lange, J. Peise, B. Lücke, I. Kruse, G. Vitagliano, I. Apellaniz, M. Kleinmann, G. Tóth and C. Klempt, Entanglement between two spatially separated atomic modes, Science 360, 416 (2018).
https://doi.org/10.1126/science.aao2035

[58] J. G. Bohnet, B. C. Sawyer, J. W. Britton, M. L. Wall, A. M. Rey, M. Foss-Feig, J. J. Bollinger, Quantum spin dynamics and entanglement generation with hundreds of trapped ions, Science 352, 1297 (2016).
https://doi.org/10.1126/science.aad9958

[59] Z. Qin, M. Gessner, Z. Ren, X. Deng, D. Han, W. Li, X. Su, A. Smerzi, and K. Peng, Characterizing the multipartite continuous-variable entanglement structure from squeezing coefficients and the Fisher information, npj Quant. Inf. 5, 3 (2019).
https://doi.org/10.1038/s41534-018-0119-6

[60] R. Takagi and Q. Zhuang, Convex resource theory of non-Gaussianity, Phys. Rev. A 97, 062337 (2018).
https://doi.org/10.1103/PhysRevA.97.062337

[61] F. Albarelli, M. G. Genoni, M. G. A. Paris, and A. Ferraro, Resource theory of quantum non-Gaussianity and Wigner negativity, Phys. Rev. A 98, 052350 (2018).
https://doi.org/10.1103/PhysRevA.98.052350

[62] A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quantum coherence as a resource, Rev. Mod. Phys. 89, 041003 (2017).
https://doi.org/10.1103/RevModPhys.89.041003

Cited by

[1] Kuan-Yi Lee, Jhen-Dong Lin, Adam Miranowicz, Franco Nori, Huan-Yu Ku, and Yueh-Nan Chen, "Steering-enhanced quantum metrology using superpositions of noisy phase shifts", Physical Review Research 5 1, 013103 (2023).

[2] Jiajie Guo, Feng-Xiao Sun, Daoquan Zhu, Manuel Gessner, Qiongyi He, and Matteo Fadel, "Detecting Einstein-Podolsky-Rosen steering in non-Gaussian spin states from conditional spin-squeezing parameters", Physical Review A 108 1, 012435 (2023).

[3] Huan-Yu Ku, Kuan-Yi Lee, Po-Rong Lai, Jhen-Dong Lin, and Yueh-Nan Chen, "Coherent activation of a steerability-breaking channel", Physical Review A 107 4, 042415 (2023).

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