Entanglement and squeezing in continuous-variable systems
QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, I-50125 Firenze, Italy
Published: | 2017-07-14, volume 1, page 17 |
Eprint: | arXiv:1702.08413v3 |
Doi: | https://doi.org/10.22331/q-2017-07-14-17 |
Citation: | Quantum 1, 17 (2017). |
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
We introduce a multi-mode squeezing coefficient to characterize entanglement in $N$-partite continuous-variable systems. The coefficient relates to the squeezing of collective observables in the $2N$-dimensional phase space and can be readily extracted from the covariance matrix. Simple extensions further permit to reveal entanglement within specific partitions of a multipartite system. Applications with nonlinear observables allow for the detection of non-Gaussian entanglement.

Popular summary
► BibTeX data
► References
[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, New York, NY, 2000).
https://doi.org/10.1017/CBO9780511976667
[2] L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Non-classical states of atomic ensembles: fundamentals and applications in quantum metrology, arXiv:1609.01609 [quant-ph] (2016).
arXiv:1609.01609
[3] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
https://doi.org/10.1103/RevModPhys.81.865
[4] F. Mintert, A. R. Carvalho, M. Kuś, and A. Buchleitner, Phys. Rep. 415, 207 (2005).
https://doi.org/10.1016/j.physrep.2005.04.006
[5] O. Gühne and G. Tóth, Phys. Rep. 474, 1 (2009).
https://doi.org/10.1016/j.physrep.2009.02.004
[6] D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heinzen, Phys. Rev. A 46, R6797 (1992).
https://doi.org/10.1103/PhysRevA.46.R6797
[7] M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993).
https://doi.org/10.1103/PhysRevA.47.5138
[8] A. Sørensen, L. M. Duan, J. I. Cirac, and P. Zoller, Nature 409, 63 (2001).
https://doi.org/10.1038/35051038
[9] A. S. Sørensen and K. Mølmer, Phys. Rev. Lett. 86, 4431 (2001).
https://doi.org/10.1103/PhysRevLett.86.4431
[10] G. Tóth, C. Knapp, O. Gühne, and H. J. Briegel, Phys. Rev. A 79, 042334 (2009).
https://doi.org/10.1103/PhysRevA.79.042334
[11] J. Ma, X. Wang, C. Sun, and F. Nori, Phys. Rep. 509, 89 (2011).
https://doi.org/10.1016/j.physrep.2011.08.003
[12] G. Tóth, C. Knapp, O. Gühne, and H. J. Briegel, Phys. Rev. Lett. 99, 250405 (2007).
https://doi.org/10.1103/PhysRevLett.99.250405
[13] G. Vitagliano, P. Hyllus, I. L. Egusquiza, and G. Tóth, Phys. Rev. Lett. 107, 240502 (2011).
https://doi.org/10.1103/PhysRevLett.107.240502
[14] P. Hyllus, L. Pezzé, A. Smerzi, and G. Tóth, Phys. Rev. A 86, 012337 (2012).
https://doi.org/10.1103/PhysRevA.86.012337
[15] R. Schmied, J.-D. Bancal, B. Allard, M. Fadel, V. Scarani, P. Treutlein, and N. Sangouard, Science 352, 441 (2016).
https://doi.org/10.1126/science.aad8665
[16] M. Gessner, L. Pezzè, and A. Smerzi, Phys. Rev. A 95, 032326 (2017).
https://doi.org/10.1103/PhysRevA.95.032326
[17] S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J.-i. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, Nat. Photon. 7, 982 (2013).
https://doi.org/10.1038/nphoton.2013.287
[18] M. Chen, N. C. Menicucci, and O. Pfister, Phys. Rev. Lett. 112, 120505 (2014).
https://doi.org/10.1103/PhysRevLett.112.120505
[19] J. Roslund, R. M. de Araújo, S. Jiang, C. Fabre, and N. Treps, Nat. Photon. 8, 109 (2014).
https://doi.org/10.1038/nphoton.2013.340
[20] C. Gross, H. Strobel, E. Nicklas, T. Zibold, N. Bar-Gill, G. Kurizki, and M. K. Oberthaler, Nature 480, 219 (2011).
https://doi.org/10.1038/nature10654
[21] C. D. Hamley, C. S. Gerving, T. M. Hoang, E. M. Bookjans, and M. S. Chapman, Nat. Phys. 8, 305 (2012).
https://doi.org/10.1038/NPHYS2245
[22] J. Peise, I. Kruse, K. Lange, B. Lücke, L. Pezzé, J. Arlt, W. Ertmer, K. Hammerer, L. Santos, A. Smerzi, and C. Klempt, Nat. Commun. 6, 8984 (2015).
https://doi.org/10.1038/ncomms9984
[23] R. Simon, Phys. Rev. Lett. 84, 2726 (2000).
https://doi.org/10.1103/PhysRevLett.84.2726
[24] L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 84, 2722 (2000).
https://doi.org/10.1103/PhysRevLett.84.2722
[25] V. Giovannetti, S. Mancini, D. Vitali, and P. Tombesi, Phys. Rev. A 67, 022320 (2003).
https://doi.org/10.1103/PhysRevA.67.022320
[26] S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77, 513 (2005).
https://doi.org/10.1103/RevModPhys.77.513
[27] G. Adesso and F. Illuminati, J. Phys. A 40, 7821 (2007).
https://doi.org/10.1088/1751-8113/40/28/S01
[28] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys. 84, 621 (2012).
https://doi.org/10.1103/RevModPhys.84.621
[29] L. Pezzé and A. Smerzi, Phys. Rev. Lett. 102, 100401 (2009).
https://doi.org/10.1103/PhysRevLett.102.100401
[30] L. Pezzè, Y. Li, W. Li, and A. Smerzi, Proc. Natl. Acad. Sci. 113, 11459 (2016).
https://doi.org/10.1073/pnas.1603346113
[31] M. Gessner, L. Pezzè, and A. Smerzi, Phys. Rev. A 94, 020101(R) (2016).
https://doi.org/10.1103/PhysRevA.94.020101
[32] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994).
https://doi.org/10.1103/PhysRevLett.72.3439
[33] M. G. Paris, Intl. J. Quant. Inf. 7, 125 (2009).
https://doi.org/10.1142/S0219749909004839
[34] V. Giovannetti, S. Lloyd, and L. Maccone, Nat. Photon. 5, 222 (2011).
https://doi.org/10.1038/nphoton.2011.35
[35] L. Pezzé and A. Smerzi, in Atom Interferometry, Proceedings of the International School of Physics "Enrico Fermi", Course 188, Varenna, edited by G. Tino and M. Kasevich (IOS Press, Amsterdam, Netherlands, 2014).
[36] H. Strobel, W. Muessel, D. Linnemann, T. Zibold, D. B. Hume, L. Pezzè, A. Smerzi, and M. K. Oberthaler, Science 345, 424 (2014).
https://doi.org/10.1126/science.1250147
[37] J. G. Bohnet, B. C. Sawyer, J. W. Britton, M. L. Wall, A. M. Rey, M. Foss-Feig, and J. J. Bollinger, Science 352, 1297 (2016).
https://doi.org/10.1126/science.aad9958
[38] P. van Loock and A. Furusawa, Phys. Rev. A 67, 052315 (2003).
https://doi.org/10.1103/PhysRevA.67.052315
[39] A. A. Valido, F. Levi, and F. Mintert, Phys. Rev. A 90, 052321 (2014).
https://doi.org/10.1103/PhysRevA.90.052321
[40] R. Y. Teh and M. D. Reid, Phys. Rev. A 90, 062337 (2014).
https://doi.org/10.1103/PhysRevA.90.062337
[41] E. Shchukin and P. van Loock, Phys. Rev. A 92, 042328 (2015).
https://doi.org/10.1103/PhysRevA.92.042328
[42] S. Gerke, J. Sperling, W. Vogel, Y. Cai, J. Roslund, N. Treps, and C. Fabre, Phys. Rev. Lett. 114, 050501 (2015).
https://doi.org/10.1103/PhysRevLett.114.050501
[43] R. F. Werner and M. M. Wolf, Phys. Rev. Lett. 86, 3658 (2001).
https://doi.org/10.1103/PhysRevLett.86.3658
[44] W. Heisenberg, Z. Phys. 43, 172 (1927).
https://doi.org/10.1007/BF01397280
[45] H. P. Robertson, Phys. Rev. 34, 163 (1929).
https://doi.org/10.1103/PhysRev.34.163
[46] S. P. Walborn, B. G. Taketani, A. Salles, F. Toscano, and R. L. de Matos Filho, Phys. Rev. Lett. 103, 160505 (2009).
https://doi.org/10.1103/PhysRevLett.103.160505
[47] Y. Huang, IEEE Trans. Inf. Theo. 59, 6774 (2013).
https://doi.org/10.1109/TIT.2013.2257936
[48] E. Shchukin and P. van Loock, Phys. Rev. Lett. 117, 140504 (2016).
https://doi.org/10.1103/PhysRevLett.117.140504
[49] A. Ferraro, S. Olivares, and M. G. A. Paris, Gaussian states in continuous variable quantum information (Bibliopolis, Napoli, 2005).
[50] T. Aoki, N. Takei, H. Yonezawa, K. Wakui, T. Hiraoka, A. Furusawa, and P. van Loock, Phys. Rev. Lett. 91, 080404 (2003).
https://doi.org/10.1103/PhysRevLett.91.080404
[51] M. Seevinck and J. Uffink, Phys. Rev. A 65, 012107 (2001).
https://doi.org/10.1103/PhysRevA.65.012107
[52] R. Medeiros de Araújo, J. Roslund, Y. Cai, G. Ferrini, C. Fabre, and N. Treps, Phys. Rev. A 89, 053828 (2014).
https://doi.org/10.1103/PhysRevA.89.053828
[53] G. Ferrini, J. Roslund, F. Arzani, Y. Cai, C. Fabre, and N. Treps, Phys. Rev. A 91, 032314 (2015).
https://doi.org/10.1103/PhysRevA.91.032314
[54] W. S. Bakr, J. I. Gillen, A. Peng, S. Folling, and M. Greiner, Nature 462, 74 (2009).
https://doi.org/10.1038/nature08482
[55] J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Nature 467, 68 (2010).
https://doi.org/10.1038/nature09378
[56] C. F. Roos, T. Monz, K. Kim, M. Riebe, H. Häffner, D. F. V. James, and R. Blatt, Phys. Rev. A 77, 040302 (2008).
https://doi.org/10.1103/PhysRevA.77.040302
[57] A. Abdelrahman, O. Khosravani, M. Gessner, H.-P. Breuer, A. Buchleitner, D. J. Gorman, R. Masuda, T. Pruttivarasin, M. Ramm, P. Schindler, and H. Häffner, Nat. Commun. 8, 15712 (2017).
https://doi.org/10.1038/ncomms15712
[58] H. Jeong, A. Zavatta, M. Kang, S.-W. Lee, L. S. Costanzo, S. Grandi, T. C. Ralph, and M. Bellini, Nat. Photon. 8, 564 (2014).
https://doi.org/10.1038/nphoton.2014.136
[59] O. Morin, K. Huang, J. Liu, H. Le Jeannic, C. Fabre, and J. Laurat, Nat. Photon. 8, 570 (2014).
https://doi.org/10.1038/nphoton.2014.137
Cited by
[1] Manuel Gessner, Augusto Smerzi, and Luca Pezzè, "Metrological Nonlinear Squeezing Parameter", Physical Review Letters 122 9, 090503 (2019).
[2] Philipp Kunkel, Maximilian Prüfer, Helmut Strobel, Daniel Linnemann, Anika Frölian, Thomas Gasenzer, Martin Gärttner, and Markus K. Oberthaler, "Spatially distributed multipartite entanglement enables EPR steering of atomic clouds", Science 360 6387, 413 (2018).
[3] Matteo Fadel, Quantum Science and Technology 57 (2021) ISBN:978-3-030-85471-3.
[4] Mattia Walschaers, "Non-Gaussian Quantum States and Where to Find Them", PRX Quantum 2 3, 030204 (2021).
[5] Matteo Fadel and Manuel Gessner, "Entanglement of Local Hidden States", Quantum 6, 651 (2022).
[6] Manuel Gessner, Luca Pezzè, and Augusto Smerzi, "Sensitivity Bounds for Multiparameter Quantum Metrology", Physical Review Letters 121 13, 130503 (2018).
[7] Matteo Fadel and Manuel Gessner, "Relating spin squeezing to multipartite entanglement criteria for particles and modes", Physical Review A 102 1, 012412 (2020).
[8] Manuel Gessner, Augusto Smerzi, and Luca Pezzè, "Multiparameter squeezing for optimal quantum enhancements in sensor networks", Nature Communications 11 1, 3817 (2020).
[9] Zhongzhong Qin, Manuel Gessner, Zhihong Ren, Xiaowei Deng, Dongmei Han, Weidong Li, Xiaolong Su, Augusto Smerzi, and Kunchi Peng, "Characterizing the multipartite continuous-variable entanglement structure from squeezing coefficients and the Fisher information", npj Quantum Information 5 1, 3 (2019).
[10] Zhi-Hong Ren, Yan Li, Yan-Na Li, and Wei-Dong Li, "Development on quantum metrology with quantum Fisher information", Acta Physica Sinica 68 4, 040601 (2019).
[11] Manuel Gessner and Augusto Smerzi, "Statistical speed of quantum states: Generalized quantum Fisher information and Schatten speed", Physical Review A 97 2, 022109 (2018).
[12] Zhihong Ren, Weidong Li, Augusto Smerzi, and Manuel Gessner, "Metrological Detection of Multipartite Entanglement from Young Diagrams", Physical Review Letters 126 8, 080502 (2021).
[13] Mingsheng Tian, Yu Xiang, Feng-Xiao Sun, Matteo Fadel, and Qiongyi He, "Characterizing Multipartite non-Gaussian Entanglement for a Three-Mode Spontaneous Parametric Down-Conversion Process", Physical Review Applied 18 2, 024065 (2022).
[14] Manuel Gessner, "Enhancement of the metrological sensitivity limit through knowledge of the average energy", Physical Review A 100 3, 032114 (2019).
[15] Gardo Blado, Francisco Herrera, and Joshuah Erwin, "Entanglement and the generalized uncertainty principle", Physics Essays 31 4, 397 (2018).
[16] Manuel Gessner, Luca Pezzè, and Augusto Smerzi, "Resolution-enhanced entanglement detection", Physical Review A 95 3, 032326 (2017).
The above citations are from Crossref's cited-by service (last updated successfully 2023-06-09 10:27:42) and SAO/NASA ADS (last updated successfully 2023-06-09 10:27:43). 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.