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Cauchy–Schwarz inequality for general measurements as an entanglement criterion

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Abstract

Considering the broadest set of measurements allowed by quantum mechanics, we demonstrate that the violation Cauchy–Schwarz inequality for any-order correlation function signals the entanglement among bosons. Our result is general—it applies to any system of bosons, even when the number of particles is not fixed, provided that there is no coherence between different number states.

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Notes

  1. The volumes of the two regions in Eqs. (8a)–(8c) can be chosen arbitrarily small, and in fact, the CSI also applies to correlations between two points. However, from the experimental point of view it is usually advantageous to increase the signal by accumulating the data from substantial volumes.

References

  1. Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935)

  2. Bell, J.S.: On the Einstein Podolsky Rosen paradox. Physics 1, 195 (1964)

    Google Scholar 

  3. Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447–452 (1966)

    Article  ADS  MATH  Google Scholar 

  4. Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V., Wehner, S.: Bell nonlocality. Rev. Mod. Phys. 86, 419–478 (2014)

    Article  ADS  Google Scholar 

  5. Freedman, S.J., Clauser, J.F.: Experimental test of local hidden-variable theories. Phys. Rev. Lett. 28, 938–941 (1972)

    Article  ADS  Google Scholar 

  6. Aspect, A., Grangier, P., Roger, G.: Experimental tests of realistic local theories via Bell’s theorem. Phys. Rev. Lett. 47, 460–463 (1981)

    Article  ADS  Google Scholar 

  7. Tittel, W., Brendel, J., Gisin, B., Herzog, T., Zbinden, H., Gisin, N.: Experimental demonstration of quantum correlations over more than 10 km. Phys. Rev. A 57, 3229–3232 (1998)

    Article  ADS  Google Scholar 

  8. Tittel, W., Brendel, J., Zbinden, H., Gisin, N.: Violation of Bell inequalities by photons more than 10 km apart. Phys. Rev. Lett. 81, 3563–3566 (1998)

    Article  ADS  Google Scholar 

  9. Weihs, G., Jennewein, T., Simon, C., Weinfurter, H., Zeilinger, A.: Violation of Bell’s inequality under strict Einstein locality conditions. Phys. Rev. Lett. 81, 5039–5043 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Pan, J.-W., Bouwmeester, D., Daniell, M., Weinfurter, H., Zeilinger, A.: Experimental test of quantum nonlocality in three-photon Greenberger–Horne–Zeilinger entanglement. Nature 403, 515–519 (2000)

    Article  ADS  Google Scholar 

  11. Kielpinski, D., Meyer, V., Sackett, C.A., Itano, W.M., Monroe, C., Wineland, D.J.: Experimental violation of a Bell’s inequality with efficient detection. Nature 409, 791–794 (2001)

    Article  ADS  Google Scholar 

  12. Gröblacher, S., Paterek, T., Kaltenbaek, R., Brukner, Č., Żukowski, M., Aspelmeyer, M., Zeilinger, A.: An experimental test of non-local realism. Nature 446, 871–875 (2007)

    Article  ADS  Google Scholar 

  13. Salart, D., Baas, A., van Houwelingen, J.A.W., Gisin, N., Zbinden, H.: Spacelike separation in a Bell test assuming gravitationally induced collapses. Phys. Rev. Lett. 100, 220404 (2008)

    Article  ADS  Google Scholar 

  14. Ansmann, M., Wang, H., Bialczak, R.C., Hofheinz, M., Lucero, E., Neeley, M., O’Connell, A.D., Sank, D., Weides, M., Wenner, J., Cleland, A.N., Martinis, J.M.: Violation of Bell’s inequality in Josephson phase qubits. Nature 461, 504–506 (2009)

    Article  ADS  Google Scholar 

  15. Giustina, M., Mech, A., Ramelow, S., Wittmann, B., Kofler, J., Beyer, J., Lita, A., Calkins, B., Gerrits, T., Nam, S.W., Ursin, R., Zeilinger, A.: Bell violation using entangled photons without the fair-sampling assumption. Nature 497, 227–230 (2013)

    Article  ADS  Google Scholar 

  16. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Bengtsson, I., Życzkowski, K.: Geometry of Quantum States. An Introduction to Quantum Entanglement. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  18. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  20. Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70, 1895 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. Bennett, C.H., Brassard, G., Popescu, S., Schumacher, B., Smolin, J.A., Wootters, W.K.: Purification of noisy entanglement and faithful teleportation via noisy channels. Phys. Rev. Lett. 76, 722 (1996)

    Article  ADS  Google Scholar 

  22. Giovannetti, V., Lloyd, S., Maccone, L.: Quantum-enhanced measurements: beating the standard quantum limit. Science 306(5700), 1330–1336 (2004)

    Article  ADS  Google Scholar 

  23. Pezzé, L., Smerzi, A.: Entanglement, nonlinear dynamics, and the Heisenberg limit. Phys. Rev. Lett. 102(10), 100401 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  24. Esteve, J., Gross, C., Weller, A., Giovanazzi, S., Oberthaler, M.K.: Squeezing and entanglement in a Bose–Einstein condensate. Nature 455(7217), 1216–1219 (2008)

    Article  ADS  Google Scholar 

  25. Appel, J., Windpassinger, P.J., Oblak, D., Hoff, U.B., Kjærgaard, N., Polzik, E.S.: Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit. PNAS 106(27), 10960–10965 (2009)

    Article  ADS  Google Scholar 

  26. Gross, C., Zibold, T., Nicklas, E., Esteve, J., Oberthaler, M.K.: Nonlinear atom interferometer surpasses classical precision limit. Nature 464(7292), 1165–1169 (2010)

    Article  ADS  Google Scholar 

  27. Leroux, I.D., Schleier-Smith, M.H., Vuletić, V.: Orientation-dependent entanglement lifetime in a squeezed atomic clock. Phys. Rev. Lett. 104(25), 250801 (2010)

    Article  ADS  Google Scholar 

  28. Riedel, M.F., Böhi, P., Li, Y., Hänsch, T.W., Sinatra, A., Treutlein, P.: Atom-chip-based generation of entanglement for quantum metrology. Nature 464(7292), 1170–1173 (2010)

    Article  ADS  Google Scholar 

  29. Chen, Z., Bohnet, J.G., Sankar, S.R., Dai, J., Thompson, J.K.: Conditional spin squeezing of a large ensemble via the vacuum Rabi splitting. Phys. Rev. Lett. 106(13), 133601 (2011)

    Article  ADS  Google Scholar 

  30. Berrada, T., van Frank, S., Bücker, R., Schumm, T., Schaff, J.-F., Schmiedmayer, J.: Integrated Mach–Zehnder interferometer for Bose–Einstein condensates. Nat. Commun. 4, 2077 (2013)

  31. Strobel, H., Muessel, W., Linnemann, D., Zibold, T., Hume, D.B., Pezzé, L., Smerzi, A., Oberthaler, M.K.: Fisher information and entanglement of non-Gaussian spin states. Science 345(6195), 424–427 (2014)

    Article  ADS  Google Scholar 

  32. Lücke, B., Scherer, M., Kruse, J., Pezzé, L., Deuretzbacher, F., Hyllus, J.P., Peise, W.E., Arlt, J., Santos, L., et al.: Twin matter waves for interferometry beyond the classical limit. Science 334(6057), 773–776 (2011)

    Article  ADS  Google Scholar 

  33. Bücker, R., Grond, J., Manz, S., Berrada, T., Betz, T., Koller, C., Hohenester, U., Schumm, T., Perrin, A., Schmiedmayer, J.: Twin-atom beams. Nat. Phys. 7, 608 (2011)

    Article  Google Scholar 

  34. Sørensen, A., Duan, L.-M., Cirac, J.I., Zoller, P.: Many-particle entanglement with Bose–Einstein condensates. Nature 409(6816), 63–66 (2001)

    Article  ADS  Google Scholar 

  35. Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413–1415 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Werner, R.F.: Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277–4281 (1989)

    Article  ADS  Google Scholar 

  37. Wasak, T., Szańkowski, P., Ziń, P., Trippenbach, M., Chwedeńczuk, J.: Cauchy–Schwarz inequality and particle entanglement. Phys. Rev. A 90, 033616 (2014)

    Article  ADS  Google Scholar 

  38. Ichikawa, T., Sasaki, T., Tsutsui, I., Yonezawa, N.: Exchange symmetry and multipartite entanglement. Phys. Rev. A 78, 052105 (2008)

    Article  ADS  Google Scholar 

  39. Wei, T.-C.: Entanglement under the renormalization-group transformations on quantum states and in quantum phase transitions. Phys. Rev. A 81, 062313 (2010)

    Article  ADS  Google Scholar 

  40. Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. Publications of Scuola Normale Superiore, Pisa (2011)

    Book  MATH  Google Scholar 

  41. Tura, J., Augusiak, R., Sainz, A.B., Vértesi, T., Lewenstein, M., Acín, A.: Detecting nonlocality in many-body quantum states. Science 344, 1256 (2014)

  42. Clauser, J.F.: Experimental investigation of a polarization correlation anomaly. Phys. Rev. Lett. 36, 1223 (1976)

    Article  ADS  Google Scholar 

  43. Shchukin, E., Vogel, W.: Inseparability criteria for continuous bipartite quantum states. Phys. Rev. Lett. 95, 230502 (2005)

    Article  ADS  Google Scholar 

  44. Hillery, M., Zubairy, M.S.: Entanglement conditions for two-mode states. Phys. Rev. Lett. 96, 050503 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  45. Miranowicz, A., Bartkowiak, M., Wang, X., Liu, Y., Nori, F.: Testing nonclassicality in multimode fields: a unified derivation of classical inequalities. Phys. Rev. A 82, 013824 (2010)

    Article  ADS  Google Scholar 

  46. de Nova, J.R.M., Sols, F., Zapata, I.: Violation of Cauchy–Schwarz inequalities by spontaneous Hawking radiation in resonant boson structures. Phys. Rev. A 89, 043808 (2014)

    Article  ADS  Google Scholar 

  47. Busch, X., Carusotto, I., Parentani, R.: Spectrum and entanglement of phonons in quantum fluids of light. Phys. Rev. A 89, 043819 (2014)

    Article  ADS  Google Scholar 

  48. Englert, B.-G., Wódkiewicz, K.: Separability of two-party Gaussian states. Phys. Rev. A 65, 054303 (2002)

    Article  ADS  Google Scholar 

  49. Dall, R.G., Manning, A.G., Hodgman, S.S., RuGway, W., Kheruntsyan, K.V., Truscott, A.G.: Ideal n-body correlations with massive particles. Nat. Phys. 9(6), 341–344 (2013)

    Article  Google Scholar 

  50. Wick, G.C., Wightman, A.S., Wigner, E.P.: The intrinsic parity of elementary particles. Phys. Rev. 88, 101–105 (1952)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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Acknowledgments

M. T and T. W. acknowledge the Foundation for Polish Science International Ph.D. Projects Programme co-financed by the EU European Regional Development Fund. T. W., P. Sz. and J. Ch. were supported by the National Science Center Grant no. DEC-2011/03/D/ST2/00200.

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Authors and Affiliations

  1. Institute of Theoretical Physics, University of Warsaw, ul. Pasteura 5, 02-093, Warszawa, Poland

    Tomasz Wasak, Piotr Szańkowski, Marek Trippenbach & Jan Chwedeńczuk

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  1. Tomasz Wasak
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Wasak, T., Szańkowski, P., Trippenbach, M. et al. Cauchy–Schwarz inequality for general measurements as an entanglement criterion. Quantum Inf Process 15, 269–278 (2016). https://doi.org/10.1007/s11128-015-1181-z

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