Non-Gaussian swapping of entangled resources


We investigate the continuous-variable entanglement swapping protocol in a non-Gaussian setting, with non-Gaussian states employed either as entangled inputs and/or as swapping resources. The quality of the swapping protocol is assessed in terms of the teleportation fidelity achievable when using the swapped states as shared entangled resources in a teleportation protocol. We thus introduce a two-step cascaded quantum communication scheme that includes a swapping protocol followed by a teleportation protocol. The swapping protocol is fed by a general class of tunable non-Gaussian states, the Squeezed Bell states, which, by means of controllable free parameters, allows to pass in a continuous way from Gaussian twin beams up to maximally non-Gaussian squeezed number states. In the realistic instance, taking into account the effects of losses and imperfections, we show that as the input two-mode squeezing increases, optimized non-Gaussian swapping resources allow for a monotonically increasing enhancement of the fidelity compared to the corresponding Gaussian setting. This result suggests that the use of non-Gaussian resources can be useful to guarantee the success of continuous-variable entanglement swapping in the presence of decoherence.

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  1. 1.

    Gisin, N., Ribordy, G., Tittel, W., Zbinden, H.: Quantum cryptography. Rev. Mod. Phys. 74, 145–195 (2002)

    ADS  MATH  Google Scholar 

  2. 2.

    Briegel, H.J., Dur, W., Cirac, J.I., Zoller, P.: Quantum repeaters: the role of imperfect local operations in quantum communication. Phys. Rev. Lett. 81, 5932–5935 (1998)

    ADS  Google Scholar 

  3. 3.

    Braunstein, S.L., van Loock, P.: Quantum information with continuous variables. Rev. Mod. Phys. 77, 513–577 (2005)

    ADS  MathSciNet  MATH  Google Scholar 

  4. 4.

    Furusawa, A., Takei, N.: Quantum teleportation for continuous variables and related quantum information processing. Phys. Rep. 443, 97 (2007)

    ADS  MathSciNet  Google Scholar 

  5. 5.

    Khalique, A., Sanders, B.C.: Practical long-distance quantum key distribution through concatenated entanglement swapping with parametric down-conversion sources. J. Opt. Soc. Am. B 32, 2382–2390 (2015)

    ADS  Google Scholar 

  6. 6.

    Asjad, M., Zippilli, S., Tombesi, P., Vitali, D.: Large distance continuous variable communication with concatenated swaps. Phys. Scr. 90, 074055-1–074055-10 (2015)

    ADS  Google Scholar 

  7. 7.

    van Loock, P., Braunstein, S.L.: Unconditional teleportation of continuous-variable entanglement. Phys. Rev. A 61, 010302(R)-1-010302(R)-4 (1999)

    MathSciNet  Google Scholar 

  8. 8.

    Jia, X., Su, X., Pan, Q., Gao, J., Xie, C., Peng, K.: Experimental demonstration of unconditional entanglement swapping for continuous variables. Phys. Rev. Lett. 93, 250503-1–250503-4 (2004)

    ADS  Google Scholar 

  9. 9.

    Yang, T., Zhang, Q., Chen, T.-Y., Lu, S., Yin, J., Pan, J.-W., Wei, Z.-Y., Tian, J.-R., Zhang, J.: Experimental synchronization of independent entangled photon sources. Phys. Rev. Lett. 96, 110501-1–110501-4 (2006)

    ADS  Google Scholar 

  10. 10.

    Jin, R.-B., Takeoka, M., Takagi, U., Shimizu, R., Sasaki, M.: Highly efficient entanglement swapping and teleportation at telecom wavelength. Sci. Rep. 5(9333), 1–7 (2015)

    Google Scholar 

  11. 11.

    Takeda, S., Fuwa, M., van Loock, P., Furusawa, A.: Entanglement swapping between discrete and continuous variables. Phys. Rev. Lett. 114, 100501-1–100501-5 (2015)

    ADS  Google Scholar 

  12. 12.

    Parker, R.C., Joo, J., Razavi, M., Spiller, T.P.: Hybrid photonic loss resilient entanglement swapping. J. Opt. 19, 104004-1–104004-8 (2017)

    ADS  Google Scholar 

  13. 13.

    Hoelscher-Obermaier, J., van Loock, P.: Optimal Gaussian entanglement swapping. Phys. Rev. A 83, 012319-1–012319-9 (2011)

    ADS  Google Scholar 

  14. 14.

    Dell’Anno, F., De Siena, S., Albano, L., Illuminati, F.: Continuous-variable quantum teleportation with non-Gaussian resources. Phys. Rev. A 76, 022301-1–022301-11 (2007)

    ADS  Google Scholar 

  15. 15.

    Dell’Anno, F., De Siena, S., Illuminati, F.: Realistic continuous-variable quantum teleportation with non-Gaussian resources. Phys. Rev. A 81, 012333-1–012333-12 (2010)

    ADS  Google Scholar 

  16. 16.

    Dell’Anno, F., De Siena, S., Albano, L., Illuminati, F.: Continuous variable quantum teleportation with sculptured and noisy non-Gaussian resources. Eur. Phys. J. Spec. Top. 160, 115–126 (2008)

    Google Scholar 

  17. 17.

    Dell’Anno, F., De Siena, S., Adesso, G., Illuminati, F.: Teleportation of squeezing: optimization using non-Gaussian resources. Phys. Rev. A 82, 062329-1-062329-9 (2010)

    ADS  Google Scholar 

  18. 18.

    Kim, M.S., Son, W., Bužek, V., Knight, P.L.: Entanglement by a beam splitter: nonclassicality as a prerequisite for entanglement. Phys. Rev. A 65, 032323-1–032323-7 (2002)

    ADS  Google Scholar 

  19. 19.

    Dodonov, V.V., de Souza, L.A.: Decoherence of superpositions of displaced number states. J. Opt. B Quantum Semiclass. Opt. 7, S490–S499 (2005)

    ADS  Google Scholar 

  20. 20.

    Cerf, N.J., Krüger, O., Navez, P., Werner, R.F., Wolf, M.M.: Non-Gaussian cloning of quantum coherent states is optimal. Phys. Rev. Lett. 95, 070501-1–070501-4 (2005)

    ADS  Google Scholar 

  21. 21.

    Opatrný, T., Kurizki, G., Welsch, D.-G.: Improvement on teleportation of continuous variables by photon subtraction via conditional measurement. Phys. Rev. A 61, 032302-1–032302-7 (2000)

    ADS  Google Scholar 

  22. 22.

    Cochrane, P.T., Ralph, T.C., Milburn, G.J.: Teleportation improvement by conditional measurements on the two-mode squeezed vacuum. Phys. Rev. A 65, 062306-1–062306-6 (2002)

    ADS  Google Scholar 

  23. 23.

    Olivares, S., Paris, M.G.A., Bonifacio, R.: Teleportation improvement by inconclusive photon subtraction. Phys. Rev. A 67, 032314-1–032314-5 (2003)

    ADS  Google Scholar 

  24. 24.

    Kitagawa, A., Takeoka, M., Sasaki, M., Chefles, A.: Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states. Phys. Rev. A 73, 042310-1–042310-12 (2006)

    ADS  Google Scholar 

  25. 25.

    Yang, Y., Li, F.L.: Entanglement properties of non-Gaussian resources generated via photon subtraction and addition and continuous-variable quantum-teleportation improvement. Phys. Rev. A 80, 022315-1–022315-9 (2009)

    ADS  Google Scholar 

  26. 26.

    Adesso, G., Dell’Anno, F., De Siena, S., Illuminati, F., Souza, L.A.M.: Optimal estimation of losses at the ultimate quantum limit with non-Gaussian states. Phys. Rev. A 79, 040305(R)-1–040305(R)-4 (2009)

    ADS  Google Scholar 

  27. 27.

    Menicucci, N.C., van Loock, P., Gu, M., Weedbrook, C., Ralph, T.C., Nielsen, M.A.: Universal quantum computation with continuous-variable cluster states. Phys. Rev. Lett. 97, 110501-1–110501-4 (2006)

    ADS  Google Scholar 

  28. 28.

    Bartley, T.J., Walmsley, I.A.: Directly comparing entanglement-enhancing non-Gaussian operations. New J. Phys. 17, 023038-1-023038-26 (2015)

    ADS  Google Scholar 

  29. 29.

    Wang, S., Hou, L.-L., Chen, X.-F., Xu, X.-F.: Continuous-variable quantum teleportation with non-Gaussian entangled states generated via multiple-photon subtraction and addition. Phys. Rev. A 91, 063832-1-063832-12 (2015)

    ADS  Google Scholar 

  30. 30.

    Seshadreesan, K.P., Dowling, J.P., Agarwal, G.S.: Non-Gaussian entangled states and quantum teleportation of Schroedinger-cat states. Phys. Scr. 90, 074029-1-074029-12 (2015)

    ADS  Google Scholar 

  31. 31.

    Ottaviani, C., Lupo, C., Ferraro, A., Paternostro, M., Pirandola, S.: Multipartite Entanglement Swapping and Mechanical Cluster States. arXiv:1712.08085v1 [quant-ph] 1-5 (2017)

  32. 32.

    Hu, L., Liao, Z., Zubairy, M.S.: Continuous-variable entanglement via multiphoton catalysis. Phys. Rev. A 95, 012310–01325 (2017)

    ADS  Google Scholar 

  33. 33.

    Fan, L., Zubairy, M.S.: Quantum illumination using non-Gaussian states generated by photon subtraction and photon addition. Phys. Rev. A 98, 012319–012326 (2018)

    ADS  Google Scholar 

  34. 34.

    Wolf, M.M., Giedke, G., Cirac, J.I.: Extremality of Gaussian quantum states. Phys. Rev. Lett 96, 080502-1–080502-4 (2006)

    ADS  MathSciNet  Google Scholar 

  35. 35.

    Eisert, J., Scheel, S., Plenio, M.B.: Distilling Gaussian states with Gaussian operations is impossible. Phys. Rev. Lett. 89, 137903-1–137903-4 (2002)

    ADS  MathSciNet  MATH  Google Scholar 

  36. 36.

    Ohliger, M., Kieling, K., Eisert, J.: Limitations of quantum computing with Gaussian cluster states. Phys. Rev. A 82, 042336-1–042336-12 (2010)

    ADS  Google Scholar 

  37. 37.

    Bartlett, S., Sanders, B., Braunstein, S., Nemoto, K.: Efficient classical simulation of continuous variable quantum information processes. Phys. Rev. Lett. 88, 097904 (2002)

    ADS  Google Scholar 

  38. 38.

    Eisert, J., Scheel, S., Plenio, M.B.: Distilling Gaussian states with Gaussian operations is impossible. Phys. Rev. Lett. 89, 137903 (2002)

    ADS  MathSciNet  MATH  Google Scholar 

  39. 39.

    Hage, B., Samblowski, A., Di Guglielmo, J., Franzen, A., Fiurášek, J., Schnabel, R.: Preparation of distilled and purified continuous-variable entangled states. Nat. Phys. 4, 915–918 (2008)

    Google Scholar 

  40. 40.

    Gomes, R.M., Salles, A., Toscano, F., Souto Ribeiro, P.H., Walborn, S.P.: Quantum entanglement beyond Gaussian criteria. PNAS 106(51), 21517–21520 (2009)

    ADS  Google Scholar 

  41. 41.

    Tyc, T., Korolkova, N.: Highly non-Gaussian states created via cross-Kerr nonlinearity. New J. Phys. 10, 023041-1–023041-13 (2008)

    ADS  Google Scholar 

  42. 42.

    Dell’Anno, F., De Siena, S., Illuminati, F.: Structure of multiphoton quantum optics. I. Canonical formalism and homodyne squeezed states. Phys. Rev. A 69, 033812-1–033812-11 (2004)

    ADS  Google Scholar 

  43. 43.

    Dell’Anno, F., De Siena, S., Illuminati, F.: Structure of multiphoton quantum optics. II. Bipartite systems, physical processes, and heterodyne squeezed states. Phys. Rev. A 69, 033813-1–033813-16 (2004)

    ADS  Google Scholar 

  44. 44.

    Agarwal, G.S., Tara, K.: Nonclassical properties of states generated by the excitations on a coherent state. Phys. Rev. A 43, 492–497 (1991)

    ADS  Google Scholar 

  45. 45.

    Bjork, G., Yamamoto, Y.: Generation of nonclassical photon states using correlated photon pairs and linear feedforward. Phys. Rev. A 37, 4229–4239 (1988)

    ADS  Google Scholar 

  46. 46.

    Zhang, Z., Fan, H.: Properties of states generated by excitations on a squeezed vacuum state. Phys. Lett. A 165, 14–18 (1992)

    ADS  MathSciNet  Google Scholar 

  47. 47.

    Dakna, M., Anhut, T., Opatrný, T., Knöll, L., Welsch, D.G.: Generating Schrdinger-cat-like states by means of conditional measurements on a beam splitter. Phys. Rev. A 55, 3184–3194 (1997)

    ADS  Google Scholar 

  48. 48.

    Kim, M.S., Park, E., Knight, P.L., Jeong, H.: Nonclassicality of a photon-subtracted Gaussian field. Phys. Rev. A 71, 043805-1–043805-6 (2005)

    ADS  Google Scholar 

  49. 49.

    Menzies, D., Filip, R.: Gaussian-optimized preparation of non-Gaussian pure states. Phys. Rev. A 79, 012313-1–012313-7 (2009)

    ADS  Google Scholar 

  50. 50.

    Lee, S.-Y., Nha, H.: Quantum state engineering by a coherent superposition of photon subtraction and addition. Phys. Rev. A 82, 053812-1-053812-7 (2010)

    ADS  Google Scholar 

  51. 51.

    Genoni, M.G., Beduini, F.A., Allevi, A., Bondani, M., Olivares, S., Paris, M.G.A.: Non-Gaussian states by conditional measurements. Phys. Scr. T140, 014007-1–014007-5 (2010)

    ADS  Google Scholar 

  52. 52.

    Lee, S.-Y., Ji, S.-W., Kim, H.-J., Nha, H.: Enhancing quantum entanglement for continuous variables by a coherent superposition of photon subtraction and addition. Phys. Rev. A 84, 012302-1-012302-6 (2011)

    ADS  Google Scholar 

  53. 53.

    Xu, X.-X., Yuan, H.-C., Fan, H.-Y.: Generating Hermite polynomial excited squeezed states by means of conditional measurements on a beam splitter. J. Opt. Soc. Am. B 32, 1146–1154 (2015)

    ADS  Google Scholar 

  54. 54.

    Bose, S., Kumar, M.S.: Quantitative study of beam-splitter-generated entanglement from input states with multiple nonclassicality-inducing operations. Phys. Rev. A 95, 012330–012338 (2017)

    ADS  Google Scholar 

  55. 55.

    Zavatta, A., Viciani, S., Bellini, M.: Quantum-to-classical transition with single-photon-added coherent states of light. Science 306, 660–662 (2004)

    ADS  Google Scholar 

  56. 56.

    Lvovsky, A.I., Babichev, S.A.: Synthesis and tomographic characterization of the displaced Fock state of light. Phys. Rev. A 66, 011801(R)-1–011801(R)-4 (2002)

    ADS  Google Scholar 

  57. 57.

    Wenger, J., Tualle-Brouri, R., Grangier, P.: Non-Gaussian statistics from individual pulses of squeezed light. Phys. Rev. Lett. 92, 153601-1–153601-4 (2004)

    ADS  Google Scholar 

  58. 58.

    D’Auria, V., Chiummo, A., De Laurentis, M., Porzio, A., Solimeno, S., Paris, M.G.A.: Tomographic characterization of OPO sources close to threshold. Opt. Expr. 13, 948–956 (2005)

    ADS  Google Scholar 

  59. 59.

    Ourjoumtsev, A., Dantan, A., Tualle-Brouri, R., Grangier, P.: Increasing entanglement between Gaussian states by coherent photon subtraction. Phys. Rev. Lett. 98, 030502-1–030502-4 (2007)

    ADS  Google Scholar 

  60. 60.

    Parigi, V., Zavatta, A., Kim, M., Bellini, M.: Probing quantum commutation rules by addition and subtraction of single photons to/from a light field. Science 317, 1890–1893 (2007)

    ADS  Google Scholar 

  61. 61.

    Ourjoumtsev, A., Jeong, H., Tualle-Brouri, R., Grangier, P.: Generation of optical Schrdinger cats from photon number states. Nature 448, 784–786 (2007)

    ADS  Google Scholar 

  62. 62.

    D’Auria, V., de Lisio, C., Porzio, A., Solimeno, S., Anwar, J., Paris, M.G.A.: Non-Gaussian states produced by close-to-threshold optical parametric oscillators: role of classical and quantum fluctuations. Phys. Rev. A 81, 033846-1-033846-9 (2010)

    ADS  Google Scholar 

  63. 63.

    Etesse, J., Bouillard, M., Kanseri, B., Tualle-Brouri, R.: Experimental generation of squeezed cat states with an operation allowing iterative growth. Phys. Rev. Lett. 114, 193602-1–193602-5 (2015)

    ADS  Google Scholar 

  64. 64.

    Huang, K., Le Jeannic, H., Verma, V.B., Shaw, M.D., Marsili, F., Nam, S.W., Wu, E., Zeng, H., Morin, O., Laurat, J.: Experimental quantum state engineering with time-separated heraldings from a continuous-wave light source: a temporal-mode analysis. Phys. Rev. A 93, 013838-1–013838-8 (2016)

    ADS  Google Scholar 

  65. 65.

    Mari, A., Kieling, K., Nielsen, B.M., Polzik, E.S., Eisert, J.: Directly estimating nonclassicality. Phys. Rev. Lett. 106, 010403–010407 (2011)

    ADS  Google Scholar 

  66. 66.

    Miranowicz, A., Bartkowiak, M., Wang, X., Liu, Y.-X., Nori, F.: Testing nonclassicality in multimode fields: a unified derivation of classical inequalities. Phys. Rev. A 82, 013824–013838 (2010)

    ADS  Google Scholar 

  67. 67.

    Ivan, J.S., Chaturvedi, S., Ercolessi, E., Marmo, G., Morandi, G., Mukunda, N., Simon, R.: Entanglement and nonclassicality for multimode radiation-field states. Phys. Rev. A 83, 032118–032138 (2011)

    ADS  Google Scholar 

  68. 68.

    Park, J., Zhang, J., Lee, J., Ji, S.-W., Um, M., Lv, D., Kim, K., Nha, H.: Testing nonclassicality and non-Gaussianity in phase space. Phys. Rev. Lett. 114, 190402–190407 (2015)

    ADS  Google Scholar 

  69. 69.

    Kühn, B., Vogel, W.: Quantum non-Gaussianity and quantification of nonclassicality. Phys. Rev. A 97, 053823–053830 (2018)

    ADS  Google Scholar 

  70. 70.

    Ivan, J.S., Kumar, M.S., Simon, R.: A measure of non-Gaussianity for quantum states. Quantum Inf. Process. 11, 853 (2012)

    MathSciNet  MATH  Google Scholar 

  71. 71.

    Genoni, M.G., Paris, M.G.A., Banaszek, K.: Measure of the non-Gaussian character of a quantum state. Phys. Rev. A 76, 042327–042333 (2007)

    ADS  Google Scholar 

  72. 72.

    Genoni, M.G., Paris, M.G.A., Banaszek, K.: Quantifying the non-Gaussian character of a quantum state by quantum relative entropy. Phys. Rev. A 78, 060303–060307 (2008)

    ADS  MathSciNet  Google Scholar 

  73. 73.

    Genoni, M.G., Paris, M.G.A.: Quantifying non-Gaussianity for quantum information. Phys. Rev. A 82, 052341–052360 (2010)

    ADS  Google Scholar 

  74. 74.

    Barbieri, M., Spagnolo, N., Genoni, M.G., Ferreyrol, F., Blandino, R., Paris, M.G.A., Grangier, P., Tualle-Brouri, R.: Non-Gaussianity of quantum states: an experimental test on single-photon-added coherent states. Phys. Rev. A 82, 063833–063838 (2010)

    ADS  Google Scholar 

  75. 75.

    Marian, P., Marian, T.A.: Relative entropy is an exact measure of non-Gaussianity. Phys. Rev. A 88, 012322–012328 (2013)

    ADS  Google Scholar 

  76. 76.

    Hughes, C., Genoni, M.G., Tufarelli, T., Paris, M.G.A., Kim, M.S.: Quantum non-Gaussianity witnesses in phase space. Phys. Rev. A 90, 013810–013819 (2014)

    ADS  Google Scholar 

  77. 77.

    Dell’Anno, F., De Siena, S., Illuminati, F.: Multiphoton quantum optics and quantum state engineering. Phys. Rep. 428, 53–168 (2006)

    ADS  MathSciNet  Google Scholar 

  78. 78.

    Dell’Anno, F., Buono, D., Nocerino, G., Porzio, A., Solimeno, S., De Siena, S., Illuminati, F.: Tunable non-Gaussian resources for continuous-variable quantum technologies. Phys. Rev. A 88, 043818-1–043818-13 (2013)

    ADS  Google Scholar 

  79. 79.

    Xiang, S.-H., Song, K.-H.: Quantum non-Gaussianity of single-mode Schrdinger cat states based on Kurtosis. EPJD 69, 260 (2015)

    ADS  Google Scholar 

  80. 80.

    Park, J., Lee, J., Ji, S.-W., Nha, H.: Quantifying non-Gaussianity of quantum-state correlation. Phys. Rev. A 96, 052324–052333 (2017)

    ADS  Google Scholar 

  81. 81.

    Leonhardt, U., Paul, H.: Realistic optical homodyne measurements and quasiprobability distributions. Phys. Rev. A 48, 4598–4604 (1993)

    ADS  Google Scholar 

  82. 82.

    Marian, P., Marian, T.A.: Continuous-variable teleportation in the characteristic-function description. Phys. Rev. A 74, 042306-1–042306-5 (2006)

    ADS  Google Scholar 

  83. 83.

    Vahlbruch, H., Mehmet, M., Chelkowski, S., Hage, B., Franzen, A., Lastzka, N., Gossler, S., Danzmann, K., Schnabel, R.: Observation of squeezed light with 10-dB quantum-noise reduction. Phys. Rev. Lett 100, 033602-1–033602-4 (2008)

    ADS  Google Scholar 

  84. 84.

    Vahlbruch, H., Khalaidovski, A., Lastzka, N., Gräf, C., Danzmann, K., Schnabel, R.: The GEO600 squeezed light source. Class. Quantum Grav. 27, 084027-1–084027-8 (2010)

    ADS  Google Scholar 

  85. 85.

    Eberle, T., Steinlechner, S., Bauchrowitz, J., Händchen, V., Vahlbruch, H., Mehmet, M., Müller-Ebhardt, H., Schnabel, R.: Quantum enhancement of the zero-area sagnac interferometer topology for gravitational wave detection. Phys. Rev. Lett. 104, 251102-1–251102-4 (2010)

    ADS  Google Scholar 

  86. 86.

    Mehmet, M., Ast, S., Eberle, T., Steinlechner, S., Vahlbruch, H., Schnabel, R.: Squeezed light at 1550 nm with a quantum noise reduction of 12.3 dB. Opt. Express 19, 25764–25772 (2011)

    ADS  Google Scholar 

  87. 87.

    Vollmer, C.E., Baune, C., Samblowski, A., Eberle, T., Händchen, V., Fiurásek, J., Schnabel, R.: Quantum up-conversion of squeezed vacuum states from 1550 to 532 nm. Phys. Rev. Lett. 112, 073602-1-073602-5 (2014)

    ADS  Google Scholar 

  88. 88.

    Vahlbruch, H., Mehmet, M., Danzmann, K., Schnabel, R.: Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency. Phys. Rev. Lett. 117, 110801-1–110801-5 (2016)

    ADS  Google Scholar 

  89. 89.

    Andersen, U.L., Gehring, T., Marquardt, C., Leuchs, G.: 30 Years of squeezed light generation. Phys. Scr. 91, 053001-1–053001-11 (2016)

    ADS  Google Scholar 

  90. 90.

    Hu, L.-Y., Liao, Z., Ma, S., Zubairy, M.S.: Optimal fidelity of teleportation with continuous variables using three tunable parameters in a realistic environment. Phys. Rev. A 93, 033807-1–033807-10 (2016)

    ADS  Google Scholar 

  91. 91.

    Walls, D., Milburn, G.: Quantum Optics. Springer, Berlin (1994)

    Google Scholar 

  92. 92.

    Serafini, A., Paris, M.G.A., Illuminati, F., De Siena, S.: Quantifying decoherence in continuous variable systems. J. Opt. B Quantum Semiclass. Opt. 7, R19–R36 (2005)

    ADS  Google Scholar 

  93. 93.

    Bowen, W.P., Treps, N., Buchler, B.C., Schnabel, R., Ralph, T.C., Symul, T., Lam, P.K.: Unity gain and nonunity gain quantum teleportation. IEEE J. Sel. Top. Quant. 9, 1519–1532 (2003)

    Google Scholar 

  94. 94.

    Adesso, G., Illuminati, F.: Equivalence between entanglement and the optimal fidelity of continuous variable teleportation. Phys. Rev. Lett. 95, 150503-1–150503-4 (2005)

    ADS  Google Scholar 

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We acknowledge the EU FP7 Cooperation STREP Project EQuaM—Emulators of Quantum Frustrated Magnetism, Grant Agreement No. 323714. We also acknowledge financial support from the Italian Minister of Scientific Research (MIUR) under the national PRIN program.

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Correspondence to Silvio De Siena.


A CV entanglement swapping protocol in the characteristic function representation

The characteristic function representation provides a most elegant and compact description of the CV teleportation protocol [82]. Such description proves to be particularly convenient in the instance of non-Gaussian resources [14,15,16,17] and has been generalized to include the non-ideal case [15]. In this appendix, we apply this formalism to the description of the realistic CV entanglement swapping protocol, schematically illustrated in Fig. 1.

Let \(\rho _{12}^{A}\) and \(\rho _{34}^{B}\) denote the density matrices associated with the two-mode entangled input pure state of modes 1 and 2, and the two-mode entangled pure resource of modes 3 and 4, respectively. The global four-mode initial state is the biseparable state \(\rho _{0}= \rho _{12}^{A} \otimes \rho _{34}^{B}\) and the corresponding initial global characteristic function \(\chi _{0}\) associated with \(\rho _{0}\) reads:

$$\begin{aligned} \chi _{0}(\alpha _1;\alpha _2;\alpha _3;\alpha _4)= Tr\left[ \prod _{j=1}^{4} D_j (\alpha _j) \rho _{0}\right] = \chi _{12}(\alpha _1;\alpha _2) \; \chi _{34}(\alpha _3;\alpha _4) \,, \end{aligned}$$

where Tr denotes the trace operation, \(D_j (\alpha _j)\) denotes the displacement operator of mode j \((j=1,\ldots ,4)\), \(\chi _{12}\) is the characteristic function of the two-mode input state, and \(\chi _{34}\) is the characteristic function of the two-mode resource. By introducing the quadrature operators \(X_{j}=\frac{1}{\sqrt{2}}(a_{j}+a_{j}^{\dag })\) and \(P_{j}=\frac{i}{\sqrt{2}}(a_{j}^{\dag }-a_{j})\), and the corresponding phase space variables \(x_{j}= \frac{1}{\sqrt{2}}(\alpha _{j}+\alpha _{j}^{*})\) and \(p_{j}=\frac{i}{\sqrt{2}}( \alpha _{j}^{*}-\alpha _{j})\), the characteristic function can be written in terms of \(x_j\), \(p_j\), i.e., \(\chi _{0}(\alpha _1;\alpha _2;\alpha _3;\alpha _4)\equiv \chi _{0}(x_1,p_1;x_2,p_2;x_3,p_3;x_4,p_4)\).

The first step of the protocol consists of a Bell measurement at the first user’s location. The modes 2 and 3 are mixed at a balanced beam splitter; the effects of photon losses and the inefficiencies of the photodetectors are simulated by two additional fictitious beam splitters placed in front of the detectors, characterized by the transmissivities \(T_{j}^2\) (reflectivity \(R_{j}^2=1-T_{j}^2\)), \(j=2,3\). Let us denote by \({\tilde{x}}\) and \({\tilde{p}}\) the homodyne measurements of the first quadrature of the mode 3 and of the second quadrature of the mode 2, respectively. The description of realistic Bell measurements using the formalism of the characteristic function is discussed in full detail in Ref. [15]. Here we just provide the final expression of the characteristic function \(\chi _{Bm}(x_1,p_1;x_4,p_4)\) associated with the entire measurement process:

$$\begin{aligned} \chi _{Bm}(x_1,p_1;x_4,p_4)= & {} \frac{{\mathcal {P}}^{-1}({\tilde{p}},{\tilde{x}}) }{(2\pi )^{2}} \int \hbox {d}\xi \hbox {d}\upsilon \, e^{i\xi {\tilde{p}} - i {\tilde{x}} \upsilon } \exp \left[ -\frac{R_2^2}{4} \xi ^2 -\frac{R_3^2}{4} \upsilon ^2 \right] \nonumber \\&\times \chi _{12} \left( x_1,p_1; \frac{T_2 \xi }{\sqrt{2}},\frac{T_3 \upsilon }{\sqrt{2}} \right) \chi _{34} \left( \frac{T_2 \xi }{\sqrt{2}},-\frac{T_3 \upsilon }{\sqrt{2}}; x_4,p_4\right) \,, \nonumber \\ \end{aligned}$$

where the function \({\mathcal {P}}({\tilde{p}},{\tilde{x}})\) is the distribution of the measurement outcomes \({\tilde{p}}\) and \({\tilde{x}}\), that is:

$$\begin{aligned} {\mathcal {P}}({\tilde{p}},{\tilde{x}})= & {} \frac{1 }{(2\pi )^{2}} \int \hbox {d}\xi \hbox {d}\upsilon \, e^{i\xi {\tilde{p}} - i {\tilde{x}} \upsilon } \exp \left[ -\frac{R_2^2}{4} \xi ^2 -\frac{R_3^2}{4} \upsilon ^2 \right] \nonumber \\&\times \chi _{12} \left( 0,0; \frac{T_2 \xi }{\sqrt{2}},\frac{T_3 \upsilon }{\sqrt{2}} \right) \chi _{34} \left( \frac{T_2 \xi }{\sqrt{2}},-\frac{T_3 \upsilon }{\sqrt{2}};0,0\right) . \end{aligned}$$

After measurement, modes 1 and 4 propagate in noisy channels (e.g., optical fibers) toward Alice’s and Bob’s locations, respectively. The dynamics of a multimode system subject to decoherence is described, in the interaction picture, by the following master equation for the density operator \(\rho \) [91, 92]:

$$\begin{aligned} \partial _{t} \rho \,=\, \sum _{i=1,4} \frac{\Upsilon _i}{2} \left\{ n_{th,i} L[a_{i}^{\dag }] \rho + (n_{th,i}+1) L[a_{i}] \rho \right\} \,, \end{aligned}$$

where the Lindblad superoperators are defined as \(L[{\mathcal {O}}] \rho \equiv 2 {\mathcal {O}} \rho \mathcal {O^{\dag }} - \mathcal {O^{\dag }} {\mathcal {O}} \rho - \rho \mathcal {O^{\dag }} {\mathcal {O}}\), \(\Upsilon _i\) is the mode damping rate, and \(n_{th,i}\) is the number of thermal photons in mode i. Because of decoherence due to propagation in the noisy channels, the characteristic function (11) can be rewritten in the following form:

$$\begin{aligned} \chi _{t}(x_{1},p_{1};x_{4},p_{4}) \,=\,&\chi _{Bm}(e^{-\frac{1}{2}\Upsilon _1 t}x_{1},e^{-\frac{1}{2}\Upsilon _1 t}p_{1};e^{-\frac{1}{2}\Upsilon _4 t}x_{4},e^{-\frac{1}{2}\Upsilon _4 t}p_{4}) \nonumber \\&\times \; e^{-\frac{1}{2} \sum _{i=1,4} (1-e^{-\Upsilon _i t})\left( \frac{1}{2}+n_{th,i}\right) (x_{i}^{2}+p_{i}^{2})}. \end{aligned}$$

The description of the technical features of the experimental apparatus, e.g., the efficiency of the photodetectors, and characteristics as the length of the channels (fibers), and the temperature of the environment, is complete once the quantities \(T_{j}\) (equivalently \(R_{j}\), \(j=2,3\)), \(\Upsilon _i\), and \(n_{th,i}\) (\(i=1,4\)) are specified and fixed at certain given values.

In the last step of the protocol, two local unitary displacements \(\lambda _1\) and \(\lambda _4\) are performed at Alice’s and Bob’s locations; a local unitary displacement \(\lambda _1 = - g_1 ( {\tilde{x}} - i{\tilde{p}})\) is performed on mode 1, and a local unitary displacement \(\lambda _4 = g_4 ( {\tilde{x}} + i{\tilde{p}})\) is performed on mode 4. The real parameters \(g_1\) and \(g_4\) denote the gain factors of the displacement transformations [93]. After such local unitary operations, the characteristic function reads:

$$\begin{aligned} \chi _D(x_1,p_1;x_4,p_4) = e^{-i\sqrt{2}{\tilde{x}}(g_1 p_1 - g_4 p_4)-i\sqrt{2}{\tilde{p}}(g_1 x_1 + g_4 x_4)} \; \chi _{t}(x_1,p_1;x_4,p_4) . \end{aligned}$$

Finally, in order to obtain the output characteristic function \(\chi _{out}(x_1,p_1;x_4,p_4)\), describing the output two-mode entangled state of the entanglement swapping protocol, one must take the average over all the possible outcomes \({\tilde{p}}\) and \({\tilde{x}}\) of the Bell measurements:

$$\begin{aligned} \chi _\mathrm{out}^{(\mathrm swapp)}(x_1,p_1;x_4,p_4) = \int \hbox {d}{\tilde{x}} \hbox {d}{\tilde{p}} {\mathcal {P}}({\tilde{p}},{\tilde{x}})\chi _D(x_1,p_1;x_4,p_4), \end{aligned}$$

where \(\tau _i = \Upsilon _i t\). The above integral yields the final expression (3) for the characteristic function associated with the swapped resource.

The core mathematical task is thus the explicit evaluation of the characteristic function \(\chi _\mathrm{out}^{(\mathrm swapp)}(x_1,p_1;x_4,p_4)\) associated with the output of the realistic swapping protocol, as expressed by Eq. (3). We have determined its analytical expression for the most general non-Gaussian setting, that is the entanglement swapping of SB input states using SB states as resources. As the class of SB states contains as special cases both the Gaussian TB states and the non-Gaussian PS states, the general expression of the output characteristic function reduces to the explicit expression for these special Gaussian and non-Gaussian cases as well. We do not report here the explicit analytical expression of \(\chi _\mathrm{out}^{(\mathrm swapp)}(x_1,p_1;x_4,p_4)\), as it is exceedingly long and cumbersome and does not yield any particularly useful physical insight. On the other hand, having obtained the explicit expression of the output characteristic function of the swapping protocol, it is straightforward to compute the output characteristic function \(\chi _\mathrm{out}^{(\mathrm telep)}(x_{4},p_{4})\) of the subsequent ideal teleportation protocol, Eq. (5).

Finally, we have derived the analytical expression for the teleportation fidelity \({\mathcal {F}}_{X^{sw}Y}\), Eq. (6), which in the most general instance is \({\mathcal {F}}_{\mathrm{SB}^{sw}\mathrm{SB}}\). Such a fidelity depends on the following parameters: the squeezing amplitudes and phases \(r_{12}\), \(\phi _{12}\), \(r_{34}\), \(\phi _{34}\) and the angles and phases \(\delta _{12}\), \(\theta _{12}\), \(\delta _{34}\), \(\theta _{34}\) of the input states and of the resources; the parameters associated with the experimental apparatus are listed in Table 2. Without loss of generality, as already verified in Refs. [14, 15], one can obtain some significant simplifications by fixing the phases of the SB states. Specifically, we set the non-Gaussian phases \(\theta _{12}=\theta _{34}=0\) and the squeezing phases \(\phi _{12}=\phi _{34}=\pi \) in Eq. (1). With such choice, the teleportation fidelity depends on the two gains \(g_i\) \((i=1,4)\) through the total gain parameter \({\tilde{g}}=g_1+g_4\), both in the ideal and in the realistic protocols.

B Performance criteria for the swapping protocol

In the paper, to check efficiency a specific criterion have been exploited, based on a cascaded scheme, i.e., a swapping protocol followed by a teleportation protocol. We mentioned in the introduction, in fact, that, unlike the Gaussian case where only the entanglement content is relevant [94], when non-Gaussian resources are exploited the intrinsic nonlinearity of interaction leads to different results depending on the specific quantity chosen to be optimized.

We now show this aspect by optimizing three different objects:

  1. (a)

    the squeezing content of the output swapped state,

  2. (b)

    the fidelity between the input and the output two-mode states,

  3. (c)

    the fidelity of teleportation of a coherent state, following the cascade scheme adopted in this paper.

For a better understanding, here we resort to a simplified experimental setup: we consider an ideal swapping protocol, with a Gaussian twin beam as the input two-mode entangled state, and a Bell-like state as the two-mode entangled resource, which are described, respectively, by the following characteristic functions:

$$\begin{aligned}&\chi _{12}^{(TwB)}(x_1,p_1;x_2,p_2)= e^{-\frac{1}{4}\cosh 2r_{12}(x_1^2+p_1^2+x_2^2+p_2^2)+\frac{1}{2}\sinh 2r_{12}(x_1 x_2-p_1 p_2)} \,, \end{aligned}$$
$$\begin{aligned}&\chi _{34}^{(Bell)}(x_3,p_3;x_4,p_4)= e^{-\frac{1}{4}(x_3^2+p_3^2+x_4^2+p_4^2)} \Big \{\cos ^2 \delta _{34} +(x_3 x_4 -p_3 p_4) \sin \delta _{34} \cos \delta _{34} \nonumber \\&\quad +\left( \frac{x_3^2+p_3^2}{2}-1\right) \left( \frac{x_4^2+p_4^2}{2}-1\right) \sin ^2 \delta _{34} \Big \} \,. \end{aligned}$$

With the above simplified choice of the resource, the non-Gaussian character is preserved together with the presence of parameters (angles) that can be optimized, but it is possible to explicitly compute all the quantities one is interested in.

According to Eq. (4) for the swapping output, in this case we have a swapped state described by the characteristic function:

$$\begin{aligned} \chi _{14}^{(sw)}(x_1,p_1;x_4,p_4)= & {} e^{-\frac{1}{4}\cosh 2r_{12}(x_1^2+p_1^2+x_4^2+p_4^2)-\frac{1}{2}(x_4^2+p_4^2) +\frac{1}{2}\sinh 2r_{12}(x_1x_4-p_1p_4)} \nonumber \\&\times \left\{ 1+ (\cos 2\delta _{34}-1) \left( \frac{x_4^2+p_4^2}{2}-\frac{x_4^4+p_4^4}{8}- \frac{x_4^2 p_4^2}{4} \right) \right. \nonumber \\&\quad \left. +\sin 2\delta _{34}\frac{x_4^2- p_4^2}{2} \right\} . \end{aligned}$$

Now we perform the optimization procedure in the three different cases.

  1. (a)

    Optimization of the squeezing content of the output swapped state. Given the generalized two-mode quadratures \(X_{hk}=\frac{1}{\sqrt{2}}(a_h+a_k +h.c.)\) and \(P_{hk}=\frac{i}{\sqrt{2}}(a_h^{\dag }+a_k^{\dag } - h.c.)\), it is straightforward to compute the variances \(\Delta X_{hk}^2\) and \(\Delta P_{hk}^2\) by exploiting the above analytical expressions for characteristic functions. The squeezed quadrature for the input twin beam Eq. (17) is \(P_{12}\), i.e., \(\Delta P_{12}^2 = e^{-2r_{12}}\), and the optimization procedure implies minimization over the non-Gaussian parameter \(\delta _{34}\) of the difference between the second quadrature variances associated with the swapped state and the input state:

    $$\begin{aligned} \min _{\delta _{34}} \left\{ (\Delta P_{14}^2)^{(sw)}-(\Delta P_{12}^2)^{(in)} \right\} \,. \end{aligned}$$

    Minimization leads to the optimal value (independent of \(r_{12}\)) for the angle \(\delta _{34}^{(\mathrm opt)}\) that is reported in the first row of Table 3.

  2. (b)

    Optimization of the fidelity between the input and the output two-mode states. In this case we must compute:

    $$\begin{aligned} \max _{\delta _{34}} {\mathcal {F}}(r_{12},\delta _{34}), \end{aligned}$$


    $$\begin{aligned}&{\mathcal {F}}(r_{12},\delta _{34}) \end{aligned}$$
    $$\begin{aligned}&= \frac{1}{(2\pi )^2} \int \hbox {d}x_1 \hbox {d}p_1 \hbox {d}x_4 \hbox {d}p_4 \chi _{12}^{(\mathrm TB)}(x_1,p_1;x_4,p_4) \chi _{14}^{(sw)}(-x_1,-p_1;-x_4,-p_4). \end{aligned}$$

    Maximization leads to the optimal value for the angle \(\delta _{34}^{(\mathrm opt)}\) that is reported in the second row of Table 3.

  3. (c)

    Optimization of the fidelity of teleportation of a coherent state, following the cascade scheme adopted in this paper.

The cascaded scheme here described provides in this simplified case the optimal angle \(\delta _{34}^{(\mathrm opt)}\) reported in the third row of Table 3.

Thus, the optimized angles do not coincide, identifying different optimized resources for the three different cases.

Table 3 Optimized angles

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Dell’Anno, F., Buono, D., Nocerino, G. et al. Non-Gaussian swapping of entangled resources. Quantum Inf Process 18, 20 (2019).

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  • Entanglement
  • Swapping
  • Non-Gaussianity