Maximally entangled states in discrete and Gaussian regimes

  • Youngrong Lim
  • Jaewan Kim
  • Soojoon Lee
  • Kabgyun JeongEmail author


We study a relation between discrete-variable quantum states and continuous-variable (especially, restricted on Gaussian) ones. In the previous work, we have investigated an information-theoretic correspondence between the Gaussian maximally mixed states and their purifications as Gaussian maximally entangled states in Jeong and Lim (Phys Lett A 380:3607, 2016). We here compare the purified continuous-variable maximally entangled state with a two-mode squeezed vacuum state, which is a conventional entangled state in Gaussian regime, by the explicit calculation of quantum fidelities between those states and an \(N\times N\)-dimensional maximally entangled state in the finite Hilbert space. Consequently, we naturally conclude that the purified maximally entangled state is more suitable to the Gaussian maximally entangled state than the two-mode squeezed vacuum state, in a sense that it might be useful for continuous-variable quantum information tasks in which entangled states are needed.


Gaussian maximally entangled (mixed)state Two-mode squeezed vacuum state Dimension-mode matching Qutrit Bell test Photon number entangled state 



This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2017R1A6A3A01007264) and the Ministry of Science and ICT (NRF-2016R1A2B4014928). J.K. appreciates the financial support by the KIST Institutional Program (Project No. 2E26680-16-P025). K.J. acknowledges financial support by the National Research Foundation of Korea (NRF) through a grant funded by the Ministry of Science and ICT (NRF-2017R1E1A1A03070510 and NRF-2017R1A5A1015626) and the Ministry of Education (NRF-2018R1D1A1B07047512).


  1. 1.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  2. 2.
    Wilde, M.M.: Quantum Information Theory. Cambridge University Press, Cambridge (2013)CrossRefGoogle Scholar
  3. 3.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26, 1484 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325 (1997)ADSCrossRefGoogle Scholar
  5. 5.
    Bennett, C.H., Brassard, G.: Quantum cryptography: public key distribution and coin tossing. In: Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, vol. 175, p. 8, New York (1984)Google Scholar
  6. 6.
    Ekert, A.K.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661 (1991)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bennett, C.H.: Quantum cryptography using any two nonorthogonal states. Phys. Rev. Lett. 68, 3121 (1992)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Smith, G., Yard, J.: Quantum communication with zero-capacity channels. Science 321, 1812 (2008)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Hastings, M.B.: Superadditivity of communication capacity using entangled inputs. Nat. Phys. 5, 255 (2009)CrossRefGoogle Scholar
  10. 10.
    Li, K., Winter, A., Zou, X., Guo, G.: Private capacity of quantum channels is not additive. Phys. Rev. Lett. 103, 120501 (2009)ADSCrossRefGoogle Scholar
  11. 11.
    Einstein, A., Podolsky, B., Rogen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935)ADSCrossRefGoogle Scholar
  12. 12.
    Eisert, J., Wolf, M.M.: Continuous-variable quantum information science. In: Leuchs, G., Cerf, N., Polzik, E. (eds.) Quantum Information with Continuous Variables, Part II: Optical continuous variables. Imperial College Press, London (2005)Google Scholar
  13. 13.
    Weedbrook, C., Pirandola, S., García-Patrón, R., Cerf, N.J., Ralph, T.C., Shapiro, J.H., Lloyd, S.: Gaussian quantum information. Rev. Mod. Phys. 84, 621 (2012)ADSCrossRefGoogle Scholar
  14. 14.
    Brukner, C̆., Kim, M.S., Pan, J.-W., Zeilinger, A.: Correspondence between continuous-variable and discrete quantum systems of arbitrary dimensions. Phys. Rev. A 68, 062105 (2003)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    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)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Braunstein, S.L., Kimble, H.J.: Teleportation of continuous quantum variables. Phys. Rev. Lett. 80, 869 (1998)ADSCrossRefGoogle Scholar
  17. 17.
    Briegel, H.-J., Dür, W., Cirac, J.I., Zoller, P.: Quantum repeaters: the role of imperfect local operations in quantum communication. Phys. Rev. Lett. 81, 5932 (1998)ADSCrossRefGoogle Scholar
  18. 18.
    Vollbrecht, K.G.H., Muschik, C.A., Cirac, J.I.: Entanglement distillation by dissipation and continuous quantum repeaters. Phys. Rev. Lett. 107, 120502 (2011)ADSCrossRefGoogle Scholar
  19. 19.
    Hillery, M., Bužek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A 59, 1829 (1999)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Tyc, T., Sanders, B.C.: How to share a continuous-variable quantum secret by optical interferometry. Phys. Rev. A 65, 042310 (2002)ADSCrossRefGoogle Scholar
  21. 21.
    Banaszek, K., Wódkiewicz, K.: Nonlocality of the Einstein–Podolsky–Rosen state in the phase space. Acta Phys. Slov. 49, 491 (1999)zbMATHGoogle Scholar
  22. 22.
    Facchi, P., Florio, G., Lupo, C., Mancini, S., Pascazio, S.: Gaussian maximally multipartite entangled states. Phys. Rev. A 80, 062311 (2009)ADSCrossRefGoogle Scholar
  23. 23.
    Jeong, K., Lim, Y.: Purification of Gaussian maximally mixed states. Phys. Lett. A 380, 3607 (2016)ADSCrossRefGoogle Scholar
  24. 24.
    Brádler, K.: Continuous-variable private quantum channel. Phys. Rev. A 72, 042313 (2005)ADSCrossRefGoogle Scholar
  25. 25.
    Jeong, K., Kim, J., Lee, S.-Y.: Gaussian private quantum channel with squeezed coherent states. Sci. Rep. 5, 13974 (2015)ADSCrossRefGoogle Scholar
  26. 26.
    de Palma, G., Mari, A., Giovannetti, V., Holevo, A.S.: Normal form decomposition for Gaussian-to-Gaussian superoperators. J. Math. Phys. 56, 052202 (2015)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Collins, D., Gisin, N., Linden, N., Massar, S., Popescu, S.: Bell inequalities for arbitrarily high-dimensional systems. Phys. Rev. Lett. 88, 040404 (2002)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Fu, L.-B., Chen, J.-L., Zhao, X.-G.: Maximal violation of the Clauser–Horne–Shimony–Holt inequality for two qutrits. Phys. Rev. A 68, 022323 (2003)ADSCrossRefGoogle Scholar
  29. 29.
    Kaszlikowski, D., Kwek, L.C., Chen, J.L., Żukowski, M., Oh, C.H.: Clauser–Horne inequality for three-state systems. Phys. Rev. A 65, 032118 (2002)ADSCrossRefGoogle Scholar
  30. 30.
    Chen, J.-L., Kaszlikowski, D., Kwek, L.C., Oh, C.H.: Wringing out new Bell inequalities for three-dimensional systems (qutrits). Mod. Phys. Lett. A 17, 2231 (2002)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Źukowski, M., Zeilinger, A., Horne, M.A.: Realizable higher-dimensional two-particle entanglements via multiport beam splitters. Phys. Rev. A 55, 2564 (1997)ADSCrossRefGoogle Scholar
  32. 32.
    Kurochkin, Y., Prasad, A.S., Lvovsky, A.I.: Distillation of the two-mode squeezed state. Phys. Rev. Lett. 112, 070402 (2014)ADSCrossRefGoogle Scholar
  33. 33.
    Tombesi, P., Mecozzi, A.: Generation of macroscopically distinguishable quantum states and detection by the squeezed-vacuum technique. J. Opt. Soc. Am. B 4, 1700 (1987)ADSCrossRefGoogle Scholar
  34. 34.
    Sanders, B.C.: Entangled coherent states. Phys. Rev. A 45, 6811 (1992)ADSCrossRefGoogle Scholar
  35. 35.
    Lee, S.-Y., Park, J., Lee, H.-W., Nha, H.: Generating arbitrary photon-number entangled states for continuous-variable quantum informatics. Opt. Express 20, 14221 (2012)ADSCrossRefGoogle Scholar
  36. 36.
    Adesso, G., Serafini, A., Illuminati, F.: Multipartite entanglement in three-mode Gaussian states of continuous-variable systems: quantification, sharing structure, and decoherence. Phys. Rev. A 73, 032345 (2006)ADSCrossRefGoogle Scholar
  37. 37.
    Werner, R.F., Wolf, M.M.: Bound entangled Gaussian states. Phys. Rev. Lett. 86, 3658 (2001)ADSCrossRefGoogle Scholar
  38. 38.
    Hioe, F.T., Eberly, J.H.: \(N\)-level coherence vector and higher conservation laws in quantum optics and quantum mechanics. Phys. Rev. Lett. 47, 838 (1981)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    van Enk, S.: Entanglement capabilities in infinite dimensions: multidimensional entangled coherent states. Phys. Rev. Lett. 91, 017902 (2003)ADSCrossRefGoogle Scholar
  40. 40.
    Cheong, Y.W., Lee, J.: Generation of entangled coherent states. J. Korean Phys. Soc. 51, 1513 (2007)ADSCrossRefGoogle Scholar
  41. 41.
    Kim, J., Lee, J., Ji, S.-W., Nha, H., Anisimov, P.M., Dowling, J.P.: Coherent-state optical qudit cluster state generation and teleportation via homodyne detection. Opt. Commun. 337, 79 (2015)ADSCrossRefGoogle Scholar
  42. 42.
    Wang, C., et al.: A Schrödinger cat living in two boxes. Science 352, 1087 (2016)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    Xu, Y., et al.: Geometrically manipulating photonic Schrödinger cat states and realizing cavity phase gates. arXiv:1810.04690
  44. 44.
    Hirota, O., Sasaki, M.: Entangled state based on nonorthogonal state. In: Quantum Communication, Measurement, and Computing, vol. 3, pp. 359–366. Springer, New York (2001)Google Scholar
  45. 45.
    Lupo, C., Mancini, S., de Pasquale, A., Facchi, P., Florio, G., Pascazio, S.: Invariant measures on multimode quantum Gaussian states. J. Math. Phys. 53, 122209 (2012)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    Strang, G.: Introduction to Linear Algebra. Wellesley-Cambridge Press, Wellesley (1993)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and Research Institute for Basic SciencesKyung Hee UniversitySeoulKorea
  2. 2.IMDARC, Department of Mathematical SciencesSeoul National UniversitySeoulKorea
  3. 3.School of Computational SciencesKorea Institute for Advanced StudySeoulKorea

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