# Maximally entangled states in discrete and Gaussian regimes

## Abstract

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.

## Keywords

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

### Acknowledgements

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).

## References

- 1.Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
- 2.Wilde, M.M.: Quantum Information Theory. Cambridge University Press, Cambridge (2013)CrossRefGoogle Scholar
- 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.Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett.
**79**, 325 (1997)ADSCrossRefGoogle Scholar - 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.Ekert, A.K.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett.
**67**, 661 (1991)ADSMathSciNetCrossRefGoogle Scholar - 7.Bennett, C.H.: Quantum cryptography using any two nonorthogonal states. Phys. Rev. Lett.
**68**, 3121 (1992)ADSMathSciNetCrossRefGoogle Scholar - 8.Smith, G., Yard, J.: Quantum communication with zero-capacity channels. Science
**321**, 1812 (2008)ADSMathSciNetCrossRefGoogle Scholar - 9.Hastings, M.B.: Superadditivity of communication capacity using entangled inputs. Nat. Phys.
**5**, 255 (2009)CrossRefGoogle Scholar - 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.Einstein, A., Podolsky, B., Rogen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev.
**47**, 777 (1935)ADSCrossRefGoogle Scholar - 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.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.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.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.Braunstein, S.L., Kimble, H.J.: Teleportation of continuous quantum variables. Phys. Rev. Lett.
**80**, 869 (1998)ADSCrossRefGoogle Scholar - 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.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.Hillery, M., Bužek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A
**59**, 1829 (1999)ADSMathSciNetCrossRefGoogle Scholar - 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.Banaszek, K., Wódkiewicz, K.: Nonlocality of the Einstein–Podolsky–Rosen state in the phase space. Acta Phys. Slov.
**49**, 491 (1999)zbMATHGoogle Scholar - 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.Jeong, K., Lim, Y.: Purification of Gaussian maximally mixed states. Phys. Lett. A
**380**, 3607 (2016)ADSCrossRefGoogle Scholar - 24.Brádler, K.: Continuous-variable private quantum channel. Phys. Rev. A
**72**, 042313 (2005)ADSCrossRefGoogle Scholar - 25.Jeong, K., Kim, J., Lee, S.-Y.: Gaussian private quantum channel with squeezed coherent states. Sci. Rep.
**5**, 13974 (2015)ADSCrossRefGoogle Scholar - 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.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.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.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.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.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.Kurochkin, Y., Prasad, A.S., Lvovsky, A.I.: Distillation of the two-mode squeezed state. Phys. Rev. Lett.
**112**, 070402 (2014)ADSCrossRefGoogle Scholar - 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.Sanders, B.C.: Entangled coherent states. Phys. Rev. A
**45**, 6811 (1992)ADSCrossRefGoogle Scholar - 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.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.Werner, R.F., Wolf, M.M.: Bound entangled Gaussian states. Phys. Rev. Lett.
**86**, 3658 (2001)ADSCrossRefGoogle Scholar - 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.van Enk, S.: Entanglement capabilities in infinite dimensions: multidimensional entangled coherent states. Phys. Rev. Lett.
**91**, 017902 (2003)ADSCrossRefGoogle Scholar - 40.Cheong, Y.W., Lee, J.: Generation of entangled coherent states. J. Korean Phys. Soc.
**51**, 1513 (2007)ADSCrossRefGoogle Scholar - 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.Wang, C., et al.: A Schrödinger cat living in two boxes. Science
**352**, 1087 (2016)ADSMathSciNetCrossRefGoogle Scholar - 43.Xu, Y., et al.: Geometrically manipulating photonic Schrödinger cat states and realizing cavity phase gates. arXiv:1810.04690
- 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.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.Strang, G.: Introduction to Linear Algebra. Wellesley-Cambridge Press, Wellesley (1993)zbMATHGoogle Scholar