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Geometry of quantum state space and entanglement

  • Prasenjit DebEmail author
  • Pratapaditya Bej
Article
  • 37 Downloads

Abstract

Recently, an explicit relation between a measure of entanglement and a geometric entity has been reported in Deb (Quantum Inf Process 15:1629–1638, 2016). It has been shown that if a qubit gets entangled with another ancillary qubit, then negativity, up to a constant factor, is equal to the square root of a specific Riemannian metric defined on the metric space corresponding to the state space of the qubit. In this article, we consider different class of bipartite entangled states and show explicit relation between two measures of entanglement and Riemannian metric.

Keywords

Entanglement Riemannian metric State space Negativity Concurrence 

Notes

Acknowledgements

The authors would like to acknowledge Scientific and Engineering Research Board, Govt. of India, for financial support. The authors also acknowledge Bose Institute for providing the research facilities.

References

  1. 1.
    Amari, S., Nagaoka, H.: Methods of information geometry. Am. Math. Soc. 191 (2007)Google Scholar
  2. 2.
    Morozova, E.A., C̆encov, N.N.: Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya. Itogi Nauki Tekh. 36, 69–102 (1990)Google Scholar
  3. 3.
    Petz, D.: Monotone metrics on matrix spaces. Linear Algebra Appl. 244, 81–96 (1996)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Petz, D., Hasegawa, H.: On the Riemannian metric of \(\alpha \)-entropies of density matrices. Lett. Math. Phys. 38, 221–225 (1996)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Tóth, G., Petz, D.: External properties of the variance and the quantum Fisher information. Phys. Rev. A 87, 032324-1–032324-11 (2013)ADSGoogle Scholar
  6. 6.
    Petz, D.: Covariance and Fisher information in quantum mechanics. J. Phys. A 35, 929–939 (2002)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Petz, D., Sudár, C.: Geometries of quantum states. J. Math. Phys. 37, 2662–2673 (1996)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Gibilisco, P., Isola, T.: Wigner–Yanase information on quantum state space: the geometric approach. J. Math. Phys. 44, 3752–3762 (2003)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Gibilisco, P., Isola, T.: A characterisation of Wigner–Yanase skew information among statistically monotone metrics. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4, 553–557 (2001)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Wigner, E.P., Yanase, M.M.: Information contents of distribution. Proc. Natl. Acad. Sci. USA 49, 910–918 (1963)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Wigner, E.P., Yanase, M.M.: On the positive semidefinite nature of certain matrix expressions. Can. J. Math. 16, 397–406 (1964)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Wootters, W.K.: Statistical distance and Hilbert space. Phys. Rev. D 23, 357–362 (1981)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Braunstein, S.L., Caves, C.M.: Statistical distance and the geometry of quantum states. Phys. Rev. Lett. 72, 3439–3443 (1994)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Pires, D.P., Céleri, L.C., Soares-Pinto, D.O.: Geometric lower bound for a quantum coherence measure. Phys. Rev. A 91, 042330 (2015)ADSCrossRefGoogle Scholar
  15. 15.
    Berry, M.V.: Quantum phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984)ADSCrossRefGoogle Scholar
  16. 16.
    Schrodinger, E.: The present situation in quantum mechanics. Naturwissenschaften 23, 807–812 (1935)ADSCrossRefGoogle Scholar
  17. 17.
    Werner, R.F.: Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277–4281 (1989)ADSCrossRefGoogle Scholar
  18. 18.
    Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Bennett, C.H., Brassard, G., Crpeau, 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, 1895–1899 (1993)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Bennett, C.H., Wiesner, S.J.: Communication via one-and two-particle operators on Einstein–Podolsky–Rosen states. Phys. Rev. Lett. 69, 2881–2884 (1992)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Ekert, A.K.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661–663 (1991)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Bennett, C.H., Brassard, G., Mermin, N.D.: Quantum cryptography without Bell’s theorem. Phys. Rev. Lett. 68, 557–560 (1992)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Horodecki, M., Oppenheim, J., Winter, A.: Partial quantum information. Nature 436, 673–676 (2005)ADSCrossRefGoogle Scholar
  24. 24.
    Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2000)Google Scholar
  25. 25.
    Deb, P.: Geometry of quantum state space and quantum correlations. Quantum Inf. Process. 15, 1629–1638 (2016)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998)ADSCrossRefGoogle Scholar
  27. 27.
    Życzkowski, K., Horodecki, P., Sanpera, A., Lewenstein, M.: Volume of the set of separable states. Phys. Rev. A 58, 883–892 (1998)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Vidal, G., Werner, R.F.: Computable measure of entanglement. Phys. Rev. A 65, 032314-1–032314-11 (2002)ADSCrossRefGoogle Scholar
  29. 29.
    Ishizaka, S., Hiroshima, T.: Maximally entangled mixed states under nonlocal unitary operations in two qubits. Phys. Rev. A 62, 022310 (2000)ADSCrossRefGoogle Scholar
  30. 30.
    Munro, W.J., James, D.F.V., White, A.G., Kwiat, P.G.: Maximizing the entanglement of two mixed qubits. Phys. Rev. A 64, 030302 (2001)ADSCrossRefGoogle Scholar
  31. 31.
    Wei, T.-C., Nemoto, K., Goldbart, P.M., Kwiat, P.G., Munro, W.J., Verstraete, F.: Maximal entanglement versus entropy for mixed quantum states. Phys. Rev. A 67, 022110 (2003)ADSCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Physics and Center for Astroparticle Physics and Space ScienceBose InstituteBidhan Nagar, KolkataIndia

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