Advertisement

Foundations of Physics Letters

, Volume 14, Issue 3, pp 199–212 | Cite as

Entanglement of Formation of an Arbitrary State of Two Rebits

  • Carlton M. Caves
  • Christopher A. Fuchs
  • Pranaw Rungta
Article

Abstract

We consider entanglement for quantum states defined in vector spaces over the real numbers. Such real entanglement is different from entanglement in standard quantum mechanics over the complex numbers. The differences provide insight into the nature of entanglement in standard quantum theory. Wootters [Phys. Rev. Lett.80, 2245 (1998)] has given an explicit formula for the entanglement of formation of two qubits in terms of what he calls the concurrence of the joint density operator. We give a contrasting formula for the entanglement of formation of an arbitrary state of two “rebits,” a rebit being a system whose Hilbert space is a 2-dimensional real vector space.

quantum entanglement quantum information real quantum mechanics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    Introduction to Quantum Computation and Information, H.-K. Lo, S. Popescu and T. Spiller, eds. (World Scientific, Singapore, 1998).Google Scholar
  2. 2.
    C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895 (1993).ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    P. W. Shor, “Scheme for reducing decoherence in quantum computer memory,” Phys. Rev. A 52, R2493 (1995).ADSCrossRefGoogle Scholar
  4. 4.
    A. M. Steane, “Error correcting codes in quantum theory,” Phys. Rev. Lett. 77, 793 (1996).ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245 (1998).ADSCrossRefGoogle Scholar
  6. 6.
    W. K. Wootters, “Local accessibility of quantum states,” in Complexity, Entropy and the Physics of Information, W. H. Zurek, ed. (Addison-Wesley, Redwood City, 1990), pp. 39-46.Google Scholar
  7. 7.
    S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields (Oxford University Press, New York, 1995).MATHGoogle Scholar
  8. 8.
    E. C. G. Stueckelberg, “Quantum theory in real Hilbert space,” Helv. Phys. Acta 33, 727 (1960).MathSciNetMATHGoogle Scholar
  9. 9.
    S. Weinberg, “Testing quantum mechancs,” Ann. Phys. 194, 336 (1989).ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett. 77, 1413 (1996).ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: Necessary and sufficient conditions,” Phys. Lett. A 223, 1 (1996).ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046 (1996).ADSCrossRefGoogle Scholar
  13. 13.
    C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wooters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824 (1996).ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    M. Horodecki, P. Horodecki, and R. Horodecki, “Mixed-state entanglement and distillation: Is there a ‘bound’ entanglement in nature?” Phys. Rev. Lett. 80, 5239 (1998).ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    P. M. Hayden, M. Horodecki, and B. M. Terhal, “The asymptotic entanglement cost of preparing a quantum state,” unpublished, arXiv:quant-ph/0008134.Google Scholar
  16. 16.
    S. Hill and W. K. Wootters, “Entanglement of a pair of quantum bits,” Phys. Rev. Lett. 78, 5022 (1997).ADSCrossRefGoogle Scholar
  17. 17.
    E. Schrödinger, “Probability relations between separated systems,” Proc. Camb. Phil. Soc. 32, 446 (1936).ADSCrossRefMATHGoogle Scholar
  18. 18.
    E. T. Jaynes, “Information theory and statistical mechanics. II,” Phys. Rev. 108, 171 (1957).ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    L. P. Hughston, R. Jozsa, and W. K. Wootters, “A complete classification of quantum ensembles having a given density matrix,” Phys. Lett. A 183, 14 (1993).ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, Dordrecht, 1995).MATHGoogle Scholar
  21. 21.
    K. Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, “Volume of the set of separable states,” Phys. Rev. A 58, 883 (1998).ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    S. L. Braunstein, C. M. Caves, R. Jozsa, N. Linden, S. Popescu, and R. Schack, “Separability of very noisy mixed states and implications for NMR quantum computing,” Phys. Rev. Lett. 83, 1054 (1999).ADSCrossRefGoogle Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • Carlton M. Caves
    • 1
  • Christopher A. Fuchs
    • 2
  • Pranaw Rungta
    • 3
  1. 1.Center for Advanced Studies, Department of Physics and AstronomyUniversity of New MexicoAlbuquerque
  2. 2.Los Alamos National Laboratory, MS-B285Los Alamos
  3. 3.Center for Advanced Studies, Department of Physics and AstronomyUniversity of New MexicoAlbuquerque

Personalised recommendations