Skip to main content
SpringerLink
Log in

Local Hidden Variables Underpinning of Entanglement and Teleportation

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Entangled states whose Wigner functions are non-negative may be viewed as being accounted for by local hidden variables (LHV). Recently, there were studies of Bell’s inequality violation (BIQV) for such states in conjunction with the well known theorem of Bell that precludes BIQV for theories that have LHV underpinning. We extend these studies to teleportation which is also based on entanglement. We investigate if, to what extent, and under what conditions may teleportation be accounted for via LHV theory. Our study allows us to expose the role of various quantum requirements. These are, e.g., the uncertainty relation among non-commuting operators, and the no-cloning theorem which forces the complete elimination of the teleported state at its initial port.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Einstein A., Podolsky B., Rosen N., (1935). “Can quantum-mechanical description of physical reality be considered complete?”. Phys. Rev. 47, 777

    Article  MATH  ADS  Google Scholar 

  2. Bell J.S., (1964). “On the Einstein-Podolsky-Rosen paradox”. Physics 1, 195

    Google Scholar 

  3. Bell J.S., (1966). “On the problem of hidden variables in quantum theory”. Rev. Mod. Phys. 38, 447

    Article  MATH  ADS  Google Scholar 

  4. Bohm D., Quantum Theory (Prentice-Hall, Englewood Cliffs, NJ, 1951), Chap. 22.

  5. Gisin N., (1991). “Bell’s inequality holds for all non-product states”. Phys. Lett. A 154, 201

    Article  ADS  MathSciNet  Google Scholar 

  6. J. S. Bell, “EPR correlations and EPW distributions,” in Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1987). The first (i.e., prior to Bell) to note that the Wigner function of the maximally entangled state, the so-called Rosen’s (or EPR) state: ψ(q 1,q 2) ∝ δ(q 1−q 2), viz. W(q 1,p 1,q 2,p 2) ∝ δ(q 1−q 2)δ(p 1 + p 2), may be viewed as a statistical distribution for “hidden variables” were A. M Cetto, L. de la Pena, and E. Santos, “A Bell inequality involving position, momentum and energy,” Phys. Lett. A 113, 304 (1985). Some of the early discussion is given in the article by U. Leonhardt and J. A. Vaccaro, “Quantum-optical Bell correlations in phase space,” in The Dilemma of Einstein, Podolsky and Rosen—60 Years Later (Institute of Physics Publishing, Bristol and Philadelphia, and The Israel Physical Society, Jerusalem, 1996).

  7. Wigner E.P., (1932). “On the quantum correction for thermodynamic equilibrium”. Phys. Rev. 40, 749

    Article  MATH  ADS  Google Scholar 

  8. K. Banaszek and K. Wodkiewicz, “Nonlocality of the Einstein-Podolsky-Rosen state in the Wigner representation,” Phys. Rev. A 58, 4345 (1998); “Testing quantum nonlocality in phase space,” Phys. Rev. Lett. 82, 2009 (1999). The interest in achieving BIQV for the EPR state is, of course, not just due to Bell’s claim noted above.

    Google Scholar 

  9. Chen Z., Pan J., Hou G., Zhang Y., (2002). “Maximal violation of Bell’s inequalities for continuous variable systems”. Phys. Rev. Lett. 88: 040406

    Article  ADS  Google Scholar 

  10. Gour G., Khanna F.C., Mann A., Revzen M., (2004). “Optimization of Bell’s inequality violation for continuous variable systems”. Phys. Lett. A 324, 415

    Article  ADS  MathSciNet  Google Scholar 

  11. Revzen M., Mello P.A., Mann A., Johansen L.M., (2005). “Bell’s inequality violation with non-negative Wigner functions”. Phys. Rev. A 71: 022103

    Article  ADS  MathSciNet  Google Scholar 

  12. Bennett C.H., Brassard G., Crepeau C., Jozsa R., Peres A., Wootters W., (1993). “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels”. Phys. Rev. Lett. 70: 1895

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. Vaidman L., (1994). “Teleportation of quantum states”. Phys. Rev. A 49: 1473

    Article  ADS  MathSciNet  Google Scholar 

  14. Braunstein S.L., Kimble H.J., (1998). “Teleportation of continuous quantum variables”. Phys. Rev. Lett. 80, 869

    Article  ADS  Google Scholar 

  15. Bouwmeester D., Pan J.M., Mattle K., Eibl M., Weinfurther H., Zeilinger A., (1997). “Experimental quantum teleportation”. Nature (London) 390, 575

    Article  ADS  Google Scholar 

  16. Furusawa A., Sorensen J.L., Braunstein S.L., Fuchs C.A., Kimble H.J., Polzik E.S., (1998). “Unconditional quantum teleportation”. Science 282, 706

    Article  ADS  Google Scholar 

  17. Bowen W.P., Treps N., Buchler B.C., Schnabel R., Ralph T.C., Bachor H.A., Symul T., Lam P.K., (2003). “Experimental investigation of continuous-variable quantum teleportation”. Phys. Rev. A 67: 032302

    Article  ADS  Google Scholar 

  18. Zhang T.C., Goh K.W., Chou C.W., Lodahl P., Kimble H.J., (2003). “Quantum teleportation of light beams”. Phys. Rev. A 67: 033802

    Article  ADS  Google Scholar 

  19. Braunstein S.L., Fuchs C.A., Kimble H.J., van Loock P., (2001). “Quantum versus classical domains for teleportation with continuous variables”. Phys. Rev. A 64: 022321

    Article  ADS  Google Scholar 

  20. Grosshans F., Grangier P., (2001). “Quantum cloning and teleportation criteria for continuous quantum variables”. Phys. Rev. A 64: 010301

    Article  ADS  MathSciNet  Google Scholar 

  21. Hudson R.L., (1974). “When is the Wigner quasi-probability density non-negative?”. Rep. Math. Phys. 6, 249

    Article  MATH  MathSciNet  Google Scholar 

  22. Schleich W., (2001). Quantum Optics in Phase Space. Wiley-VCH, Berlin

    Book  MATH  Google Scholar 

  23. Mann A., Revzen M., (1993). “Gaussian density matrices: quantum analogs of classical states”. Fortschr. Phys. 41, 431

    MathSciNet  Google Scholar 

  24. Englert B.-G., Wodkiewicz K., (2003). “Tutorial notes on one-party and two-party gaussian states”. Int. J. Q. Inf. 1, 153

    Article  MATH  Google Scholar 

  25. Shannon C.E., “A mathematical theory of communication,” Bell Syst. Tech. J. 27, pp. 379–423 and 623–656 (1948).

  26. von Neumann J., (1955). Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton

    MATH  Google Scholar 

  27. Popper K.R., (1959). “The propensity interpretation of probability”. Br. J. Philos. Sci. 10, 25

    Article  Google Scholar 

  28. W. K. Wootters, “A Wigner function formulation of finite-state quantum mechanics,” Ann. Phys. (N.Y.) 176, 1 (1987). For a detailed review on the progress in quantum information based on continuous quantum variables, with emphasis on quantum optical implementations in terms of the quadrature amplitudes of the electromagnetic field, see S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513 (2005).

    Google Scholar 

  29. Baker G.A., (1958). “Formulation of quantum mechanics based on the quasi-probability distribution induced on phase space”. Phys. Rev. 109: 2198

    Article  ADS  MathSciNet  Google Scholar 

  30. Hillery M., Oconnell R.F., Scully M.O., Wigner E.P., (1984). “Distribution functions in physics: Fundamentals”. Phys. Rep. 106, 121

    Article  ADS  MathSciNet  Google Scholar 

  31. Duan L.M, Giedke G., Cirac J.I., Zoller P., (2000). “Inseparability criterion for continuous variable systems”. Phys. Rev. Lett. 84: 2722

    Article  ADS  Google Scholar 

  32. Simon R., (2000). “Peres-Horodecki separability criterion for continuous variable systems”. Phys. Rev. Lett. 84: 2726

    Article  ADS  Google Scholar 

  33. Wootters W.K., Zurek W.H., (1982). “A single quantum cannot be cloned”. Nature 299, 802

    Article  ADS  Google Scholar 

  34. Dieks D., (1982). “Communication by EPR devices”. Phys. Lett. A 92, 271

    Article  ADS  Google Scholar 

  35. Barnum H., Caves C.M., Fuchs C.A., Jozsa R., Schumacher B., (1996). “Noncommuting mixed states cannot be broadcast”. Phys. Rev. Lett. 76: 2818

    Article  ADS  Google Scholar 

  36. Daffertshofer A., Plastino A.R., Plastino A., (2002). “Classical no-cloning theorem”. Phys. Rev. Lett. 88: 210601

    Article  ADS  Google Scholar 

  37. We ignore the practical difficulty of realizing the pure EPR state. Its realization requires an unbounded energy resource.

  38. Peres A., (1996). “Separability criterion for density matrices”. Phys. Rev. Lett. 77: 1413

    Article  MATH  ADS  MathSciNet  Google Scholar 

  39. S. L. Braunstein, private communication.

  40. Koniorczyk M., Kiss T., Janszky J., (2001). “Teleportation: from probability distributions to quantum states”. J. Phys. A 34: 6949

    Article  MATH  ADS  MathSciNet  Google Scholar 

  41. Mor T., (2006). “On classical teleportation and classical nonlocality”. Int. J. Quant. Inf. 4, 161

    Article  MATH  Google Scholar 

  42. Caves C.M., Wodkiewicz K., (2004). “Classical phase-space descriptions of continuous-variable teleportation”. Phys. Rev. Lett. 93: 040506

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Technion-Israel Institute of Technology, Haifa, 32000, Israel

    A. Kalev, A. Mann & M. Revzen

Authors
  1. A. Kalev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Mann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Revzen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Kalev.

Additional information

Communicated by Alwyn van der Merwe

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kalev, A., Mann, A. & Revzen, M. Local Hidden Variables Underpinning of Entanglement and Teleportation. Found Phys 37, 125–143 (2007). https://doi.org/10.1007/s10701-006-9092-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-006-9092-6

Keywords

Search

Navigation