Skip to main content
SpringerLink
Log in

Relating Local Time Evolutions with Bipartite States: An Exact Map Manifested by Weak Measurements

  • Conference paper
Quantum Theory: A Two-Time Success Story

  • 1884 Accesses

Abstract

We suggest a natural mapping between bipartite states and quantum evolutions of local states, which is a Jamiolkowski map. It is shown that spatial correlations of weak measurements in bi-partite systems precisely coincide with temporal correlations of local systems. This mapping has several practical and conceptual implications on the correspondence between Bell and Leggett-Garg inequalities, the statistical properties of evolutions in large systems, temporal decoherence and computational gain, in evaluation of spatial correlations of large systems. [Editor’s note: for a video of the talk given by Prof. Reznik at the Aharonov-80 conference in 2012 at Chapman University, see quantum.chapman.edu/talk-23.]

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Throughout the paper the term correlation corresponds to the expectation of the product of operators, sometimes regarded as correlator, without subtracting the first moments as required in the statistical definition of the term.

References

  1. A. Jamiolkowski, Rep. Math. Phys. 3, 275 (1972)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. K. Kraus, States, Effects and Operations: Fundamental Notions of Quantum Theory (Springer, Berlin, 1983)

    Book  MATH  Google Scholar 

  3. V. Coffman, J. Kundu, W.K. Wootters, Phys. Rev. A 61, 052306 (2000)

    Article  ADS  Google Scholar 

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

    MATH  Google Scholar 

  5. Y. Aharonov, D.Z. Albert, A. Casher, L. Vaidman, Phys. Lett. A 124, 199–203 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  6. J.S. Bell, Physics 1, 195–200 (1964)

    Google Scholar 

  7. A.J. Leggett, A. Garg, Phys. Rev. Lett. 54, 857–860 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  8. A.N. Korotkov, D.V. Averin, Phys. Rev. B 64, 165310 (2001)

    Article  ADS  Google Scholar 

  9. K.J. Resch, A.M. Steinberg, Phys. Rev. Lett. 92, 130402 (2004)

    Article  ADS  Google Scholar 

  10. G. Mitchison, R. Jozsa, S. Popescu, Phys. Rev. A 76, 062105 (2007)

    Article  ADS  Google Scholar 

  11. J.S. Lundeen, A.M. Steinberg, Phys. Rev. Lett. 102, 020404 (2009)

    Article  ADS  Google Scholar 

  12. L.M. Johansen, P.A. Mello, Phys. Lett. A 372, 5760–5764 (2008)

    Article  ADS  MATH  Google Scholar 

  13. S. Marcovitch, B. Reznik. arXiv:1005.3236

  14. H.M. Wiseman, Phys. Rev. A 65, 032111 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  15. G. Lindblad, Commun. Math. Phys. 48, 199 (1976)

    Article  MathSciNet  Google Scholar 

  16. J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt, Phys. Rev. Lett. 23, 880–884 (1969)

    Article  ADS  Google Scholar 

  17. B.S. Cirel’son, Lett. Math. Phys. 4, 93–100 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  18. R. Ruskov, A.N. Korotkov, A. Mizel, Phys. Rev. Lett. 96, 200404 (2006)

    Article  ADS  Google Scholar 

  19. A.N. Jordan, A.N. Korotkov, M. Büttiker, Phys. Rev. Lett. 97, 026805 (2006)

    Article  ADS  Google Scholar 

  20. R.F. Werner, Phys. Rev. A 40, 4277 (1989)

    Article  ADS  Google Scholar 

  21. A. Acin, T. Durt, N. Gisin, J.I. Latorre, Phys. Rev. A 65, 052325 (2002)

    Article  ADS  Google Scholar 

  22. D. Collins, N. Gisin, N. Linden, S. Massar, S. Popescu, Phys. Rev. Lett. 88, 040404 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  23. D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, A. Zeilinger, Phys. Rev. Lett. 85, 4418 (2000)

    Article  ADS  Google Scholar 

  24. D. Collins, N. Gisin, J. Phys. A, Math. Gen. 37, 1775 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. T.H. Yang, M. Navascués, L. Sheridan, V. Scarani, Phys. Rev. A 83, 022105 (2011)

    Article  ADS  Google Scholar 

  26. T. Vidick, S. Wehner. arXiv:1011.5206

  27. P. Hayden, D.W. Leung, A. Winter, Commun. Math. Phys. 265, 95 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. C. Brukner, S. Taylor, S. Cheung, V. Vedral. arXiv:quant-ph/0402127

  29. M.S. Leifer, Phys. Rev. A 74, 042310 (2006)

    Article  ADS  Google Scholar 

  30. R.B. Griffiths, Phys. Rev. A 71, 042337 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  31. F. Verstraete, H. Verschelde. arXiv:quant-ph/0202124v2

  32. P. Arrighi, C. Patricot, Ann. Phys. 311, 26–52 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. W. Dür, J.I. Cirac, Phys. Rev. A 64, 012317 (2001)

    Article  ADS  Google Scholar 

  34. M.L. Nowakowski1, P. Horodecki. arXiv:quant-ph/0503070

  35. J. Oppenheim, B. Reznik, Phys. Rev. A 70, 022312 (2004)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We are deeply grateful to Y. Aharonov whose insights initiated this work. We also thank A. Botero and P. Skrzypczyk. This work has been supported by the Israel Science Foundation grant number 920/09, the German-Israeli foundation, and the European Commission (PICC).

Author information

Authors and Affiliations

  1. School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv, 69978, Israel

    S. Marcovitch & B. Reznik

Authors
  1. S. Marcovitch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Reznik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Reznik .

Editor information

Editors and Affiliations

  1. Schmid College of Science and Technology, Chapman University, Orange, USA

    Daniele C. Struppa

  2. Schmid College of Science and Technology, Chapman University, Orange, USA

    Jeffrey M. Tollaksen

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Italia

About this paper

Cite this paper

Marcovitch, S., Reznik, B. (2014). Relating Local Time Evolutions with Bipartite States: An Exact Map Manifested by Weak Measurements. In: Struppa, D., Tollaksen, J. (eds) Quantum Theory: A Two-Time Success Story. Springer, Milano. https://doi.org/10.1007/978-88-470-5217-8_22

Download citation

Publish with us

Policies and ethics

Search

Navigation