Skip to main content
SpringerLink
Log in

Local deterministic simulation of equatorial Von Neumann measurements on tripartite GHZ state

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Experimental free-will or measurement independence is one of the crucial assumptions in derivation of any nonlocal theorem. Any nonlocal correlation obtained in quantum world can have a local deterministic explanation if there is no experimental free-will in choosing the measurement settings. Recently, Hall (Phys Rev Lett 105:250404, 2010) has shown that to obtain a local deterministic description for singlet state correlation one does not need to give up measurement independence completely, but a partial measurement dependence suffices. In three party scenario considering Greenberger–Horne–Zeilinger (GHZ) correlation one can exhibit absolute contradiction between quantum theory and local realism. In this paper we show that such correlation also has local deterministic description if measurement independence is given up, even if not completely. We provide a local deterministic model for equatorial Von Neumann measurements on tripartite GHZ state by sacrificing measurement independence partially.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935). doi:10.1103/PhysRev.47.777

    Article  ADS  MATH  Google Scholar 

  2. Bell, J.S.: On the Einstein–Podolsky–Rosen paradox. Physics 1(3), 195–200 (1964)

    Google Scholar 

  3. Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  4. Aspect, A., Grangier, P., Roger, G.: Experimental realization of Einstein–Podolsky–Rosen–Bohm Gedanken experiment: a new violation of Bell’s inequalities. Phys. Rev. Lett. 49, 91–94 (1982). doi:10.1103/PhysRevLett.49.91

    Article  ADS  Google Scholar 

  5. Aspect, A., Dalibard, J., Roger, G.: Experimental test of Bell’s inequalities using time-varying analyzers. Phys. Rev. Lett. 49, 1804–1807 (1982). doi:10.1103/PhysRevLett.49.1804

    Article  ADS  MathSciNet  Google Scholar 

  6. Aspect, A.: Bell’s inequality test: more ideal than ever. Nat. (Lond.) 398, 189–190 (1999). http://www.nature.com/nature/journal/v398/n6724/full/398189a0.html

  7. Greenberger, D.M., Horne, M.A., Zeilinger, A.: Bell’s theorem without inequalities. Am. J. Phys. 58(12), 1131–1143 (1990). http://www.physik.uni-bielefeld.de/yorks/qm12/ghsz.pdf

  8. Pan, J.W., Bouwmeester, D., Daniell, M., Weinfurter, H., Zeilinger, A.: Experimental test of quantum nonlocality in three-photon GreenbergerHorneZeilinger entanglement. Nature 403, 515–519 (2000). http://www.nature.com/nature/journal/v403/n6769/abs/403515a0.html

  9. Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V., Wehner, S.: Bell nonlocality. Rev. Mod. Phys. 86, 419 (2014). doi:10.1103/RevModPhys.86.419

    Article  ADS  Google Scholar 

  10. Barrett, J., Hardy, L., Kent, A.: No signaling and quantum key distribution. Phys. Rev. Lett. 95, 010503 (2005). doi:10.1103/PhysRevLett.95.010503

    Article  ADS  Google Scholar 

  11. Acín, A., Brunner, N., Gisin, N., Massar, S., Pironio, S., Scarani, V.: Device-independent security of quantum cryptography against collective attacks. Phys. Rev. Lett. 98, 230501 (2007). doi:10.1103/PhysRevLett.98.230501

    Article  ADS  Google Scholar 

  12. Pironio, S., Acín, A, Brunner, N., Gisin, N., Massar, S., Scarani, V.: Device-independent quantum key distribution secure against collective attacks. New J. Phys. 11, 045021 (2009). http://iopscience.iop.org/1367-2630/11/4/045021

  13. Vazirani, U., Vidick, T.: Fully device independent quantum key distribution. arXiv:1210.1810

  14. Colbeck, R., Kent, A.: Private randomness expansion with untrusted devices. J. Phys. A 44, 095305 (2011). doi:10.1088/1751-8113/44/9/095305

    Article  ADS  MathSciNet  Google Scholar 

  15. Pironio, S., et al.: Random numbers certified by Bells theorem. Nat. (Lond.) 464, 1021–1024 (2010). http://www.nature.com/nature/journal/v464/n7291/full/nature09008.html

  16. Banik, M.: Measurement-device-independent randomness certification by local entangled states. arXiv:1401.1338

  17. Maudlin, T.: In: Hull, D., Forbes, M., Okruhlik, K. (eds.) Proceedings of the 1992 Meeting of the Philosophy of Science Association, vol. 1, pp. 404–417 (Philosophy of Science Association, East Lansing, MI) (1992)

  18. Brassard, G., Cleve, R., Tapp, A.: Cost of exactly simulating quantum entanglement with classical communication. Phys. Rev. Lett. 83, 1874–1877 (1999). doi:10.1103/PhysRevLett.83.1874

    Article  ADS  Google Scholar 

  19. Steiner, M.: Towards quantifying non-local information transfer: finite-bit non-locality. Phys. Lett. A 270, 239244 (2000). doi:10.1016/S0375-9601(00)00315-7

    Article  Google Scholar 

  20. Toner, B.F., Bacon, D.: Communication cost of simulating Bell correlations. Phys. Rev. Lett. 91, 187904 (2003). doi:10.1103/PhysRevLett.91.187904

    Article  ADS  MathSciNet  Google Scholar 

  21. Tessier, T.E., Caves, C.M., Deutsch, I.H., Eastin, B.: Optimal classical-communication-assisted local model of n-qubit Greenberger–Horne–Zeilinger correlations. Phys. Rev. A 72, 032205 (2005). doi:10.1103/PhysRevA.72.032305

    Article  ADS  Google Scholar 

  22. Broadbent, A., Chouha, P.R., Tapp, A.: In: Proceedings of the Third International Conference on Quantum, Nano and Micro Technologies (ICQNM 2009)

  23. Branciard, C., Gisin, N.: Quantifying the nonlocality of Greenberger–Horne–Zeilinger quantum correlations by a bounded communication simulation protocol. Phys. Rev. Lett. 107, 020401 (2011). doi:10.1103/PhysRevLett.107.020401

    Article  ADS  Google Scholar 

  24. Brassard, G., Devroye, L., Gravel, C.: Exact simulation for the GHZ distribution. arXiv:1303.5942 (2013)

  25. Shimony, A., Horne, M.A., Clauser, J.F.: An exchange on local beables. Dialectica 39, 97–102 (1985)

    Article  MathSciNet  Google Scholar 

  26. Hall, M.J.W.: Local deterministic model of singlet state correlations based on relaxing measurement independence. Phys. Rev. Lett. 105, 250404 (2010). doi:10.1103/PhysRevLett.105.250404

    Article  ADS  Google Scholar 

  27. Hall, M.J.W.: Relaxed Bell inequalities and Kochen–Specker theorems. Phys. Rev. A 84, 022102 (2011). doi:10.1103/PhysRevA.84.022102

    Article  ADS  Google Scholar 

  28. Barrett, J., Gisin, N.: How much measurement independence is needed to demonstrate nonlocality? Phys. Rev. Lett. 106, 100406 (2011). doi:10.1103/PhysRevLett.106.100406

    Article  ADS  Google Scholar 

  29. Brans, C.H.: Bell’s theorem does not eliminate fully causal hidden variables. Int. J. Theor. Phys. 27(2), 219–226 (1988). doi:10.1007/BF00670750

    Article  Google Scholar 

  30. Mermin, N.D.: Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys. Rev. Lett. 65, 1990 (1838). doi:10.1103/PhysRevLett.65.1838

    Google Scholar 

  31. Hillery, M., Bužek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A 59, 1999 (1829). doi:10.1103/PhysRevA.59.1829

    Google Scholar 

  32. Zhao-Yang, T., Le-Man, K.: Broadcasting of entanglement in three-particle Greenberger–Horne–Zeilinger state via quantum copying. Chin. Phys. Lett. 17(7), 469–471 (2000). http://cpl.iphy.ac.cn/EN/Y2000/V17/I7/469

  33. Jin, X.-R., et al.: Three-party quantum secure direct communication based on GHZ states. Phys. Lett. A 354(1–2), 67–70 (2006). doi:10.1016/j.physleta.2006.01.035

    Article  ADS  Google Scholar 

  34. Banik, M., Gazi, M.D.R., Das, S., Rai, A., Kunkri, S.: Optimal free will on one side in reproducing the singlet correlation. J. Phys. A Math. Theor. 45, 205301 (2012). doi:10.1088/1751-8113/45/20/205301

    Article  ADS  Google Scholar 

Download references

Acknowledgments

It is a pleasure to thank Guruprasad Kar for many stimulating discussions. MB acknowledge C. Branciard for fruitful discussions. AM acknowledges support from CSIR, Govt. of India (File No.09/093(0148)/ 2012-EMR-I).

Author information

Authors and Affiliations

  1. Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata, 700108, India

    Arup Roy, Amit Mukherjee, Some Sankar Bhattacharya & Manik Banik

  2. S.N. Bose National Center for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata, 700098, India

    Subhadipa Das

Authors
  1. Arup Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amit Mukherjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Some Sankar Bhattacharya
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Manik Banik
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Subhadipa Das
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manik Banik.

Appendices

Appendix 1: Calculation of measurement dependency

The degree of measurement dependency is quantified as Eq. (8). To find \(M\) we have to maximize the difference between the densities of hidden variable (HV) for all pairs of measurement settings over the HV space. The difference will be maximized if the upper value of the distribution of HV corresponding to one measurement setting overlaps by maximum amount with the lower value of the other measurement setting. The distribution of HV for each measurement setting consists of three pair of regions comprising two opposite spherical sectors ({\(R_{1+},R_{1-}\)},{\(R_{2+},R_{2-}\)},{\(R_{3+},R_{3-}\)}) and a pair of delta functions (\(D_+,D_-\)) as defined in Eq. (13)and shown in Fig. 2. The distribution of HV for measurement setting \(\{\hat{m}_X\}\) and \(\{\hat{m}_X^{\prime }\}\) are denoted by prime and unprimed region, respectively (see Fig. 3). The maximum of right hand side of Eq. (8) occurs when \(R_{1+}\) contains \(D_-^{\prime }\) and the region \(R_{3-}^{\prime }\) completely and the region \(R_{2-}^{\prime }\) partially as shown in Fig. 3. It becomes that \(M=1.43\) and thus we have \(F:=1-\frac{M}{2}\simeq \) 28.5 %.

Fig. 2
figure 2

(Color on-line). \(R_{1+} (R_{1-})\) is the region where \(s(\hat{m}_A.\varvec{\lambda })=s(\hat{m}_B.\varvec{\lambda })= s(\hat{m}_C.\varvec{\lambda })=+1 (-1)\); \(R_{2+} (R_{2-})\) is the region where \(s(\hat{m}_A.\varvec{\lambda })=-s(\hat{m}_B.\varvec{\lambda })= -s(\hat{m}_C.\varvec{\lambda })=+1 (-1)\) and \(R_{3+} (R_{3-})\) is the region where \(-s(\hat{m}_A.\varvec{\lambda })=-s(\hat{m}_B. \varvec{\lambda })=s(\hat{m}_C.\varvec{\lambda })=+1 (-1)\). \(D_+\) (\(D_{-}\)) denotes the fixed vector \(\varvec{\lambda _0}\) (-\(\varvec{\lambda _0}\))

Full size image
Fig. 3
figure 3

(Color on-line). This figure shows distribution of HV corresponding to measurement settings \(\{\hat{m}_X\}\) and \(\{\hat{m}_X^{\prime }\}\). The inner circle represents the distribution corresponding to the measurement setting \(\{\hat{m}_X^{\prime }\}\) and the outer circle corresponding to the measurement setting \(\{\hat{m}_X\}\). The maximum of the right hand side of Eq. (8) occurs when \(R_{1+}\) contains \(D_-^{\prime }\) and the region \(R_{3-}^{\prime }\) completely and the region \(R_{2-}^{\prime }\) partially

Full size image

Appendix 2: Simulation of GHZ expectation value

In this case also Alice, Bob and Charlie share a variable \(\varvec{\lambda }\) chosen from unit circle and given a measurement direction from equatorial plane Alice, Bob and Charlie give there answer like Eq. (10). The distribution of the variable \(\varvec{\lambda }\) in this case is given by

$$\begin{aligned} \vartheta (\varvec{\lambda }|\{\hat{m}_X\})&:= \vartheta ^{\prime }(\varvec{\lambda } |\{\hat{m}_X\})\varTheta (\phi _{AB}-\phi _{AC})\\&\quad +\vartheta ^{\prime \prime }(\varvec{\lambda }|\{\hat{m}_X\})\varTheta (\phi _{AC}-\phi _{AB}), \end{aligned}$$

where

$$\begin{aligned} \vartheta ^{\prime }(\varvec{\lambda }|\{\hat{m}_X\}):&= \frac{1+\beta \cos (\phi _A+\phi _B+\phi _C)}{6(\pi -\phi _{AB})}\nonumber \\&\text{ if } s(\hat{m}_A.\varvec{\lambda })=s(\hat{m}_B.\varvec{\lambda }) =s(\hat{m}_C.\varvec{\lambda })=\beta \nonumber \\ :&= \frac{1+\beta \cos (\phi _A+\phi _B+\phi _C)}{6(\phi _{AB}- \phi _{AC})}\nonumber \\&\text{ if } -s(\hat{m}_A.\varvec{\lambda })=-s(\hat{m}_B. \varvec{\lambda })=s(\hat{m}_C.\varvec{\lambda })=\beta \nonumber \\ :&= \frac{1+\beta \cos (\phi _A+\phi _B+\phi _C)}{6\phi _{AB}}\nonumber \\&\text{ if } s(\hat{m}_A.\varvec{\lambda })=-s(\hat{m}_B. \varvec{\lambda })=-s(\hat{m}_C.\varvec{\lambda })=\beta \end{aligned}$$

with \(\beta \in \{+1,-1\}\); and

$$\begin{aligned} \vartheta ^{\prime \prime }(\varvec{\lambda }|\{\hat{m}_X\}):&= \frac{1+\beta \cos (\phi _A+\phi _B+\phi _C)}{6(\pi -\phi _{AC})}\nonumber \\&\text{ if } s(\hat{m}_A.\varvec{\lambda })=s(\hat{m}_B.\varvec{\lambda }) =s(\hat{m}_C.\varvec{\lambda })=\beta \nonumber \\ :&= \frac{1+\beta \cos (\phi _A+\phi _B+\phi _C)}{6\phi _{AC}}\nonumber \\&\text{ if } s(\hat{m}_A.\varvec{\lambda })=-s(\hat{m}_B.\varvec{\lambda }) =-s(\hat{m}_C.\varvec{\lambda })=\beta \nonumber \\ :&= \frac{1+\beta \cos (\phi _A+\phi _B+\phi _C)}{6(\phi _{AC}- \phi _{AB})}\nonumber \\&\text{ if } -s(\hat{m}_A.\varvec{\lambda })=s(\hat{m}_B.\varvec{\lambda }) =-s(\hat{m}_C.\varvec{\lambda })=\beta . \end{aligned}$$

In this case it becomes \(F=\) 37.5 %. Therefore 62.5 % lack of measurement independence for each party is sufficient to simulate statistic of the equatorial Von Neumann measurements on GHZ state.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Roy, A., Mukherjee, A., Bhattacharya, S.S. et al. Local deterministic simulation of equatorial Von Neumann measurements on tripartite GHZ state. Quantum Inf Process 14, 217–228 (2015). https://doi.org/10.1007/s11128-014-0832-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-014-0832-9

Keywords

Search

Navigation