Skip to main content

Advertisement

SpringerLink
Log in

Absolute non-violation of a three-setting steering inequality by two-qubit states

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Steerability is a characteristic non-local trait of quantum states lying in between entanglement and Bell non-locality. A given quantum state is considered to be steerable if it violates a suitably chosen steering inequality. A quantum state which otherwise satisfies a certain inequality can violate the inequality under a global change of basis, i.e., if the state is transformed by a non-local unitary operation. Intriguingly, there are states which preserve their non-violation (pertaining to the said inequality) under any global unitary operation. The present work explores the effect of global unitary operations on the steering ability of a quantum state which lives in two qubits. We characterize such states in terms of a necessary and sufficient condition on their spectrum. Such states are also characterized in terms of some analytic characteristics of the set to which they belong. Looking back at steerability the present work also provides a relation between steerability and quantum teleportation together with the derivation of a result related to the optimal violation of steering inequality. An analytic estimation of the size of such non-violating states in terms of purity is also obtained. Interestingly, the estimation in terms of purity also gives a necessary and sufficient condition in terms of Bloch parameters of the state. Illustrations from some signature class of quantum states further underscore our observations.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V., Wehner, S.: Bell nonlocality. Rev. Mod. Phys. 86, 839 (2014)

    Article  ADS  Google Scholar 

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

    Google Scholar 

  4. Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press, Cambridge (1987)

    MATH  Google Scholar 

  5. Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880 (1969)

    Article  ADS  MATH  Google Scholar 

  6. Bennett, C.H., Brassard, G., Crépeau, 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 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Bennett, C.H., Wiesner, S.J.: Communication via one- and two-particle operators on Einstein–Podolsky–Rosen states. Phys. Rev. Lett. 69, 2881 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Pironio, S., Acín, A., Massar, S., Boyer de la Giroday, A., Matsukevich, D.N., Maunz, P., Olmschenk, S., Hayes, D., Luo, L., Manning, T.A., Monroe, C.: Random numbers certified by Bell’s theorem. Nature 464, 1021 (2010)

    Article  ADS  Google Scholar 

  9. Colbeck, R., Renner, R.: Free randomness can be amplified. Nat. Phys. 8, 450 (2012)

    Article  Google Scholar 

  10. Chaturvedi, A., Banik, M.: Measurement-device-independent randomness from local entangled states. EPL 112, 30003 (2015)

    Article  ADS  Google Scholar 

  11. Barrett, J., Hardy, L., Kent, A.: No signaling and quantum key distribution. Phys. Rev. Lett. 95, 010503 (2005)

    Article  ADS  Google Scholar 

  12. Acín, A., Gisin, N., Masanes, L.: From Bell’s theorem to secure quantum key distribution. Phys. Rev. Lett. 97, 120405 (2006)

    Article  ADS  MATH  Google Scholar 

  13. Brunner, N., Pironio, S., Acín, A., Gisin, N., Methot, A.A., Scarani, V.: Testing the dimension of Hilbert spaces. Phys. Rev. Lett. 100, 210503 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Gallego, R., Brunner, N., Hadley, C., Acín, A.: Device independent tests of classical and quantum dimensions. Phys. Rev. Lett. 105, 230501 (2010)

    Article  ADS  Google Scholar 

  15. Das, S., Banik, M., Rai, A., Gazi, M.D.R., Kunkri, S.: Hardy’s nonlocality argument as a witness for postquantum correlations. Phys. Rev. A 87, 012112 (2013)

    Article  ADS  Google Scholar 

  16. Mukherjee, A., Roy, A., Bhattacharya, S.S., Das, S., Gazi, M.R., Banik, M.: Hardy’s test as a device-independent dimension witness. Phys. Rev. A 92, 022302 (2015)

    Article  ADS  Google Scholar 

  17. Brunner, N., Linden, N.: Connection between Bell nonlocality and Bayesian game theory. Nat. Commun. 4, 2057 (2013)

    ADS  Google Scholar 

  18. Pappa, A., et al.: Nonlocality and conflicting interest games. Phys. Rev. Lett. 114, 020401 (2015)

    Article  ADS  Google Scholar 

  19. Roy, A., Mukherjee, A., Guha, T., Ghosh, S., Bhattacharya, S.S., Banik, M.: Nonlocal correlations: fair and unfair strategies in Bayesian game. Phys. Rev. A 94, 032120 (2016)

    Article  ADS  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  21. Schrödinger, E.: Discussion of probability relations between separated systems. Proc. Camb. Philos. Soc. 31, 553 (1935)

    Article  MATH  Google Scholar 

  22. Schrödinger, E.: Discussion of probability relations between separated systems. Proc. Camb. Philos. Soc. 32, 446 (1936)

    Article  ADS  MATH  Google Scholar 

  23. Wiseman, H.M., Jones, S.J., Doherty, A.C.: Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox. Phys. Rev. Lett. 98, 140402 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Jones, S.J., Wiseman, H.M., Doherty, A.C.: Entanglement, Einstein–Podolsky–Rosen correlations, bell nonlocality, and steering. Phys. Rev. A 76, 052116 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Milne, A., Jevtic, S., Jennings, D., Wiseman, H., Rudolph, T.: Quantum steering ellipsoids, extremal physical states and monogamy. N. J. Phys. 16, 083017 (2014)

    Article  Google Scholar 

  26. Jevtic, S., Pusey, M., Jennings, D., Rudolph, T.: Quantum steering ellipsoids. Phys. Rev. Lett. 113, 020402 (2014)

    Article  ADS  Google Scholar 

  27. Jevtic, S., Hall, M.J.W., Anderson, M.R., Zwierz, M., Wiseman, H.M.: arXiv:1411.1517v1 (2014)

  28. Bowles, J., Vértesi, T., Quintino, M.T., Brunner, N.: One-way Einstein–Podolsky–Rosen steering. Phys. Rev. Lett. 112, 200402 (2014)

    Article  ADS  Google Scholar 

  29. Wittmann, B., Ramelow, S., Steinlechner, F., Langford, N.K., Brunner, N., Wiseman, H., Ursin, R., Zeilinger, A.: Loophole-free Einstein–Podolsky–Rosen experiment via quantum steering. N. J. Phys. 14, 053030 (2012)

    Article  Google Scholar 

  30. Evans, D.A., Wiseman, H.M.: Optimal measurements for tests of Einstein–Podolsky–Rosen steering with no detection loophole using two-qubit Werner states. Phys. Rev. A 90, 012114 (2014)

    Article  ADS  Google Scholar 

  31. Reid, M.D.: Demonstration of the Einstein–Podolsky–Rosen paradox using nondegenerate parametric amplification. Phys. Rev. A 40, 913 (1989)

    Article  ADS  Google Scholar 

  32. Ou, Z.Y., Pereira, S.F., Kimble, H.J., Peng, K.C.: Realization of the Einstein–Podolsky–Rosen paradox for continuous variables. Phys. Rev. Lett. 68, 3663 (1992)

    Article  ADS  Google Scholar 

  33. Cavalcanti, E.G., Jones, S.J., Wiseman, H.M., Reid, M.D.: Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox. Phys. Rev. A 80, 032112 (2009)

    Article  ADS  Google Scholar 

  34. Gurvits, L.: In: Larmore, L.L., Goemans, M.X. (eds.) Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, vol. 10 (2003)

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223, 1 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Horodecki, P.: Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A 232, 333 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Horodecki, M., Horodecki, P., Horodecki, R.: Mixed-state entanglement and distillation: is there a “bound” entanglement in nature? Phys. Rev. Lett. 80, 5239 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Terhal, B.M.: Bell inequalities and the separability criterion. Phys. Lett. A 271, 319 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Guhne, O., Toth, G.: Entanglement detection. Phys. Rep. 474, 1 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  41. Holmes, R.B.: Geometric Functional Analysis and its Applications. Springer, Berlin (1975)

    Book  MATH  Google Scholar 

  42. Barbieri, M., De Martini, F., Di Nepi, G., Mataloni, P., D’Ariano, G.M., Macchiavello, C.: Detection of entanglement with polarized photons: experimental realization of an entanglement witness. Phys. Rev. Lett. 91, 227901 (2003)

    Article  ADS  Google Scholar 

  43. Wieczorek, W., Schmid, C., Kiesel, N., Pohlner, R., Guhne, O., Weinfurter, H.: Experimental observation of an entire family of four-photon entangled states. Phys. Rev. Lett. 101, 010503 (2008)

    Article  ADS  Google Scholar 

  44. Ganguly, N., Adhikari, S., Majumdar, A.S., Chatterjee, J.: Entanglement witness operator for quantum teleportation. Phys. Rev. Lett. 107, 270501 (2011)

    Article  Google Scholar 

  45. Adhikari, S., Ganguly, N., Majumdar, A.S.: Construction of optimal teleportation witness operators from entanglement witnesses. Phys. Rev. A 86, 032315 (2012)

    Article  ADS  Google Scholar 

  46. Zhao, M.-J., Fei, S.-M., Li-Jost, X.: Complete entanglement witness for quantum teleportation. Phys. Rev. A 85, 054301 (2012)

    Article  ADS  Google Scholar 

  47. Hyllus, P., Guhne, O., Bruß, D., Lewenstein, M.: Relations between entanglement witnesses and Bell inequalities. Phys. Rev. A 72, 012321 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  48. Horodecki, R., Horodecki, P., Horodecki, M.: Violating Bell inequality by mixed spin-1/2 states: necessary and sufficient condition. Phys. Lett. A 200, 340 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. Verstraete, F., Audenaert, K., Moor, B.D.: Maximally entangled mixed states of two qubits. Phys. Rev. A 64, 012316 (2001)

    Article  ADS  Google Scholar 

  50. Johnston, N.: Separability from spectrum for qubit-qudit states. Phys. Rev. A 88, 062330 (2013)

    Article  ADS  Google Scholar 

  51. Hildebrand, R.: Positive partial transpose from spectra. Phys. Rev. A 76, 052325 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  52. Ganguly, N., Chatterjee, J., Majumdar, A.S.: Witness of mixed separable states useful for entanglement creation. Phys. Rev. A 89, 052304 (2014)

    Article  ADS  Google Scholar 

  53. Horodecki, R., Horodecki, M., Horodecki, P.: Teleportation, Bell’s inequalities and inseparability. Phys. Lett. A 222, 21–25 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. Costa, A.C.S., Angelo, R.M.: Quantification of Einstein–Podolski–Rosen steering for two-qubit states. Phys. Rev. A 93, 020103(R) (2016)

    Article  ADS  Google Scholar 

  55. Wolf, M.M., Perez-Garcia, D., Fernandez, C.: Measurements incompatible in quantum theory cannot be measured jointly in any other no-signaling theory. Phys. Rev. Lett. 103, 230402 (2003)

    Article  Google Scholar 

  56. Quintino, M.T., Vértesi, T., Brunner, N.: Joint measurability, Einstein–Podolsky–Rosen Steering, and Bell nonlocality. Phys. Rev. Lett. 113, 160402 (2014)

    Article  ADS  Google Scholar 

  57. Uola, R., Moroder, T., Gühne, O.: Joint measurability of generalized measurements implies classicality. Phys. Rev. Lett. 113, 160403 (2014)

    Article  ADS  Google Scholar 

  58. Banik, M.: Measurement incompatibility and Schrödinger–Einstein–Podolsky–Rosen steering in a class of probabilistic theories. J. Math. Phys. 56, 052101 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  59. Girdhar, P., Cavalcanti, E.G.: All two-qubit states that are steerable via Clauser–Horne–Shimony–Holt-type correlations are Bell nonlocal. Phys. Rev. A 94, 032317 (2016)

    Article  ADS  Google Scholar 

  60. Roy, A., Bhattacharya, S.S., Mukherjee, A., Ganguly, N.: Violation of Bell-CHSH Inequality Under Global Unitary Operations. arXiv:1608.04099

  61. Ganguly, N., Mukherjee, A., Roy, A., Bhattacharya, S.S., Paul, B., Mukherjee, K.: Bell-CHSH Violation Under Global Unitary Operations: Necessary and Sufficient Conditions. arXiv:1611.05586

  62. Patro, S., Chakrabarty, I., Ganguly, N.: Non-negativity of conditional von Neumann entropy and global unitary operations. arXiv:1703.01059

  63. Verstraete, F., Wolf, M.M.: Entanglement versus Bell violations and their behavior under local filtering operations. Phys. Rev. Lett. 89, 170401 (2002)

    Article  ADS  Google Scholar 

  64. Verstraete, F., Audenaert, K., De Moor, B.: Maximally entangled mixed states of two qubits. Phys. Rev. A 64, 012316 (2001)

    Article  ADS  Google Scholar 

  65. Życzkowski, K., Horodecki, P., Sanpera, A., Lewenstein, M.: Volume of the set of separable states. Phys. Rev. A 58, 883 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  66. Werner, R.F.: Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277 (1989)

    Article  ADS  MATH  Google Scholar 

  67. Gisin, N.: Hidden quantum nonlocality revealed by local filters. Phys. Lett. A 210(3), 151–156 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  68. Yu, T., Eberly, J.H.: Evolution from entanglement to decoherence of bipartite mixed “X” states. Quantum Inf. Comput. 7, 459 (2007)

    MathSciNet  MATH  Google Scholar 

  69. Cavalcanti, E.G., Hall, M.J.W., Wiseman, H.M.: Entanglement verification and steering when Alice and Bob cannot be trusted. Phys. Rev. A 87, 032306 (2013)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We would like to gratefully acknowledge fruitful discussions with Prof. Guruprasad Kar. AM acknowledges support from the CSIR Project 09/093(0148)/2012-EMR-I. CJ acknowledges support through the Project SR/S2/LOP-08/2013 of the DST, Govt. of India.

Author information

Authors and Affiliations

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

    Some Sankar Bhattacharya, Amit Mukherjee, Arup Roy & Nirman Ganguly

  2. Department of Mathematics, South Malda College, Malda, West Bengal, India

    Biswajit Paul

  3. Department of Mathematics, Government Girls General Degree College, Ekbalpur, Kolkata, India

    Kaushiki Mukherjee

  4. Center for Security, Theory and Algorithmic Research, International Institute of Information Technology-Hyderabad, Gachibowli, Telangana, 500032, India

    Indranil Chakrabarty

  5. S. N. Bose National Centre for Basic Sciences, Salt Lake, Kolkata, 700 098, India

    C. Jebaratnam

Authors
  1. Some Sankar Bhattacharya
    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. Arup Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Biswajit Paul
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Kaushiki Mukherjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Indranil Chakrabarty
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. C. Jebaratnam
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Nirman Ganguly
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Some Sankar Bhattacharya.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bhattacharya, S.S., Mukherjee, A., Roy, A. et al. Absolute non-violation of a three-setting steering inequality by two-qubit states. Quantum Inf Process 17, 3 (2018). https://doi.org/10.1007/s11128-017-1734-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-017-1734-4

Keywords

Search

Navigation