Advertisement

Constructing Entanglement Witnesses for Infinite-Dimensional Systems

  • Jinchuan HouEmail author
  • Wenli Wang
Article
  • 3 Downloads

Abstract

We generalize the results in Yu and Liu (Phys. Rev. Lett. 95, 150504, 2005) and Hou and Guo (Int. J. Theor. Phys. 50, 1245–1254, 2011) to infinite-dimensional systems and answer a problem raised in the second paper. Consider a bipartite system HK with dimH = dimK = . We show that (1) for any orthonormal sequences\(\{E_{k}\}_{k = 1}^{\infty }\) and\(\{F_{k}\}_{k = 1}^{\infty }\) consist of observables respectively in\(\mathcal {C}_{2}(H)\) and\(\mathcal {C}_{2}(K)\), if\({\sum }_{k} E_{k} \otimes F_{k}\) converges under the weak operator topology and if\(W=I-{\sum }_{k} E_{k}\otimes F_{k}\) is not positive, then W is a decomposable entanglement witness; (2) every state ρ of system HK has a Schmidt decomposition\(\rho = {\sum }_{k} \delta _{k} E_{k} \otimes F_{k}\) with {Ek} and {Fk} orthonormal sequences of observables.

Keywords

Infinite-dimensional systems Entanglement states PPT states Entanglement witnesses 

Notes

Acknowledgments

This work is partially supported by National Natural Science Foundation of China (11671294).

References

  1. 1.
    Nielsen, M.A., Chuang, I.L.: Quantum Computatation and Quantum Information. Cambridge University Press, Cambridge (2000)Google Scholar
  2. 2.
    Chen, K., Wu, L.-A.: A matrix realignment method for recognizing entanglement. Quantum Inf. Comput. 3, 193 (2003)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Blanchard, Ph., Brüning, E.: Remarks on the structure of states of composite quantum systems and envariance. Phys. Lett. A 335, 180–187 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Guo, Y., Qi, X.-F., Hou, J.-C.: Sufficient and necessary conditions of separability for bipartite pure states in infinite-dimensional systems. Chinese Sci. Bull. 56(9), 840–846 (2011)ADSCrossRefGoogle Scholar
  5. 5.
    Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 233, 1 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Horodecki, P.: Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A 232, 333–339 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hou, J.-C.: A characterization of positive linear maps and criteria for entangled quantum states. J. Phys. A: Math. Theor. 43, 385201 (2010)ADSCrossRefzbMATHGoogle Scholar
  8. 8.
    Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Rudolph, O.: Further results on the cross norm criterion for separability. Quantum Inf. Process. 4, 219 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hou, J.-C., Guo, Y.: When different entanglement witesses detect the same entangled states. Phys. Rev. A 82, 052301 (2010)ADSCrossRefGoogle Scholar
  11. 11.
    Yu, S., Liu, N.-L.: Entanglement detection by local orthogonal observables. Phys. Rev. Lett. 95, 150504 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hou, J.-C., Guo, Y.: Constructing entanglement witnesses for states in infinite-dimensional bipartite quantum systems. Int. J. Theor. Phys. 50, 1245–1254 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Størmer, E.: Separable states and positive maps. J. Funct. Anal. 254, 2303–2312 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Arveson, W.: Maximal vectors in Hilbert spaces and quantum entanglement. J. Funct. Anal. 256, 1476–1510 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Tóth, G., Gühne, O.: Separability criteria and entanglement witnesses for symmetric quantum states. Appl. Phys. B 98(4), 617–622 (2010)ADSCrossRefGoogle Scholar
  16. 16.
    Hou, J.-C., Qi, X.-F.: Constructing entanglement witnesses for infinite-dimensional systems. Phys. Rev. A 81, 062351 (2010)ADSCrossRefGoogle Scholar
  17. 17.
    Guo, Y., Hou, J.-C.: Realignment operation and CCNR criterion of separability for states in infinite-dimensional quantum systems. Rep. Math. Phys. 72(1), 25–40 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jafarizadeh, M.A., Mahdian, M.: Quatifying entanglement of two relativistic particles via decomposable optimal entanglement witnesses. arXiv:0911.2307vl [quant-ph] (2009)
  19. 19.
    Conway, J.B.: A Course in Functional Analysis. Springer, New York (1985)CrossRefzbMATHGoogle Scholar
  20. 20.
    Qi, X.-F., Hou, J.-C.: Characterization of optimal entanglement witnesses. Phys. Rev. A 85, 022334 (2012)ADSCrossRefGoogle Scholar
  21. 21.
    Qi, X.-F., Hou, J.-C.: Optimality of a class of entanglement witnesses for 33 systems. Inter. J. Theor. Phys. 52, 3474–3 (2013)CrossRefzbMATHGoogle Scholar
  22. 22.
    Qi, X.-F., Hou, J.-C.: Optimality of entanglement witnesses constructed from arbitrary permutations. Quantum Inf. Process. 14, 2499–2515 (2015)ADSCrossRefzbMATHGoogle Scholar
  23. 23.
    Qi, X.-F., Hou, J.-C.: Indecomposability of entanglement witnesses constructed from any permutations. Quantum Inf. Comput. 15(5,6), 0478–0488 (2015)MathSciNetGoogle Scholar
  24. 24.
    Augusiak, R., Tura, J., Lewenstein, M.: A note on the optimality of decomposable entanglement witnesses and completely entangled subspaces. arXiv:1012.3786v3 [qunt-ph] (2011)
  25. 25.
    Lewenstein, M., Kraus, B., Cirac, J.I., Horodecki, P.: Optimization of entanlement witnesses. Phys. Rev. A 62, 052310 (2000)ADSCrossRefGoogle Scholar
  26. 26.
    Hou, J.-C.: On the tensor products of operators. Acta. Math. Sinica (New Ser.) 9, 195–202 (1993)CrossRefzbMATHGoogle Scholar
  27. 27.
    Megginson, R.E.: A Introduction to Banach Space Theory Graduate Texts in Mathematics, vol. 183. Springer, New York (1998)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsTaiyuan University of TechnologyTaiyuanPeople’s Republic of China

Personalised recommendations