Local distinguishability of generalized Bell states

  • Ying-Hui Yang
  • Cai-Hong Wang
  • Jiang-Tao Yuan
  • Xia Wu
  • Hui-Juan Zuo
Article
  • 69 Downloads

Abstract

We investigate the distinguishability of orthogonal generalized Bell states (GBSs) in \(d\otimes d\) system by local operations and classical communication (LOCC), where d is a prime. We show that |S| is no more than \(d+1\) for any l GBSs, i.e., \(|S|\le d+1\), where S is maximal set which is composed of pairwise noncommuting pairs in \({\varDelta } U\). If \(|S|\le d\), then the l GBSs can be distinguished by LOCC according to our main Theorem. Compared with the results (Fan in Phys Rev Lett 92:177905, 2004; Tian et al. in Phys Rev A 92:042320, 2015), our result is more general. It can determine local distinguishability of \(l (> k)\) GBSs, where k is the number of GBSs in Fan’s and Tian’s results. Only for \(|S|=d+1\), we do not find the answer. We conjecture that any l GBSs cannot be distinguished by one-way LOCC if \(|S|=d+1\). If this conjecture is right, the problem about distinguishability of GBSs with one-way LOCC is completely solved in \(d\otimes d\).

Keywords

Local distinguishability Maximally entangled states Generalized Bell states Local operations and classical communication 

Notes

Acknowledgements

This work is supported by NSFC (Grant Nos. 61601171, 61402148, 61701553), Project of Science and Technology Department of Henan Province of China (172102210275), Foundation of Doctor of Henan Polytechnic University (B2017-48), Natural Science Foundation of Hebei Province (F2015205114).

References

  1. 1.
    Bennett, C.H., DiVincenzo, D.P., Fuchs, C.A., Mor, T., Rains, E., Shor, P.W., Smolin, J.A., Wootters, W.K.: Quantum nonlocality without entanglement. Phys. Rev. A 59, 1070 (1999)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Zhang, Z.-C., Gao, F., Tian, G.-J., Cao, T.-Q., Wen, Q.-Y.: Nonlocality of orthogonal product basis quantum states. Phys. Rev. A 90, 022313 (2014)ADSCrossRefGoogle Scholar
  3. 3.
    Wang, Y.-L., Li, M.-S., Zheng, Z.-J., Fei, S.-M.: Nonlocality of orthogonal product-basis quantum states. Phys. Rev. A 92, 032313 (2015)ADSCrossRefGoogle Scholar
  4. 4.
    Zhang, Z.-C., Gao, F., Cao, Y., Qin, S.-J., Wen, Q.-Y.: Local indistinguishability of orthogonal product states. Phys. Rev. A 93, 012314 (2016)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Xu, G.-B., Wen, Q.-Y., Qin, S.-J., Yang, Y.-H., Gao, F.: Quantum nonlocality of multipartite orthogonal product states. Phys. Rev. A 93, 032341 (2016)ADSCrossRefGoogle Scholar
  6. 6.
    Zhang, Z.-C., Zhang, K.-J., Gao, F., Wen, Q.-Y., Oh, C.H.: Construction of nonlocal multipartite quantum states. Phys. Rev. A 95, 052344 (2017)ADSCrossRefGoogle Scholar
  7. 7.
    Wang, Y.-L., Li, M.-S., Zheng, Z.-J., Fei, S.-M.: The local indistinguishability of multipartite product states. Quantum Inf. Process. 16, 5 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Walgate, J., Short, A.J., Hardy, L., Vedral, V.: Local distinguishability of multipartite orthogonal quantum states. Phys. Rev. Lett. 85, 4972 (2000)ADSCrossRefGoogle Scholar
  9. 9.
    Chen, P.X., Li, C.Z.: Distinguishing the elements of a full product basis set needs only projective measurements and classical communication. Phys. Rev. A 70, 022306 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Feng, Y., Shi, Y.Y.: Characterizing locally indistinguishable orthogonal product states. IEEE Trans. Inf. Theory 55, 2799 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Duan, R.Y., Feng, Y., Xin, Y., Ying, M.S.: Distinguishability of quantum states by separable operations. IEEE Trans. Inf. Theory 55, 1320 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Duan, R.Y., Xin, Y., Ying, M.S.: Locally indistinguishable subspaces spanned by three-qubit unextendible product bases. Phys. Rev. A 81, 032329 (2010)ADSCrossRefGoogle Scholar
  13. 13.
    Yu, N.K., Duan, R.Y., Ying, M.S.: Any \(2\otimes n\) subspace is locally distinguishable. Phys. Rev. A 84, 012304 (2011)ADSCrossRefGoogle Scholar
  14. 14.
    Bennett, C.H., DiVincenzo, D.P., Mor, T., Shor, P.W., Smolin, J.A., Terhal, B.M.: Unextendible product bases and bound entanglement. Phys. Rev. Lett. 82, 5385 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    DiVincenzo, D.P., Mor, T., Shor, P.W., Smolin, J.A., Terhal, B.M.: Unextendible product bases, uncompletable product bases and bound entanglement. Commun. Math. Phys. 238, 379 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Johnston, N.: The structure of qubit unextendible product bases. J. Phys. A: Math. Theor. 47, 424034 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Fan, H.: Distinguishability and indistinguishability by local operations and classical communication. Phys. Rev. Lett. 92, 177905 (2004)ADSCrossRefGoogle Scholar
  18. 18.
    Nathanson, M.: Distinguishing bipartitite orthogonal states using LOCC: best and worst cases. J. Math. Phys. 46, 062103 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Bandyopadhyay, S., Ghosh, S., Kar, G.: LOCC distinguishability of unilaterally transformable quantum states. New J. Phys. 13, 123013 (2011)ADSCrossRefGoogle Scholar
  20. 20.
    Yu, N.K., Duan, R.Y., Ying, M.S.: Four locally indistinguishable ququad–ququad orthogonal maximally entangled states. Phys. Rev. Lett. 109, 020506 (2012)ADSCrossRefGoogle Scholar
  21. 21.
    Nathanson, M.: Three maximally entangled states can require two-way local operations and classical communication for local discrimination. Phys. Rev. A 88, 062316 (2013)ADSCrossRefGoogle Scholar
  22. 22.
    Cosentino, A.: Positive-partial-transpose-indistinguishable states via semidefinite programming. Phys. Rev. A 87, 012321 (2013)ADSCrossRefGoogle Scholar
  23. 23.
    Cosentino, A., Russo, V.: Small sets of locally indistinguishable orthogonal maximally entangled states. Quantum Inf. Comput. 14, 1098 (2014)MathSciNetGoogle Scholar
  24. 24.
    Zhang, Z.-C., Wen, Q.-Y., Gao, F., Tian, G.-J., Cao, T.-Q.: One-way LOCC indistinguishability of maximally entangled states. Quantum Inf. Process. 13, 795 (2014)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Zhang, Z.-C., Feng, K.-Q., Gao, F., Wen, Q.-Y.: Distinguishing maximally entangled states by one-way local operations and classical communication. Phys. Rev. A 91, 012329 (2015)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Tian, G.-J., Yu, S.-X., Gao, F., Wen, Q.-Y., Oh, C.H.: Local discrimination of qudit lattice states via commutativity. Phys. Rev. A 92, 042320 (2015)ADSCrossRefGoogle Scholar
  27. 27.
    Tian, G.-J., Yu, S.-X., Gao, F., Wen, Q.-Y., Oh, C.H.: Local discrimination of four or more maximally entangled states. Phys. Rev. A 91, 052314 (2015)ADSCrossRefGoogle Scholar
  28. 28.
    Tian, G.-J., Wu, X., Cao, Y., Gao, F., Wen, Q.-Y.: General existence of locally distinguishable maximally entangled states only with two-way classical communication. Sci. Rep. 6, 30181 (2016)ADSCrossRefGoogle Scholar
  29. 29.
    Tian, G.-J., Yu, S.-X., Gao, F., Wen, Q.-Y., Oh, C.H.: Classification of locally distinguishable and indistinguishable sets of maximally entangled states. Phys. Rev. A 94, 052315 (2016)ADSCrossRefGoogle Scholar
  30. 30.
    Wang, Y.-L., Li, M.-S., Zheng, Z.-J., Fei, S.-M.: On small set of one-way LOCC indistinguishability of maximally entangled states. Quantum Inf. Process. 15, 1661 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Wang, Y.-L., Li, M.-S., Fei, S.-M., Zheng, Z.-J.: The local distinguishability of any three generalized Bell states. Quantum Inf. Process. 16, 126 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Walgate, J., Hardy, L.: Nonlocality, asymmetry, and distinguishing bipartite states. Phys. Rev. Lett. 89, 147901 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Duan, R.Y., Feng, Y., Ji, Z.F., Ying, M.S.: Distinguishing arbitrary multipartite basis unambiguously using local operations and classical communication. Phys. Rev. Lett. 98, 230502 (2007)ADSCrossRefGoogle Scholar
  34. 34.
    Yang, Y.-H., Gao, F., Tian, G.-J., Cao, T.-Q., Wen, Q.-Y.: Local distinguishability of orthogonal quantum states in a \(2\otimes 2\otimes 2\) system. Phys. Rev. A 88, 024301 (2013)ADSCrossRefGoogle Scholar
  35. 35.
    Yang, Y.-H., Gao, F., Tian, G.-J., Cao, T.-Q., Zuo, H.-J., Wen, Q.-Y.: Bound on local unambiguous discrimination between multipartite quantum states. Quantum Inf. Process. 14, 731 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    DiVincenzo, D.P., Leung, D.W., Terhal, B.M.: Quantum data hiding. IEEE Trans. Inf. Theory 48, 580 (2002)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Rahaman, R., Parker, M.G.: Quantum scheme for secret sharing based on local distinguishability. Phys. Rev. A 91, 022330 (2015)ADSCrossRefGoogle Scholar
  38. 38.
    Yang, Y.-H., Gao, F., Wu, X., Qin, S.-J., Zuo, H.-J., Wen, Q.-Y.: Quantum secret sharing via local operations and classical communication. Sci. Rep. 5, 16967 (2015)ADSCrossRefGoogle Scholar
  39. 39.
    Gao, F., Liu, B., Huang, W., Wen, Q.-Y.: Postprocessing of the oblivious key in quantum private query. IEEE. J. Sel. Top. Quantum 21, 6600111 (2015)Google Scholar
  40. 40.
    Wei, C.-Y., Wang, T.-Y., Gao, F.: Practical quantum private query with better performance in resisting joint-measurement attack. Phys. Rev. A 93, 042318 (2016)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceHenan Polytechnic UniversityJiaozuoChina
  2. 2.School of InformationCentral University of Finance and EconomicsBeijingChina
  3. 3.Mathematics and Information Science CollegeHebei Normal UniversityShijiazhuangChina

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