logo

SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 62 , Issue 3 : 030311(2019) https://doi.org/10.1007/s11433-018-9279-4

Characterizing Bell nonlocality and EPR steering

More info
  • ReceivedMay 23, 2018
  • AcceptedJul 19, 2018
  • PublishedSep 30, 2018
PACS numbers

Abstract

Bell nonlocality and Einstein-Podolsky-Rosen (EPR) steering are very important quantum correlations in composite quantum systems. Bell nonlocality of a bipartite state is observed in some local quantum measurements, while EPR steering was first observed by Schr$\ddot{\rm{o}}$dinger in the context of famous EPR paradox. In this paper, we discuss the Bell nonlocality and EPR steering of bipartite states, including mathematical definitions and characterizations of these two quantum correlations, the convexity as well as the closedness of the sets of all Bell local states and all EPR unsteerable states, respectively. We also derive sufficient conditions for a state to be steerable; these conditions imply that Alice can steer Bob's state whenever Alice has two POV measurements such that the sets of Bob's normalized conditional states become two disjoint sets of pure states, or whenever she has one POV measurement such that Bob's normalized conditional states become a linearly independent set of pure states.


Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11871318, 11771009, 11571213, and 11601300), and the Fundamental Research Funds for the Central Universities (Grant Nos. GK201703093, and GK201801011). This manuscript was first put in the Los-Alamos eprint server on Jan 30, 2018, labeled arXiv: 1801.09891 and named “Some remarks on Bell non-locality and Einstein-Podolsky-Rosen steering of bipartite states."


References

[1] Blok M. S., Kalb N., Reiserer A., Taminiau T. H., Hanson R.. Faraday Discuss., 2015, 184: 173-182 CrossRef PubMed ADS Google Scholar

[2] Deng D. L., Li X., Das Sarma S.. Phys. Rev. X, 2017, 7: 021021 CrossRef ADS arXiv Google Scholar

[3] Chiribella G., D'Ariano G. M., Perinotti P.. Phys. Rev. A, 2009, 80: 022339 CrossRef ADS arXiv Google Scholar

[4] D. Mulder, and G. Bianconi,. arXiv Google Scholar

[5] Wang C., Deng F. G., Li Y. S., Liu X. S., Long G. L.. Phys. Rev. A, 2005, 71: 44305 CrossRef ADS Google Scholar

[6] Zhao S. M., Shen Z. G., Xiao H., Wang L.. Sci. China-Phys. Mech. Astron., 2018, 61: 090323 CrossRef ADS Google Scholar

[7] Bartlett S. D., Rudolph T., Spekkens R. W.. Rev. Mod. Phys., 2007, 79: 555-609 CrossRef ADS Google Scholar

[8] Marvian I., Spekkens R. W.. New J. Phys., 2013, 15: 033001 CrossRef ADS arXiv Google Scholar

[9] Gong Z. R., Zhang Z. W., Xu D. Z., Zhao N., Sun C. P.. Sci. China-Phys. Mech. Astron., 2018, 61: 040311 CrossRef ADS Google Scholar

[10] Li B., Jiang S. H., Fei S. M., Li-Jost X. Q.. Sci. China-Phys. Mech. Astron., 2018, 61: 040321 CrossRef ADS arXiv Google Scholar

[11] DiVincenzo D. P.. Science, 1995, 270: 255-261 CrossRef ADS Google Scholar

[12] Zhou J., Liu B. J., Hong Z. P., Xue Z. Y.. Sci. China-Phys. Mech. Astron., 2018, 61: 010312 CrossRef ADS arXiv Google Scholar

[13] M. A. Nielsen, and I. L. Chuang, Quantum Computation and Quantum Information, 2nd ed. (Cambridge University Press, Cambridge, 2010). Google Scholar

[14] Luo S.. Phys. Rev. A, 2008, 77: 022301 CrossRef ADS Google Scholar

[15] Guo Z., Cao H., Chen Z.. J. Phys. A-Math. Theor., 2012, 45: 145301 CrossRef ADS Google Scholar

[16] Wu Y. C., Guo G. C.. Phys. Rev. A, 2011, 83: 062301 CrossRef ADS Google Scholar

[17] Guo Z., Cao H., Qu S.. Inf. Sci., 2014, 289: 262-272 CrossRef Google Scholar

[18] Bell J. S.. Phys. Physique Fizika, 1964, 1: 195-200 CrossRef Google Scholar

[19] Brunner N., Cavalcanti D., Pironio S., Scarani V., Wehner S.. Rev. Mod. Phys., 2014, 86: 419-478 CrossRef ADS arXiv Google Scholar

[20] Clauser J. F., Shimony A.. Rep. Prog. Phys., 1978, 41: 1881-1927 CrossRef ADS Google Scholar

[21] Home D., Selleri F.. Riv. Nuovo Cim., 1991, 14: 1-95 CrossRef ADS Google Scholar

[22] Khalfin L. A., Tsirelson B. S.. Found Phys, 1992, 22: 879-948 CrossRef ADS Google Scholar

[23] B. S. Tsirelson, Hadronic J. Suppl. 8, 329 (1993). Google Scholar

[24] Zeilinger A.. Rev. Mod. Phys., 1999, 71: S288-S297 CrossRef ADS Google Scholar

[25] Werner R. F., Wolf M. M.. Phys. Rev. A, 2001, 64: 032112 CrossRef ADS Google Scholar

[26] Genovese M.. Phys. Rep., 2005, 413: 319-396 CrossRef ADS Google Scholar

[27] Buhrman H., Cleve R., Massar S., de Wolf R.. Rev. Mod. Phys., 2010, 82: 665-698 CrossRef ADS arXiv Google Scholar

[28] Zhao L. J., Guo Y. M., Li-Jost X. Q., Fei S. M.. Sci. China-Phys. Mech. Astron., 2018, 61: 070321 CrossRef ADS Google Scholar

[29] Yang Y., Cao H., Chen L., Huang Y.. Int. J. Theor. Phys., 2018, 57: 1498-1515 CrossRef ADS Google Scholar

[30] Long G. L., Qin W., Yang Z., Li J. L.. Sci. China-Phys. Mech. Astron., 2018, 61: 030311 CrossRef ADS Google Scholar

[31] Schr?dinger E., Born M.. Math. Proc. Camb. Phil. Soc., 1935, 31: 555 CrossRef ADS Google Scholar

[32] Einstein A., Podolsky B., Rosen N.. Phys. Rev., 1935, 47: 777-780 CrossRef ADS Google Scholar

[33] Reid M. D.. Phys. Rev. A, 1989, 40: 913-923 CrossRef ADS Google Scholar

[34] Werner R. F.. Phys. Rev. A, 1989, 40: 4277-4281 CrossRef ADS Google Scholar

[35] Ou Z. Y., Pereira S. F., Kimble H. J., Peng K. C.. Phys. Rev. Lett., 1992, 68: 3663-3666 CrossRef PubMed ADS Google Scholar

[36] Wiseman H. M., Jones S. J., Doherty A. C.. Phys. Rev. Lett., 2007, 98: 140402 CrossRef PubMed ADS Google Scholar

[37] Cavalcanti E. G., Jones S. J., Wiseman H. M., Reid M. D.. Phys. Rev. A, 2009, 80: 032112 CrossRef ADS arXiv Google Scholar

[38] Saunders D. J., Jones S. J., Wiseman H. M., Pryde G. J.. Nat. Phys, 2010, 6: 845-849 CrossRef ADS arXiv Google Scholar

[39] Bennet A. J., Evans D. A., Saunders D. J., Branciard C., Cavalcanti E. G., Wiseman H. M., Pryde G. J.. Phys. Rev. X, 2012, 2: 031003 CrossRef ADS arXiv Google Scholar

[40] H?ndchen V., Eberle T., Steinlechner S., Samblowski A., Franz T., Werner R. F., Schnabel R.. Nat. Photon, 2012, 6: 596-599 CrossRef ADS Google Scholar

[41] Branciard C., Cavalcanti E. G., Walborn S. P., Scarani V., Wiseman H. M.. Phys. Rev. A, 2012, 85: 010301(R) CrossRef ADS arXiv Google Scholar

[42] Wittmann B., Ramelow S., Steinlechner F., Langford N. K., Brunner N., Wiseman H. M., Ursin R., Zeilinger A.. New J. Phys., 2012, 14: 053030 CrossRef ADS arXiv Google Scholar

[43] Steinlechner S., Bauchrowitz J., Eberle T., Schnabel R.. Phys. Rev. A, 2013, 87: 022104 CrossRef ADS arXiv Google Scholar

[44] Reid M. D.. Phys. Rev. A, 2013, 88: 062338 CrossRef ADS arXiv Google Scholar

[45] Skrzypczyk P., Navascués M., Cavalcanti D.. Phys. Rev. Lett., 2014, 112: 180404 CrossRef PubMed ADS arXiv Google Scholar

[46] Piani M., Watrous J.. Phys. Rev. Lett., 2015, 114: 060404 CrossRef PubMed ADS arXiv Google Scholar

[47] ?ukowski M., Dutta A., Yin Z.. Phys. Rev. A, 2015, 91: 032107 CrossRef ADS arXiv Google Scholar

[48] Quintino M. T., Vértesi T., Cavalcanti D., Augusiak R., Demianowicz M., Acín A., Brunner N.. Phys. Rev. A, 2015, 92: 032107 CrossRef ADS arXiv Google Scholar

[49] Zhu H., Hayashi M., Chen L.. Phys. Rev. Lett., 2016, 116: 070403 CrossRef PubMed ADS Google Scholar

[50] Sun K., Ye X. J., Xu J. S., Xu X. Y., Tang J. S., Wu Y. C., Chen J. L., Li C. F., Guo G. C.. Phys. Rev. Lett., 2016, 116: 160404 CrossRef PubMed ADS arXiv Google Scholar

[51] Quan Q., Zhu H., Liu S. Y., Fei S. M., Fan H., Yang W. L.. Sci Rep, 2016, 6: 22025 CrossRef PubMed ADS arXiv Google Scholar

[52] Chen J. L., Ren C., Chen C., Ye X. J., Pati A. K.. Sci Rep, 2016, 6: 39063 CrossRef PubMed ADS arXiv Google Scholar

[53] Cavalcanti D., Skrzypczyk P.. Rep. Prog. Phys., 2017, 80: 024001 CrossRef PubMed ADS arXiv Google Scholar

[54] D. Das, S. Datta, C. Jebaratnam, et al,. arXiv Google Scholar

[55] Ren C., Su H. Y., Shi H., Chen J.. Phys. Rev. A, 2018, 97: 032119 CrossRef ADS Google Scholar

[56] Zheng C., Guo Z., Cao H.. Int. J. Theor. Phys., 2018, 57: 1787-1801 CrossRef ADS Google Scholar

[57] Zhao M. J., Ma T., Zhang T. G., Fei S. M.. Sci. China-Phys. Mech. Astron., 2016, 59: 120313 CrossRef ADS arXiv Google Scholar

[58] Cao H. X.. Sci. China-Phys. Mech. Astron., 2017, 60: 020332 CrossRef ADS Google Scholar

[59] Wei S. J., Xin T., Long G. L.. Sci. China-Phys. Mech. Astron., 2018, 61: 070311 CrossRef ADS arXiv Google Scholar

[60] Yu S., Chen Q., Zhang C., Lai C. H., Oh C. H.. Phys. Rev. Lett., 2012, 109: 120402 CrossRef PubMed ADS arXiv Google Scholar

[61] Sun K., Ye X. J., Xu J. S., Xu X. Y., Tang J. S., Wu Y. C., Chen J. L., Li C. F., Guo G. C.. Phys. Rev. Lett., 2016, 116: 160404 CrossRef PubMed ADS arXiv Google Scholar

[62] Wu C., Chen J. L., Ye X. J., Su H. Y., Deng D. L., Wang Z., Oh C. H.. Sci Rep, 2015, 4: 4291 CrossRef PubMed ADS arXiv Google Scholar

[63] Guo Z., Cao H., Chen Z.. J. Phys. A-Math. Theor., 2012, 45: 145301 CrossRef ADS Google Scholar

[64] Guo Z., Cao H., Qu S.. Found Phys, 2015, 45: 355-369 CrossRef ADS Google Scholar

[65] Chen J. L., Ye X. J., Wu C., Su H. Y., Cabello A., Kwek L. C., Oh C. H.. Sci Rep, 2013, 3: 2143 CrossRef PubMed ADS arXiv Google Scholar

[66] Nguyen H. C., Luoma K.. Phys. Rev. A, 2017, 95: 042117 CrossRef ADS Google Scholar

  • Figure 1

    (Color online) Sketch of a Bell experiment in which $\rho^{\text{AB}}$ denotes the shared state and $x$ and $y$ label the measurement choices of Alice and Bob, respectively, while $a$ and $b$ denote the respective outcomes when $x$ and $y$ are chosen and performed.

  • Figure 2

    (Color online) Sketch of a quantum steering from Alice to Bob, in which $\rho^{\text{AB}}$ denotes the shared state and $x$ and $a$ denote Alice's measurement choice and corresponding outcome, respectively, when the measurement $x$ is chosen and performed.

Copyright 2020  CHINA SCIENCE PUBLISHING & MEDIA LTD.  中国科技出版传媒股份有限公司  版权所有

京ICP备14028887号-23       京公网安备11010102003388号