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It is well known that Bell inequalities are derived under the assumptions of locality and realism. Bell inequalities impose strict constraints on the statistical correlations of measurements of multipartite systems. Violating each of them guarantees the existence of quantum correlations in a quantum state. A quantum state with non-vanishing entanglement may violate some Bell inequalities. Recent progress of the fields like quantum biology and quantum thermodynamics reveals a particular role of quantum coherence in quantum information processing. Quantum coherence is identified by the presence of off-diagonal terms in the density matrix. To quantify quantum coherence of a given state, Baumgratz et al. (Baumgratz T, Cramer M, Plenio M B 2014 Phys. Rev. Lett. 113 140401) provided several kinds of coherence measures such as l1-norm of coherence and relative entropy of coherence. In this paper, we propose to use quantum coherence to derive Bell inequalities. We construct the Bell inequalities of four-partite product states with l1-norm of coherence, relative entropy of coherence. In the Bell inequalities of four-partite correlations, measurement operators are products of local measurement operators. Each local operator is one of the two arbitrary observables. We consider the violations of the four-partite Bell inequalities by the four-partite general pure Greenberger-Horne-Zeilinger (GHZ) state, cluster states, W states with real coefficients. We also investigate the violations of the four-partite Bell inequalities by the four-partite GHZ class mixed states, cluster class mixed states, W class mixed states and Dicke class mixed states. It is shown that the four-partite Bell inequalities in terms of relative entropy of coherence are always violated by the four-partite general pure GHZ states, cluster states with the real coefficients. Hence there is non-vanishing entanglement for these states.
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Keywords:
- Bell inequalities /
- quantum coherence /
- multipartite quantum states
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[2] Greenberger D M, Horne M A, Shimony A, Zeilinger A, Am J 1990 Physica 58 1131
[3] Clauser J F, Horne M A, Shimony A, Holt R A 1969 Phys. Rev. Lett. 23 880
[4] Mermin N D 1990 Phys. Rev. Lett. 65 1838
[5] Ardehali M 1992 Phys. Rev. A 46 5375
[6] BelinskiiAV, Klyshko D N 1993 Phys. Usp. 36 653
[7] Peres A 1999 Found. Phys. 29 589
[8] Pitowsky I, Svozil K 2001 Phys. Rev. A 64 014102
[9] Horodecki R, Horodecki P, Horodecki M, Horodecki K 2009 Rev. Mod. Phys. 81 865
[10] Wang X Q, Lu H X, Zhao J Q 2011 Acta Phys. Sin. 60 110301 (in Chinese)[王晓芹, 逯怀新, 赵加强2011 60 110301]
[11] Chen J L, Wu C F, Kwek L C, Oh C H 2004 Phys. Rev. Lett. 93 140407
[12] Yu S X, Chen Q, Zhang C J, Lai C H, Oh C H 2012 Phys. Rev. Lett. 109 120402
[13] Gisin N 1991 Phys. Lett. A 154 201
[14] Gisin N, Peres A 1992 Phys. Lett. A 162 15
[15] Silva R, Gisin N, Guryanova Y, Popescu S 2015 Phys. Rev. Lett. 114 250401
[16] Gisin N 2015 arXiv:1509.00767
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[18] Ptz G, Aktas D, Martin A, Fedrici B, Tanzilli S, Gisin N 2016 Phys. Rev. Lett. 116 010401
[19] Xie L J, Zhang D Y, Wang X W, Zhan X G, Tang S Q, Gao F 2011 Chin. Phys. B 20 080301
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[21] Sen A, Sen U, Zukowski M 2002 Phys. Rev. A 66 062318
[22] Zhao J Q, Cao L Z, Lu H X, Wang X Q 2013 Acta Phys. Sin. 62 120301 (in Chinese)[赵加强, 曹连振, 逯怀新, 王晓芹2013 62 120301]
[23] Baumgratz T, Cramer M, Plenio M B 2014 Phys. Rev. Lett. 113 140401
[24] Girolami D 2014 Phys. Rev. Lett. 113 170401
[25] Sreltsov A, Singh U, Dhar H S, Bera M N, Adesso G 2015 Phys. Rev. Lett. 115 020403
[26] Mondal D, Pramanik T, Pati A K 2017 Phys. Rev. A 95 010301
[27] Abbott D, Davies P, Pati A K 2008 Quantum Aspects of Life (London:Imperial College Press)
[28] Plenio M B, Huelga S F 2008 New J. Phys. 10 113019
[29] Rebentrost P, Mohseni M, Aspuru-Guzik A 2009 J. Phys. Chem. B 113 9942
[30] Lloyd S 2011 J. Phys. Conf. Ser. 302 012037
[31] Huelga S, Plenio M 2013 Contemp. Phys. 54 181
[32] Rodrıguez-Rosario C A, Frauenheim T, Aspuru-GuzikA 2013 arXiv:1308.1245
[33] Lostaglio M, Jennings D, Rudolph T 2015 Nat. Commun. 6 6383
[34] Narasimhachar V, Gour G 2015 Nat. Commun. 6 7689
[35] Lostaglio M, Korzekwa K, Jennings D, Rudolph T 2015 Phys. Rev. X 5 021001
[36] Gardas B, Deffner S 2015 Phys. Rev. E 92 042126
[37] Singh U, Bera M N, Misra A, Pati A K 2015 arXiv:1506.08186
[38] Winter A, Yang D 2016 Phys. Rev. Lett. 116 120404
[39] Singh U, Bera M N, Dhar H S, Pati A K 2015 Phys. Rev. A 91 052115
[40] Kumar A 2017 Phys. Lett. A 381 991
[41] Xi Z J, Li Y M, Fan H 2015 Sci. Rep. 5 10922
[42] Yao Y, Xiao X, Ge L, Sun C P 2015 Phys. Rev. A 92 022112
[43] Cheng S, Hall M J W 2015 Phys. Rev. A 92 042101
[44] Bu K F, Kumar A, Wu J D 2016 arXiv:1603.06322
[45] Qiu L, Liu Z, Pan F 2016 arXiv:1610.07237
[46] Dr W, Vidal G, Cirac J I 2000 Phys. Rev. A 62 062314
[47] Chen X Y, Wang T T 2015 Chin. Phys. B 24 080303
[48] Ghne O, Jungnitsch B, Moroder T, Weinstein Y S 2011 Phys. Rev. A 84 052319
[49] KhosaA H, Saif F 2010 Chin. Phys. B 19 040309
[50] Kiesel N, Schmid C, Tth G, Solano E, Weinfurter H 2007 Phys. Rev. Lett. 98 063604
[51] Chen J L, Su H Y, Xu Z P, Wu Y C, Wu C F, Ye X J, Zukowski M, Kwek L C 2015 Sci. Reports 5 11624
[52] Xu J Z, Guo J B, Wen W, Bai Y K, Yan F L 2012 Chin. Phys. B 21 080305
[53] PittengerA O, Rubin M H 2000 Opt.Commun. 179 447
[54] Napoli C, Bromley T R, Cianciaruso M, Piani M, Johnston N, Adesso G 2016 Phys. Rev. Lett. 116 150502
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[1] Bell J S 1964 Physics 1 195
[2] Greenberger D M, Horne M A, Shimony A, Zeilinger A, Am J 1990 Physica 58 1131
[3] Clauser J F, Horne M A, Shimony A, Holt R A 1969 Phys. Rev. Lett. 23 880
[4] Mermin N D 1990 Phys. Rev. Lett. 65 1838
[5] Ardehali M 1992 Phys. Rev. A 46 5375
[6] BelinskiiAV, Klyshko D N 1993 Phys. Usp. 36 653
[7] Peres A 1999 Found. Phys. 29 589
[8] Pitowsky I, Svozil K 2001 Phys. Rev. A 64 014102
[9] Horodecki R, Horodecki P, Horodecki M, Horodecki K 2009 Rev. Mod. Phys. 81 865
[10] Wang X Q, Lu H X, Zhao J Q 2011 Acta Phys. Sin. 60 110301 (in Chinese)[王晓芹, 逯怀新, 赵加强2011 60 110301]
[11] Chen J L, Wu C F, Kwek L C, Oh C H 2004 Phys. Rev. Lett. 93 140407
[12] Yu S X, Chen Q, Zhang C J, Lai C H, Oh C H 2012 Phys. Rev. Lett. 109 120402
[13] Gisin N 1991 Phys. Lett. A 154 201
[14] Gisin N, Peres A 1992 Phys. Lett. A 162 15
[15] Silva R, Gisin N, Guryanova Y, Popescu S 2015 Phys. Rev. Lett. 114 250401
[16] Gisin N 2015 arXiv:1509.00767
[17] Gisin N, Tanzilli S, Tittel W 2015 Europhys. News 46 36
[18] Ptz G, Aktas D, Martin A, Fedrici B, Tanzilli S, Gisin N 2016 Phys. Rev. Lett. 116 010401
[19] Xie L J, Zhang D Y, Wang X W, Zhan X G, Tang S Q, Gao F 2011 Chin. Phys. B 20 080301
[20] Zukowski M, Brukner C 2002 Phys. Rev. Lett. 88 210401
[21] Sen A, Sen U, Zukowski M 2002 Phys. Rev. A 66 062318
[22] Zhao J Q, Cao L Z, Lu H X, Wang X Q 2013 Acta Phys. Sin. 62 120301 (in Chinese)[赵加强, 曹连振, 逯怀新, 王晓芹2013 62 120301]
[23] Baumgratz T, Cramer M, Plenio M B 2014 Phys. Rev. Lett. 113 140401
[24] Girolami D 2014 Phys. Rev. Lett. 113 170401
[25] Sreltsov A, Singh U, Dhar H S, Bera M N, Adesso G 2015 Phys. Rev. Lett. 115 020403
[26] Mondal D, Pramanik T, Pati A K 2017 Phys. Rev. A 95 010301
[27] Abbott D, Davies P, Pati A K 2008 Quantum Aspects of Life (London:Imperial College Press)
[28] Plenio M B, Huelga S F 2008 New J. Phys. 10 113019
[29] Rebentrost P, Mohseni M, Aspuru-Guzik A 2009 J. Phys. Chem. B 113 9942
[30] Lloyd S 2011 J. Phys. Conf. Ser. 302 012037
[31] Huelga S, Plenio M 2013 Contemp. Phys. 54 181
[32] Rodrıguez-Rosario C A, Frauenheim T, Aspuru-GuzikA 2013 arXiv:1308.1245
[33] Lostaglio M, Jennings D, Rudolph T 2015 Nat. Commun. 6 6383
[34] Narasimhachar V, Gour G 2015 Nat. Commun. 6 7689
[35] Lostaglio M, Korzekwa K, Jennings D, Rudolph T 2015 Phys. Rev. X 5 021001
[36] Gardas B, Deffner S 2015 Phys. Rev. E 92 042126
[37] Singh U, Bera M N, Misra A, Pati A K 2015 arXiv:1506.08186
[38] Winter A, Yang D 2016 Phys. Rev. Lett. 116 120404
[39] Singh U, Bera M N, Dhar H S, Pati A K 2015 Phys. Rev. A 91 052115
[40] Kumar A 2017 Phys. Lett. A 381 991
[41] Xi Z J, Li Y M, Fan H 2015 Sci. Rep. 5 10922
[42] Yao Y, Xiao X, Ge L, Sun C P 2015 Phys. Rev. A 92 022112
[43] Cheng S, Hall M J W 2015 Phys. Rev. A 92 042101
[44] Bu K F, Kumar A, Wu J D 2016 arXiv:1603.06322
[45] Qiu L, Liu Z, Pan F 2016 arXiv:1610.07237
[46] Dr W, Vidal G, Cirac J I 2000 Phys. Rev. A 62 062314
[47] Chen X Y, Wang T T 2015 Chin. Phys. B 24 080303
[48] Ghne O, Jungnitsch B, Moroder T, Weinstein Y S 2011 Phys. Rev. A 84 052319
[49] KhosaA H, Saif F 2010 Chin. Phys. B 19 040309
[50] Kiesel N, Schmid C, Tth G, Solano E, Weinfurter H 2007 Phys. Rev. Lett. 98 063604
[51] Chen J L, Su H Y, Xu Z P, Wu Y C, Wu C F, Ye X J, Zukowski M, Kwek L C 2015 Sci. Reports 5 11624
[52] Xu J Z, Guo J B, Wen W, Bai Y K, Yan F L 2012 Chin. Phys. B 21 080305
[53] PittengerA O, Rubin M H 2000 Opt.Commun. 179 447
[54] Napoli C, Bromley T R, Cianciaruso M, Piani M, Johnston N, Adesso G 2016 Phys. Rev. Lett. 116 150502
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