基于关联行为的隐变量理论与量子非定域性的研究
|
摘要
量子非定域性的研究不仅是量子力学理论体系中的基本问题,而且在当今的研究热点量子信息处理,如量子计算、量子通信和量子密码安全等方面发挥重要的意义。本论文的主要内容就是从关联的角度,对量子非定域性研究的历程做了一个系统的梳理介绍。从EPR佯谬引发对量子力学的质疑,到隐变量理论的创立,直至贝尔提出的检验非定域性的不等式以及后来拓展的无不等式的贝尔定理,我们沿着这样一条发展路线阐述了量子非定域性研究的发展。其中,对于推广和构造贝尔类型的不等式,我们提出了一种构造非同类不等式系统方法。而对于无不等式的贝尔定理,我们先是推广到了更一般的三粒子GHZ态和W态,进一步地,我们推广一些四粒子的特殊混态的无不等式的贝尔证明。
本论文的主要内容和结构如下:
第一章介绍了各种类型关联的定义,包括广义关联,无信号关联,定域关联,部分定域关联,和量子关联。
第二章详细的介绍了定域隐变量模型的产生过程。对于EPR佯谬,玻姆模型,冯诺依曼对于无弥散态的证明,贝尔对冯诺依曼证明的质疑,以及格里森定理都做了详细的介绍。进一步地,也给出了多体隐变量模型的定义。
第三章首先介绍了贝尔不等式以及CHSH不等式的构造过程,以及它们体现最大违反。随后介绍了墨明所设计的关于不等式的形象描述。最后详细阐述了一种构造非同类不等式的系统方法。
第四章首先关于无不等式证明的GHZ模型,其次给出了关于部分广义GHZ态和W态的无不等式的贝尔证明,最后将无不等式的贝尔证明拓展到了特殊四粒子的纠缠混态。
第五章尝试讨论了定域隐变量理论与非对易性之间的关系。
第六章结论
Quantum non-locality is not only an foundation problem of quantum theory,but also plays an important significance on quantum information processing,such as quantum computation,quantum communication and quantum cryptography security.The main content of this paper is research on LHV theory and quantum non-locality from the viewpoint of correlation.From the EPR paradox leading to oppugn for quantum mechanics,to the institution of local hidden variable theory,until Bell inequality and Bell theorem without inequalities,we track such path to describe the development of quantum non-locality.As to the promotion and construction of Bell type inequalities, we present a systematic approach to construct non-homogeneous Bell-type inequalities. For Bell theorem without inequalities,we extend such proof for generalize GHZ states and W states.Furthermore,we extend Greenberger-Horn-Zeilinger(GHZ) argument of nonlocality without inequalities to the case of four-qubit mixed states.
The thesis is structured as follows:
Chapter No.1 We introduce all types of correlations,such as,generalized correlation, no-signaling correlation,local correlation,partial-local correlation,and quantum correlation.
Chapter No.2 We give a precise description about local hidden variable(LHV) model.In this chapter,we introduce the EPR disputation,Bohm model,Von Neuman's proof about no dispersion-free state,Gleason theorem in detail.Multi-partite local hidden variable model also defined.
Chapter No.3 In this chapter,we firstly introduce Bell's oppugn about Von Neuman's view.Then,we review the construction of Bell inequality.Several famous Belltype inequalities are also referred,such as,CHSH inequality,MABK inequality,and WWZB inequality etc.The third,we present a method to construct non-homogeneous Bell-type inequalities in detail,which is one of the most important work in my doctoral career.
Chapter No.4 We introduce another kind of proof about non-locality,that is,allversus nothing violation of local realism.Followed such track,we firstly introduce GHZ theorem.Then,Hardy's model is presented in detail.We also introduce Cabello's contribution on all-versus-nothing proof,Finally,we extend all-versus-nothing proof to four-qubit mixed states.
Chapter No.5 We discuss the connection between LHV model and noncommutativity in detail.
Chapter No.6 conclusions.
本论文的主要内容和结构如下:
第一章介绍了各种类型关联的定义,包括广义关联,无信号关联,定域关联,部分定域关联,和量子关联。
第二章详细的介绍了定域隐变量模型的产生过程。对于EPR佯谬,玻姆模型,冯诺依曼对于无弥散态的证明,贝尔对冯诺依曼证明的质疑,以及格里森定理都做了详细的介绍。进一步地,也给出了多体隐变量模型的定义。
第三章首先介绍了贝尔不等式以及CHSH不等式的构造过程,以及它们体现最大违反。随后介绍了墨明所设计的关于不等式的形象描述。最后详细阐述了一种构造非同类不等式的系统方法。
第四章首先关于无不等式证明的GHZ模型,其次给出了关于部分广义GHZ态和W态的无不等式的贝尔证明,最后将无不等式的贝尔证明拓展到了特殊四粒子的纠缠混态。
第五章尝试讨论了定域隐变量理论与非对易性之间的关系。
第六章结论
Quantum non-locality is not only an foundation problem of quantum theory,but also plays an important significance on quantum information processing,such as quantum computation,quantum communication and quantum cryptography security.The main content of this paper is research on LHV theory and quantum non-locality from the viewpoint of correlation.From the EPR paradox leading to oppugn for quantum mechanics,to the institution of local hidden variable theory,until Bell inequality and Bell theorem without inequalities,we track such path to describe the development of quantum non-locality.As to the promotion and construction of Bell type inequalities, we present a systematic approach to construct non-homogeneous Bell-type inequalities. For Bell theorem without inequalities,we extend such proof for generalize GHZ states and W states.Furthermore,we extend Greenberger-Horn-Zeilinger(GHZ) argument of nonlocality without inequalities to the case of four-qubit mixed states.
The thesis is structured as follows:
Chapter No.1 We introduce all types of correlations,such as,generalized correlation, no-signaling correlation,local correlation,partial-local correlation,and quantum correlation.
Chapter No.2 We give a precise description about local hidden variable(LHV) model.In this chapter,we introduce the EPR disputation,Bohm model,Von Neuman's proof about no dispersion-free state,Gleason theorem in detail.Multi-partite local hidden variable model also defined.
Chapter No.3 In this chapter,we firstly introduce Bell's oppugn about Von Neuman's view.Then,we review the construction of Bell inequality.Several famous Belltype inequalities are also referred,such as,CHSH inequality,MABK inequality,and WWZB inequality etc.The third,we present a method to construct non-homogeneous Bell-type inequalities in detail,which is one of the most important work in my doctoral career.
Chapter No.4 We introduce another kind of proof about non-locality,that is,allversus nothing violation of local realism.Followed such track,we firstly introduce GHZ theorem.Then,Hardy's model is presented in detail.We also introduce Cabello's contribution on all-versus-nothing proof,Finally,we extend all-versus-nothing proof to four-qubit mixed states.
Chapter No.5 We discuss the connection between LHV model and noncommutativity in detail.
Chapter No.6 conclusions.
引文
[1]L.Masanes,arXiv:quantph/ 0210073 (2002).
[2]J.Barrett,N.Linden,S.Massar,S.Pironio,S.Popescu,and D.Roberts,Phys.Rev.A 71,022101 (2005).
[3]G.M.Ziegler,Lectures on Polytopes.New York:Springer-Verlag (1995).
[4]N.S.Jones,and L.Masanes,Phys.Rev.A 72,052312 (2005).
[5]N.S.Jones,N.Linden,and S.Massar,Phys.Rev.A 71,042329 (2005).
[6]I.Pitowsky,Quantum Probability-Quantum Logic.Berlin:Springer (1989).
[7]M.P.Seevinck,doctoral thesis:An Inquiry into Quantum and Classical Correlations,(2008).
[8]B.C.van Fraassen,EPR:When is a correlation not a mystery? In P.Lahti and P.Mittelstaedt (Eds.),Symposium on the foundations of modern physics,(pp.113-128).Singapore:World Sci.Publ (1985).
[9]R.F.Werner,and M.M.Wolf,Quant.Inf.Comp.1,no.3,1 (2003).
[10]I.Pitowsky,Math.Programming 50,395 (1991).
[11]D.Collins,N.Gisin,S.Popescu,D.Roberts.and V.Scarani,Phys.Rev.Lett.88,170405 (2002).
[12]G.Svetlichny,Phys.Rev.D 35,3066 (1987).
[13]B.S.Tsirelson,Hadr.J.Suppl.8,329 (1993).
[14]R.F.Werner,and M.M.Wolf,Phys.Rev.A 64,032112 (2001).
[15]L.Masanes,Phys.Rev.Lett.97,050503 (2006).
[16]L.Masanes,,A.Acin,and N.Gisin,Phys.Rev.A 73,012112 (2006).
[17]P.A.M.Dirac,The development of quantum mechanics,Acc.Naz.Lincei 56,Roma (1974).
[18]A.Einstein,B.Podolsky,and N.Rosen,Phys.Rev.Lett.47,777 (1935).
[19]D.Bohm,quantum theory,Dover Publications;New Edition (1989).
[20]J.V.Neumann,mathematical foundations of quantum mechanics,Princeton University Press;Reprint edition (1996).
[21]J.S.Bell,physics (Long Island city,N.Y)1 195 (1964).
[22]J.F.Clauser,M.A.Home,A.Shimony and R.A.Holt,Phys.Rev.Lett.26,880 (1969).
[23]N.D.Mermin,Phys.Rev.Lett.65,1838 (1990);M.Ardehali,Phys.Rev.A 46,5375 (1992);A.V.Belinskii and D.N.Klyshko,Phys.Usp.36,653 (1993).
[24]M.Zukowski and C.Brukner,Phys.Rev.Lett.88,210401 (2002).
[25]S.L.Braunstein,A.Mann,et al.,Phys.Rev.Lett.68,3259 (1992).
[26]W.Tittel,J.Brendel,et al.,Phys.Rev.Lett.81,3563 (1998).
[27]Collins,phys.Rev.Lett 88,170405 (2002).
[28]D.Kaszlikowski,Phys.Rev.A 65,032118 (2002).
[29]M.Seevinck,and G.Svetlichny,Phys.Rev.Lett.89,060401 (2002).
[30]Sliwa,phys.Lett.A 317,165 (2003).
[31]C.Emary,and C.W.J.Beenakker,Phys.Rev.A.69,032317 (2004).
[32]O.Guhne et al.,Phys.Rev.Lett 95,120405 (2005).
[33]Y.C.Liang,and A.C.Doherty,Phys.Rev.A 73,052116 (2006).
[34]J.L.Chen,C.F.Wu,L.C.Kwek,C.H.Oh,and M.L.Ge,Phys.Rev.A 74,032106 (2006).
[35]E.G.Cavalcanti,and C.J.Foster et al.,Phys.Rev.Lett 99,210405 (2007).
[36]S.W.Lee,Phys.Rev.A 76,032108 (2007).
[37]M.Wiesniak,P.Badziag,et al.,Phys.Rev.A 76,012110 (2007).
[38]C.Wildfeuer,A.P.Lund,et al,Phys.Rev.A 76,052101 (2007).
[39]T.Vertesi,Phys.Rev.A 78,032112 (2008).
[40]J.L.Chen,C.W.Wu,L.C.Kwek,and C.H.Oh,Phys.Rev.A 78,032107 (2008).
[41]F.De Zela,Phys.Rev.A 79,022102 (2009).
[42]J.L.Chen and D.L.Deng,Phys.Rev.A 79,012115 (2009).
[43]J.L.Chen and D.L.Deng,Phys.Rev.A 79,012111 (2009).
[44]S.Yu,Z.B.Chen,J.W.Pan,and Y.D.Zhang,Phys.Rev.Lett.90,080401 (2003).
[45]S.Yu,J.W.Pan,Z.B.Chen,Z-B,and Y.D.Zhang,Phys.Rev.Lett.91,217903 (2003).
[46]S.Yu,N.L.Liu,Phys.Rev.Lett.95,150504 (2005).
[47]B.Zeng,D.L.Zhou,P.Zhang,Z.Xu,and L.You,Phys.Rev.A 68,042316 (2003).
[48]C.J.Zhang,Y.S.Zhang,S.Zhang,and G.C.Guo,Phys.Rev.A.76,012334 (2007).
[49]Z.Zhao,T.Yang,Y-A.Chen,A.N.Zhang,M.Zukowski,and J.W.Pan,Phys.Rev.Lett.91,180401 (2003).
[50]M.Zukowski,Found.Phys.36,541 (2006).
[51]N.Gisin,Phys.Lett.A 154,201 (1991).
[52]N.Gisin,and A.Peres,Phys.Lett.A 162,15 (1992).
[53]V.Scarani and N.Gisin,J.Phys.A 34,6043 (2001).
[54]M.Zukowski,C.Brukner,W.Laskowski,and M.Wiesniak,Phys.Rev.Lett.88,210402(2002).
[55]K.Chen,S.Albeverio and S.M.Fei,Phys.Rev.A 74,050101 (R)(2006).
[56]J.-L.Chen,C.F.Wu,L.C.Kwek and C.H.Oh,Phys.Rev.Lett.93,140407 (2004).
[57]G.Toth and O.Gühne,Phys.Rev.Lett.94,060501 (2005);G.Toth,O.Guhne and H.J.Briegel,Phys.Rev.A 73,022303 (2006).
[58]V.Scarani,A.Acfn,E.Schenck and M.Aspelmeyer,Phys.Rev.A 71,042325 (2005).
[59]I.Pitowsky and K.Svozil,Phys.Rev.A 64,014102 (2001).
[60]A.Garg,Phys.Rev.D 28,785 (1983).
[61]A.Garg,and N.D.Mermin,Phys.Rev.D 35,3831(1987).
[62]P.H.Eberhard,Phys.Rev.A 47,R747 (1993).
[63]J.-A.Larsson,Phys.Rev.A 57,3304 (1998).
[64]J.-A.Larsson,Phys.Rev.A 57,R3145 (1998).
[65]J.-A.Larsson,Phys.Rev.A 59,4801 (1999).
[66]J.-A.Larsson,and J.Semitecolos Phys.Rev.A 63,022117 (2001).
[67]S.Massar,and S.Pironio,Phys.Rev.A 68,062109 (2003).
[68]S.Massar,and S.Pironio et al.,Phys.Rev.A 66,052112 (2002).
[69]N.Brunner,Phys.Rev.Lett.98,220403 (2007).
[70]A.Cabello,and J.-A.Larsson,Phys.Rev.Lett.98,220402 (2007).
[71]M.A.Nielsen,and I.L.Chuang,Quantum Computation and Quantum Information (Cambridge University Press,Cambridge,2000).
[72]P.R.Halmos,Finite-dimensional Vector Spaces (Springer,New York,1974).
[73]P.Pearle,Phys.Rev.D 2,1418 (1970).
[74]A.Aspect,P.Grangier,and G.Roger,Phys.Rev.Lett.49,91 (1982);J.G.Rarity and P.R.Tapster,Phys.Rev.Lett.64,2495 (1990);W.Tittel,J.Brendel,H.Zbinden,and N.Gisin,Phys.Rev.Lett.81,3563 (1998);G.Weihs,T.Jennewein,C.Simon,H.Weinfurter,and A.Zeilinger,Phys.Rev.Lett.81,5039 (1998).
[75]A recent experiment carried on trapped ions closed the detection loophole.But in this experiment the ions were separated by a very small distance and the locality loophole was not closed.See M.A.Rowe et al.Nature (London)409,791 (2001).
[76]D.M.Greenberger,M.A.Home,and A.Zeilinger,in Bell's Theorem,Quantum Theory,and Conceptions of the Universe,edited by M.Kafatos (Kluwer,Dordrecht,1989),p.69 (1989).
[77]D.M.Greenberger,M.A.Home,and A.Zeilinger,Am.J.Phys.58,1131 (1990).
[78]W.Dür et al,Phys.Rev.A 62,062314 (2000).
[79]J.W.Pan et al,Nature 403,515 (2000).
[80]C.Y.Lu et al,Nature physcis 3,91 (2007).
[81]L.M.Duan etal,Nature 414,413 (2001).
[82]L.M.Duan,and H.L.Kimble,Phys.Rev.Lett.90,253601.(2003).
[83]A.Karlsson and M.Bourennane,Phys.Rev.A 58,4394 (1998).
[84]B.S.Shi and A.Tomita,Phys.Lett.A 296,161 (2002).
[85]R.Cleve et al.,Phys.Rev.Lett.83,648 (1999).
[86]G.A.Durkin et al,Phys.Rev.Lett.88,187902 (2002).
[87]C.Pagonis,M.L.Redhead,and R.K.Clifton,Phys.Lett.A 155,441 (1991).
[88]J.L.Cereceda,Found.Phys.25,925 (1995).
[89]M.Zukowski and D.Kaszlikowski.Phys.Rev.A 59,3200 (1999).
[90]A.Cabello,Phys.Rev.A 63,022104 (2001).
[91]N.D.Mermin,Ann.N.Y.Acad.Sci.755,616 (1995).
[92]L.Hardy,Phys.Rev.Lett.71,1665 (1993).
[93]A.Cabello,Phys.Rev.A 65.032108 (2002).
[94]S.B.Zheng,Phys.Rev.A.66,014103 (2002).
[95]A.Cabello,Phys.Rev.Lett.97,140406 (2006).
[96]A.Cabello,Phys.Rev.Lett.87,010403 (2001).
[97]A.Cabello,Phys.Rev.Lett.91,230403 (2003).
[98]A.Cabello,Phys.Rev.Lett.86,1911 (2001).
[99]A.Cabello,Phys.Rev.A 67,032107 (2003).
[100]A.Cabello,Phys.Rev.Lett.97,140406 (2006).
[101]A.Cabello,Phys.Rev.Lett.95,210401 (2005).
[102]X.H.Wu,R.H.Xie,Phys.lett.A 211,169 (1996).
[103]X.H.Wu et al.,Phys.lett.A 276,221 (2000).
[104]R.Raussendorf,and H.J.Briegel,Phys.Rev.Lett.86 5188 (2001).
[105]M.Hein,W.Dur,J.Eisert,R.Raussendorf,M.Van Den Nest,and H.J.Briegel,quantph/0602096.
[106]P.Walther et al.,Nature (London)434,169 (2005).
[107]A.-N.Zhang,C.Y.Lu,X.Q.Zhou,Y.A.Chen,Z.Zhao,T.Yang,and J.W.Pan,Phys.Rev.A 73,022330 (2006).
[108]N.Kiesel,C.Schmidt,U.Weber,G.Toth,O.Guhne,R.Ursin,and H.Weinfurter,Phys.Rev.Lett 95,210502 (2005).
[109]G.Toth,O.Guhne,and H.J.Briegel,Phys.Rev.A 73,022033 (2006).
[110]C.Wu,Y.Yeo,L.C.Kwek,and C.H.Oh,Phys.Rev.A 75 032332 (2007).
[111]N.D.Mermin,Physics Today,43,No.6,9 (1990).
[112]V.Scarani,A.Acin,E.Schenck,and M.Aspelmeyer,Phys.Rev.A 71,042325 (2005).
[113]G.C.Ghirardi and L.Marinatto,Phys.Rev.A 73,032102 (2006).
[114]G.C.Ghirardi and L.Marinatto,Phys.Rev.A 74,022101 (2006).
[115]G.C.Ghirardi and L.Marinatto,Phys.Rev.A 74,062107 (2006).
[116]A.Cabello,A.Feito,and A.Lamas-Linares,Phys.Rev.A 72,052112 (2005).
[117]F.A.Bovino,G.Castagnoli,A.Cabello,and A.Lamas-Linares,Phys.Rev.A 73,062110(2006).
[118]Y.Yeo and W.K.Chua,Phys.Rev.Lett 96,060502 (2006).
[119]Josef M.Jauch,Foundations of quantum mechanics,Addison-Wesley Pub.Co.,1968.
[120]Bernard d'Espagnat,Conceptual foundations of quantum mechanics,Benjamin,Advanced Book Program,1976.
[121]Enrico G.Beltrametti and Gianni Cassinelli,The logic of quantum mechanics,Addison-Wesley,Advanced Book Program,1981.
[122]Jeffrey Bub,Interpreting the quantum world,Cambridge University Press,1997.
[123]Karl Svozil,Quantum logic,Singapore;Springer-Verlag,cl998.
[124]M.Luisa Dalla Chiara,R.Giuntini,and R.Greechie,Reasoning in quantum theory,Kluwer Academic Publishers,2004.
[125]A.M.Gleason,Journal of Mathematics and Mechanics 6,885 (1957).
[126]A.Fine,Physical Review Letters,48,291 (1982).
[2]J.Barrett,N.Linden,S.Massar,S.Pironio,S.Popescu,and D.Roberts,Phys.Rev.A 71,022101 (2005).
[3]G.M.Ziegler,Lectures on Polytopes.New York:Springer-Verlag (1995).
[4]N.S.Jones,and L.Masanes,Phys.Rev.A 72,052312 (2005).
[5]N.S.Jones,N.Linden,and S.Massar,Phys.Rev.A 71,042329 (2005).
[6]I.Pitowsky,Quantum Probability-Quantum Logic.Berlin:Springer (1989).
[7]M.P.Seevinck,doctoral thesis:An Inquiry into Quantum and Classical Correlations,(2008).
[8]B.C.van Fraassen,EPR:When is a correlation not a mystery? In P.Lahti and P.Mittelstaedt (Eds.),Symposium on the foundations of modern physics,(pp.113-128).Singapore:World Sci.Publ (1985).
[9]R.F.Werner,and M.M.Wolf,Quant.Inf.Comp.1,no.3,1 (2003).
[10]I.Pitowsky,Math.Programming 50,395 (1991).
[11]D.Collins,N.Gisin,S.Popescu,D.Roberts.and V.Scarani,Phys.Rev.Lett.88,170405 (2002).
[12]G.Svetlichny,Phys.Rev.D 35,3066 (1987).
[13]B.S.Tsirelson,Hadr.J.Suppl.8,329 (1993).
[14]R.F.Werner,and M.M.Wolf,Phys.Rev.A 64,032112 (2001).
[15]L.Masanes,Phys.Rev.Lett.97,050503 (2006).
[16]L.Masanes,,A.Acin,and N.Gisin,Phys.Rev.A 73,012112 (2006).
[17]P.A.M.Dirac,The development of quantum mechanics,Acc.Naz.Lincei 56,Roma (1974).
[18]A.Einstein,B.Podolsky,and N.Rosen,Phys.Rev.Lett.47,777 (1935).
[19]D.Bohm,quantum theory,Dover Publications;New Edition (1989).
[20]J.V.Neumann,mathematical foundations of quantum mechanics,Princeton University Press;Reprint edition (1996).
[21]J.S.Bell,physics (Long Island city,N.Y)1 195 (1964).
[22]J.F.Clauser,M.A.Home,A.Shimony and R.A.Holt,Phys.Rev.Lett.26,880 (1969).
[23]N.D.Mermin,Phys.Rev.Lett.65,1838 (1990);M.Ardehali,Phys.Rev.A 46,5375 (1992);A.V.Belinskii and D.N.Klyshko,Phys.Usp.36,653 (1993).
[24]M.Zukowski and C.Brukner,Phys.Rev.Lett.88,210401 (2002).
[25]S.L.Braunstein,A.Mann,et al.,Phys.Rev.Lett.68,3259 (1992).
[26]W.Tittel,J.Brendel,et al.,Phys.Rev.Lett.81,3563 (1998).
[27]Collins,phys.Rev.Lett 88,170405 (2002).
[28]D.Kaszlikowski,Phys.Rev.A 65,032118 (2002).
[29]M.Seevinck,and G.Svetlichny,Phys.Rev.Lett.89,060401 (2002).
[30]Sliwa,phys.Lett.A 317,165 (2003).
[31]C.Emary,and C.W.J.Beenakker,Phys.Rev.A.69,032317 (2004).
[32]O.Guhne et al.,Phys.Rev.Lett 95,120405 (2005).
[33]Y.C.Liang,and A.C.Doherty,Phys.Rev.A 73,052116 (2006).
[34]J.L.Chen,C.F.Wu,L.C.Kwek,C.H.Oh,and M.L.Ge,Phys.Rev.A 74,032106 (2006).
[35]E.G.Cavalcanti,and C.J.Foster et al.,Phys.Rev.Lett 99,210405 (2007).
[36]S.W.Lee,Phys.Rev.A 76,032108 (2007).
[37]M.Wiesniak,P.Badziag,et al.,Phys.Rev.A 76,012110 (2007).
[38]C.Wildfeuer,A.P.Lund,et al,Phys.Rev.A 76,052101 (2007).
[39]T.Vertesi,Phys.Rev.A 78,032112 (2008).
[40]J.L.Chen,C.W.Wu,L.C.Kwek,and C.H.Oh,Phys.Rev.A 78,032107 (2008).
[41]F.De Zela,Phys.Rev.A 79,022102 (2009).
[42]J.L.Chen and D.L.Deng,Phys.Rev.A 79,012115 (2009).
[43]J.L.Chen and D.L.Deng,Phys.Rev.A 79,012111 (2009).
[44]S.Yu,Z.B.Chen,J.W.Pan,and Y.D.Zhang,Phys.Rev.Lett.90,080401 (2003).
[45]S.Yu,J.W.Pan,Z.B.Chen,Z-B,and Y.D.Zhang,Phys.Rev.Lett.91,217903 (2003).
[46]S.Yu,N.L.Liu,Phys.Rev.Lett.95,150504 (2005).
[47]B.Zeng,D.L.Zhou,P.Zhang,Z.Xu,and L.You,Phys.Rev.A 68,042316 (2003).
[48]C.J.Zhang,Y.S.Zhang,S.Zhang,and G.C.Guo,Phys.Rev.A.76,012334 (2007).
[49]Z.Zhao,T.Yang,Y-A.Chen,A.N.Zhang,M.Zukowski,and J.W.Pan,Phys.Rev.Lett.91,180401 (2003).
[50]M.Zukowski,Found.Phys.36,541 (2006).
[51]N.Gisin,Phys.Lett.A 154,201 (1991).
[52]N.Gisin,and A.Peres,Phys.Lett.A 162,15 (1992).
[53]V.Scarani and N.Gisin,J.Phys.A 34,6043 (2001).
[54]M.Zukowski,C.Brukner,W.Laskowski,and M.Wiesniak,Phys.Rev.Lett.88,210402(2002).
[55]K.Chen,S.Albeverio and S.M.Fei,Phys.Rev.A 74,050101 (R)(2006).
[56]J.-L.Chen,C.F.Wu,L.C.Kwek and C.H.Oh,Phys.Rev.Lett.93,140407 (2004).
[57]G.Toth and O.Gühne,Phys.Rev.Lett.94,060501 (2005);G.Toth,O.Guhne and H.J.Briegel,Phys.Rev.A 73,022303 (2006).
[58]V.Scarani,A.Acfn,E.Schenck and M.Aspelmeyer,Phys.Rev.A 71,042325 (2005).
[59]I.Pitowsky and K.Svozil,Phys.Rev.A 64,014102 (2001).
[60]A.Garg,Phys.Rev.D 28,785 (1983).
[61]A.Garg,and N.D.Mermin,Phys.Rev.D 35,3831(1987).
[62]P.H.Eberhard,Phys.Rev.A 47,R747 (1993).
[63]J.-A.Larsson,Phys.Rev.A 57,3304 (1998).
[64]J.-A.Larsson,Phys.Rev.A 57,R3145 (1998).
[65]J.-A.Larsson,Phys.Rev.A 59,4801 (1999).
[66]J.-A.Larsson,and J.Semitecolos Phys.Rev.A 63,022117 (2001).
[67]S.Massar,and S.Pironio,Phys.Rev.A 68,062109 (2003).
[68]S.Massar,and S.Pironio et al.,Phys.Rev.A 66,052112 (2002).
[69]N.Brunner,Phys.Rev.Lett.98,220403 (2007).
[70]A.Cabello,and J.-A.Larsson,Phys.Rev.Lett.98,220402 (2007).
[71]M.A.Nielsen,and I.L.Chuang,Quantum Computation and Quantum Information (Cambridge University Press,Cambridge,2000).
[72]P.R.Halmos,Finite-dimensional Vector Spaces (Springer,New York,1974).
[73]P.Pearle,Phys.Rev.D 2,1418 (1970).
[74]A.Aspect,P.Grangier,and G.Roger,Phys.Rev.Lett.49,91 (1982);J.G.Rarity and P.R.Tapster,Phys.Rev.Lett.64,2495 (1990);W.Tittel,J.Brendel,H.Zbinden,and N.Gisin,Phys.Rev.Lett.81,3563 (1998);G.Weihs,T.Jennewein,C.Simon,H.Weinfurter,and A.Zeilinger,Phys.Rev.Lett.81,5039 (1998).
[75]A recent experiment carried on trapped ions closed the detection loophole.But in this experiment the ions were separated by a very small distance and the locality loophole was not closed.See M.A.Rowe et al.Nature (London)409,791 (2001).
[76]D.M.Greenberger,M.A.Home,and A.Zeilinger,in Bell's Theorem,Quantum Theory,and Conceptions of the Universe,edited by M.Kafatos (Kluwer,Dordrecht,1989),p.69 (1989).
[77]D.M.Greenberger,M.A.Home,and A.Zeilinger,Am.J.Phys.58,1131 (1990).
[78]W.Dür et al,Phys.Rev.A 62,062314 (2000).
[79]J.W.Pan et al,Nature 403,515 (2000).
[80]C.Y.Lu et al,Nature physcis 3,91 (2007).
[81]L.M.Duan etal,Nature 414,413 (2001).
[82]L.M.Duan,and H.L.Kimble,Phys.Rev.Lett.90,253601.(2003).
[83]A.Karlsson and M.Bourennane,Phys.Rev.A 58,4394 (1998).
[84]B.S.Shi and A.Tomita,Phys.Lett.A 296,161 (2002).
[85]R.Cleve et al.,Phys.Rev.Lett.83,648 (1999).
[86]G.A.Durkin et al,Phys.Rev.Lett.88,187902 (2002).
[87]C.Pagonis,M.L.Redhead,and R.K.Clifton,Phys.Lett.A 155,441 (1991).
[88]J.L.Cereceda,Found.Phys.25,925 (1995).
[89]M.Zukowski and D.Kaszlikowski.Phys.Rev.A 59,3200 (1999).
[90]A.Cabello,Phys.Rev.A 63,022104 (2001).
[91]N.D.Mermin,Ann.N.Y.Acad.Sci.755,616 (1995).
[92]L.Hardy,Phys.Rev.Lett.71,1665 (1993).
[93]A.Cabello,Phys.Rev.A 65.032108 (2002).
[94]S.B.Zheng,Phys.Rev.A.66,014103 (2002).
[95]A.Cabello,Phys.Rev.Lett.97,140406 (2006).
[96]A.Cabello,Phys.Rev.Lett.87,010403 (2001).
[97]A.Cabello,Phys.Rev.Lett.91,230403 (2003).
[98]A.Cabello,Phys.Rev.Lett.86,1911 (2001).
[99]A.Cabello,Phys.Rev.A 67,032107 (2003).
[100]A.Cabello,Phys.Rev.Lett.97,140406 (2006).
[101]A.Cabello,Phys.Rev.Lett.95,210401 (2005).
[102]X.H.Wu,R.H.Xie,Phys.lett.A 211,169 (1996).
[103]X.H.Wu et al.,Phys.lett.A 276,221 (2000).
[104]R.Raussendorf,and H.J.Briegel,Phys.Rev.Lett.86 5188 (2001).
[105]M.Hein,W.Dur,J.Eisert,R.Raussendorf,M.Van Den Nest,and H.J.Briegel,quantph/0602096.
[106]P.Walther et al.,Nature (London)434,169 (2005).
[107]A.-N.Zhang,C.Y.Lu,X.Q.Zhou,Y.A.Chen,Z.Zhao,T.Yang,and J.W.Pan,Phys.Rev.A 73,022330 (2006).
[108]N.Kiesel,C.Schmidt,U.Weber,G.Toth,O.Guhne,R.Ursin,and H.Weinfurter,Phys.Rev.Lett 95,210502 (2005).
[109]G.Toth,O.Guhne,and H.J.Briegel,Phys.Rev.A 73,022033 (2006).
[110]C.Wu,Y.Yeo,L.C.Kwek,and C.H.Oh,Phys.Rev.A 75 032332 (2007).
[111]N.D.Mermin,Physics Today,43,No.6,9 (1990).
[112]V.Scarani,A.Acin,E.Schenck,and M.Aspelmeyer,Phys.Rev.A 71,042325 (2005).
[113]G.C.Ghirardi and L.Marinatto,Phys.Rev.A 73,032102 (2006).
[114]G.C.Ghirardi and L.Marinatto,Phys.Rev.A 74,022101 (2006).
[115]G.C.Ghirardi and L.Marinatto,Phys.Rev.A 74,062107 (2006).
[116]A.Cabello,A.Feito,and A.Lamas-Linares,Phys.Rev.A 72,052112 (2005).
[117]F.A.Bovino,G.Castagnoli,A.Cabello,and A.Lamas-Linares,Phys.Rev.A 73,062110(2006).
[118]Y.Yeo and W.K.Chua,Phys.Rev.Lett 96,060502 (2006).
[119]Josef M.Jauch,Foundations of quantum mechanics,Addison-Wesley Pub.Co.,1968.
[120]Bernard d'Espagnat,Conceptual foundations of quantum mechanics,Benjamin,Advanced Book Program,1976.
[121]Enrico G.Beltrametti and Gianni Cassinelli,The logic of quantum mechanics,Addison-Wesley,Advanced Book Program,1981.
[122]Jeffrey Bub,Interpreting the quantum world,Cambridge University Press,1997.
[123]Karl Svozil,Quantum logic,Singapore;Springer-Verlag,cl998.
[124]M.Luisa Dalla Chiara,R.Giuntini,and R.Greechie,Reasoning in quantum theory,Kluwer Academic Publishers,2004.
[125]A.M.Gleason,Journal of Mathematics and Mechanics 6,885 (1957).
[126]A.Fine,Physical Review Letters,48,291 (1982).