连续变量纠缠态下CHSH不等式破坏的考察
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摘要
在量子力学提出之后,Einstein和Podolsky、Rosen在1935年发表了一篇重要的文章来说明量子力学是不完备的,这就是我们通常所说的EPR佯谬。它导致了后来Bohm提出了隐变量理论,Bell基于EPR思想以及隐变量理论沿用了经典的统计算法定量的分析了隐变量理论是否能够再现量子力学的结论,这就是我们所知的Bell不等式。
自从Bell提出了用不等式这种形式来表达定域性隐变量理论的形式之后,EPR思想得以在实验中实现。但是人们做过的许多实验中只有极少的一部分是符合Bell不等式的,大部分的实验结果是与量子力学相符合的。到后来,Bell不等式又被人们推广出了多种形式,比如后来的GHZ定理,Hardy不等式,Cabello不等式等。最初的一个就是CHSH不等式。Bell不等式和EPR佯谬一起开辟了一条验证量子力学的空间非局域性的实验研究路线,证明了量子力学的空间非局域性质。
本文在简单回顾了量子力学的背景以及Bell不等式和CHSH不等式的提出和主要相关理论,选择了连续参量的两粒子纠缠态来对CHSH不等式进行检验。之前由于有很多证明了Bell不等式是不正确的量子态大多数是离散变量的,因此我们在这里选用连续变量。连续变量系统下的纠缠态在量子信息处理方面具有很大优势,无论是理论还是实验都是有很重要的意义的。并且由于CHSH不等式比Bell不等式更加具有对称性,这将会使得我们的计算相对比较方便。我们在此要观察CHSH不等式的最大破坏度随着连续参量变化而变化的趋势。然后在这个连续参量的两粒子纠缠态中加入噪声,通常情况下噪声可以看作是几种翻转误差的叠加,因此我们在本文中就考虑了以下三种翻转误差：比特位翻转误差,相位翻转误差以及比特位-相位翻转误差。我们计算了加入了噪声之后的连续参量两粒子纠缠态的纠缠度的变化,以及加入噪声之后CHSH不等式最大破坏度随着连续参量和加入噪声概率的这两个参数的变化而变化的趋势。
After Quantum Mechanics was founded, Einstein、Podolsky and Rosen has published an important article to declare that Quantum Mechanics was not self-contained in 1935, which is EPR Paradox as we know. It leads to the establishment of hidden variable theory. Bell quantitatively analysed whether hidden variable theory can represent the conclusions of Quantum Mechanics based on EPR Paradox and hidden variable theory by using classical statistics algorithm, this is Bell inequality as we know.
The EPR Paradox has come ture since Bell has put forward express local hidden variable theory by using inequalities. But most of the outcome of the large amount of experiments was consilient to Quantum Mechanics, only a very small part of the experiments was consilient to Bell inequality. Bell inequality was generalized to many different forms by people afterwards. For example, GHZ Theory, Hardy inequality, Cabello inequality etc. The most original one is CHSH (Clauser-Horne-Shimony-Holt) inequality. Bell inequality open up an experiment research way to confirming the non-local attribute of Quantum Mechanics together with EPR Paradox.
In this thesis, we retrospect the background and main relevance theory of Quantum Mechanics and Bell inequality and CHSH inequality. We choose the two particals entangled states with continuous variable to do the research on CHSH inequality. Since most of the research which demonstrate the Bell inequality was wrong use the dissociation variable quantum state. Here we use continuous variable quantum state to do our research. Entangled states with continuous variable has tremendous advantages in dealing with the quantum information. It has a very important purport in whether the theoretical or the experimental field. Since CHSH inequality was more symmetrical than Bell inequality, this will make our calculation more convenience comparatively. Here in this thesis we should observe the variational trend of the maximum violation of CHSH inequality which is depend on the continuous variable. Then we will add three kinds of noise into the two particals entangled states with continuous variable. Generally speaking, noise should be treat as a composition of several kinds of overturned errors. As a matter of convenience, here we use three kinds of overturned errors:bit overturned errors, phase overturned errors and bit-phase overturned errors. We will observe the variational of the two particals entangled states with continuous variable which was added in noise, we will also calculate the variational of the maximum violation of CHSH inequality after we add noise in which will depend on continuous variable and the probability of the noise was added.
自从Bell提出了用不等式这种形式来表达定域性隐变量理论的形式之后,EPR思想得以在实验中实现。但是人们做过的许多实验中只有极少的一部分是符合Bell不等式的,大部分的实验结果是与量子力学相符合的。到后来,Bell不等式又被人们推广出了多种形式,比如后来的GHZ定理,Hardy不等式,Cabello不等式等。最初的一个就是CHSH不等式。Bell不等式和EPR佯谬一起开辟了一条验证量子力学的空间非局域性的实验研究路线,证明了量子力学的空间非局域性质。
本文在简单回顾了量子力学的背景以及Bell不等式和CHSH不等式的提出和主要相关理论,选择了连续参量的两粒子纠缠态来对CHSH不等式进行检验。之前由于有很多证明了Bell不等式是不正确的量子态大多数是离散变量的,因此我们在这里选用连续变量。连续变量系统下的纠缠态在量子信息处理方面具有很大优势,无论是理论还是实验都是有很重要的意义的。并且由于CHSH不等式比Bell不等式更加具有对称性,这将会使得我们的计算相对比较方便。我们在此要观察CHSH不等式的最大破坏度随着连续参量变化而变化的趋势。然后在这个连续参量的两粒子纠缠态中加入噪声,通常情况下噪声可以看作是几种翻转误差的叠加,因此我们在本文中就考虑了以下三种翻转误差：比特位翻转误差,相位翻转误差以及比特位-相位翻转误差。我们计算了加入了噪声之后的连续参量两粒子纠缠态的纠缠度的变化,以及加入噪声之后CHSH不等式最大破坏度随着连续参量和加入噪声概率的这两个参数的变化而变化的趋势。
After Quantum Mechanics was founded, Einstein、Podolsky and Rosen has published an important article to declare that Quantum Mechanics was not self-contained in 1935, which is EPR Paradox as we know. It leads to the establishment of hidden variable theory. Bell quantitatively analysed whether hidden variable theory can represent the conclusions of Quantum Mechanics based on EPR Paradox and hidden variable theory by using classical statistics algorithm, this is Bell inequality as we know.
The EPR Paradox has come ture since Bell has put forward express local hidden variable theory by using inequalities. But most of the outcome of the large amount of experiments was consilient to Quantum Mechanics, only a very small part of the experiments was consilient to Bell inequality. Bell inequality was generalized to many different forms by people afterwards. For example, GHZ Theory, Hardy inequality, Cabello inequality etc. The most original one is CHSH (Clauser-Horne-Shimony-Holt) inequality. Bell inequality open up an experiment research way to confirming the non-local attribute of Quantum Mechanics together with EPR Paradox.
In this thesis, we retrospect the background and main relevance theory of Quantum Mechanics and Bell inequality and CHSH inequality. We choose the two particals entangled states with continuous variable to do the research on CHSH inequality. Since most of the research which demonstrate the Bell inequality was wrong use the dissociation variable quantum state. Here we use continuous variable quantum state to do our research. Entangled states with continuous variable has tremendous advantages in dealing with the quantum information. It has a very important purport in whether the theoretical or the experimental field. Since CHSH inequality was more symmetrical than Bell inequality, this will make our calculation more convenience comparatively. Here in this thesis we should observe the variational trend of the maximum violation of CHSH inequality which is depend on the continuous variable. Then we will add three kinds of noise into the two particals entangled states with continuous variable. Generally speaking, noise should be treat as a composition of several kinds of overturned errors. As a matter of convenience, here we use three kinds of overturned errors:bit overturned errors, phase overturned errors and bit-phase overturned errors. We will observe the variational of the two particals entangled states with continuous variable which was added in noise, we will also calculate the variational of the maximum violation of CHSH inequality after we add noise in which will depend on continuous variable and the probability of the noise was added.
引文
[1]Koji Nagata and Jaewook Ahn, Violation of rotational invariance of local realistic models with two settings, January 11,2008.
[2]A. Eistein, B.podolsky and N. Rosen, Phys. Rev.47,77(1935).
[3]J.S. Bell, Physics (LongIsland City, N. Y.) 1,190(1964).
[4]A. Fine, Phys. Rev. Lett.48,291 (1982); A. Fine, J. Math. Phys.23,1306 (1982).
[5]J.-W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, and A. Zeilinger, Nature (London) 403,515 (2000).
[6]Aharonov, D.,1999, e-print arXiv:quant-ph/9910081.
[7]张永德,量子信息物理原理,北京：科学出版社(2006).
[8]K. Nagata, J. Phys. A:Math. Theor.40,13101 (2007).
[9]K. Nagata, W. Laskowski, M. Wie'sniak, and M. Z'ukowski, Phys. Rev. Lett. 93,230403 (2004).
[10]K. Nagata, e-print, arXiv:0710.1935.
[11]A. Peres, Quantum Theory:Concepts and Methods(Kluwer Academic, Dordrecht, The Netherlands,1993).
[12]R. F. Werner, Phys. Rev. A 40,4277 (1989).
[13]Ryszard Horodecki, Pawet Horodecki, Michal Horodecki, Karol Horodecki, Quantum entanglement, Reviews of Modern Physics,VOL 81, June,2009
[14]Adesso, G., and F. Illuminati,2005, Phys. Rev. A 72,032334.
[15]Agarwal, G. S., and A. Biswas,2005, J. Opt. B:Quantum Semiclassical Opt.7,350.
[16]曾谨言,龙桂鲁,裴寿镛,量子力学新进展(第三辑),北京：清华大学出版社,(2003).
[17]Bennett, C. H., H. J. Bernstein, S. Popescu, and B. Schumacher,1996, Phys. Rev. A 53,2046.
[18]Cabrillo, C., J. Cirac, P. Garcia-Fernandez, and P. Zoller,1999, Phys. Rev. A 59,1025.
[19]Emary, C., B. Trauzettel, and C. W. J. Beenakker,2005, Phys. Rev. Lett. 95,127401.
[20]M. Redhead, Incompleteness, Nonlocality, and Realism, (Clarendon Press, Oxford,1989),2nd ed.
[21]Fukuda, M., and M. M. Wolf,2007, J. Math. Phys.48,072101.
[22]Greenberger, D., M. Horne, and A. Zeilinger,2005a, e-printarXiv:quant-ph/0510207.
[23]Paskauskas, R., and L. You,2001, Phys. Rev. A 64,042310.
[24]Samsonowicz, J., M. Kus, and M. Lewenstein,2007, Phys. Rev. A 76,022314.
[25]Hayden, P. M., M. Horodecki, and B. M. Terhal,2001, J. Phys. A 34,6891.
[26]Nielsen, M. A.,2002, Phys. Lett. A 303,249.
[27]Torgerson, J. R., D. Branning, C. H. Monken, and L. Mandel,1995, Phys. Lett. A 204,323.
[28]Hardy, L.,1993, Phys. Rev. Lett.71,1665.
[29]Vedral, V., and M. B. Plenio,1998, Phys. Rev. A 57,1619.
[30]Peres, A.,1996a, Phys. Rev. A 54,2685.
[31]Grover, L. K.,1997, e-print arXiv:quant-ph/9704012.
[32]Verstraete, F., K. Audenaert, J. Dehaene, and B. D. Moor,2001, J. Phys. A 34,10327.
[33]Smith, G., and J. Yard,2008, Science 321,1812.
[34]Majewski, A. W.,2002, J. Phys. A 35,123.
[35]Werner, R. F., and M. M. Wolf,2001a, Phys. Rev. A 64,032112.
[36]张永德,《量子物理信息原理》；科学出版社,(2005).
[37]Jennewein, T., C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger,2000, Phys. Rev. Lett.84,4729.
[38]Schrodinger, E.,1935, Naturwiss.23,807.
[39]Tamaki, K., M. Koashi, and N. Imoto,2003, Phys. Rev. Lett.90,167904.
[40]Hillery, M., V. Buzek, and A. Berthiaume,1999, Phys. Rev. A59,1829.
[41]Yurke, B., and D. Stoler,1992a, Phys. Rev. A 46,2229.
[42]Moroder, T., M. Curty, and N. Lutkenhaus,2006b, Phys. Rev. A 73,012311.
[43]Zurek, W. H.,2003, e-print arXiv:quant-ph/0306072.
[44]Mintert, F., A. R. R. Carvalho, M. Kus', and A. Buchleitner,2005, Phys. Rep.415,207.
[2]A. Eistein, B.podolsky and N. Rosen, Phys. Rev.47,77(1935).
[3]J.S. Bell, Physics (LongIsland City, N. Y.) 1,190(1964).
[4]A. Fine, Phys. Rev. Lett.48,291 (1982); A. Fine, J. Math. Phys.23,1306 (1982).
[5]J.-W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, and A. Zeilinger, Nature (London) 403,515 (2000).
[6]Aharonov, D.,1999, e-print arXiv:quant-ph/9910081.
[7]张永德,量子信息物理原理,北京：科学出版社(2006).
[8]K. Nagata, J. Phys. A:Math. Theor.40,13101 (2007).
[9]K. Nagata, W. Laskowski, M. Wie'sniak, and M. Z'ukowski, Phys. Rev. Lett. 93,230403 (2004).
[10]K. Nagata, e-print, arXiv:0710.1935.
[11]A. Peres, Quantum Theory:Concepts and Methods(Kluwer Academic, Dordrecht, The Netherlands,1993).
[12]R. F. Werner, Phys. Rev. A 40,4277 (1989).
[13]Ryszard Horodecki, Pawet Horodecki, Michal Horodecki, Karol Horodecki, Quantum entanglement, Reviews of Modern Physics,VOL 81, June,2009
[14]Adesso, G., and F. Illuminati,2005, Phys. Rev. A 72,032334.
[15]Agarwal, G. S., and A. Biswas,2005, J. Opt. B:Quantum Semiclassical Opt.7,350.
[16]曾谨言,龙桂鲁,裴寿镛,量子力学新进展(第三辑),北京：清华大学出版社,(2003).
[17]Bennett, C. H., H. J. Bernstein, S. Popescu, and B. Schumacher,1996, Phys. Rev. A 53,2046.
[18]Cabrillo, C., J. Cirac, P. Garcia-Fernandez, and P. Zoller,1999, Phys. Rev. A 59,1025.
[19]Emary, C., B. Trauzettel, and C. W. J. Beenakker,2005, Phys. Rev. Lett. 95,127401.
[20]M. Redhead, Incompleteness, Nonlocality, and Realism, (Clarendon Press, Oxford,1989),2nd ed.
[21]Fukuda, M., and M. M. Wolf,2007, J. Math. Phys.48,072101.
[22]Greenberger, D., M. Horne, and A. Zeilinger,2005a, e-printarXiv:quant-ph/0510207.
[23]Paskauskas, R., and L. You,2001, Phys. Rev. A 64,042310.
[24]Samsonowicz, J., M. Kus, and M. Lewenstein,2007, Phys. Rev. A 76,022314.
[25]Hayden, P. M., M. Horodecki, and B. M. Terhal,2001, J. Phys. A 34,6891.
[26]Nielsen, M. A.,2002, Phys. Lett. A 303,249.
[27]Torgerson, J. R., D. Branning, C. H. Monken, and L. Mandel,1995, Phys. Lett. A 204,323.
[28]Hardy, L.,1993, Phys. Rev. Lett.71,1665.
[29]Vedral, V., and M. B. Plenio,1998, Phys. Rev. A 57,1619.
[30]Peres, A.,1996a, Phys. Rev. A 54,2685.
[31]Grover, L. K.,1997, e-print arXiv:quant-ph/9704012.
[32]Verstraete, F., K. Audenaert, J. Dehaene, and B. D. Moor,2001, J. Phys. A 34,10327.
[33]Smith, G., and J. Yard,2008, Science 321,1812.
[34]Majewski, A. W.,2002, J. Phys. A 35,123.
[35]Werner, R. F., and M. M. Wolf,2001a, Phys. Rev. A 64,032112.
[36]张永德,《量子物理信息原理》；科学出版社,(2005).
[37]Jennewein, T., C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger,2000, Phys. Rev. Lett.84,4729.
[38]Schrodinger, E.,1935, Naturwiss.23,807.
[39]Tamaki, K., M. Koashi, and N. Imoto,2003, Phys. Rev. Lett.90,167904.
[40]Hillery, M., V. Buzek, and A. Berthiaume,1999, Phys. Rev. A59,1829.
[41]Yurke, B., and D. Stoler,1992a, Phys. Rev. A 46,2229.
[42]Moroder, T., M. Curty, and N. Lutkenhaus,2006b, Phys. Rev. A 73,012311.
[43]Zurek, W. H.,2003, e-print arXiv:quant-ph/0306072.
[44]Mintert, F., A. R. R. Carvalho, M. Kus', and A. Buchleitner,2005, Phys. Rep.415,207.