量子纠错码以及量子非定域性的相关理论研究
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摘要
量子理论描述了微观尺度下体系的运动规律。虽然最早在上世纪初其框架已经建立了,但即使到今天,其中还留有很多值得研究的没有彻底解决的课题,比如多体系统的非定域性等。此外,将量子理论与信息处理过程和计算结合,给了我们一个新的角度来看待信息论以及其本质,也导致了我们生活中很多新词汇和新的意思的产生,比如“量子信息”,“量子比特”,“纠缠”等等。
本论文主要是关于量子纠错码和量子非定域性这两个方向相关一些细节理论的研究：高维图态纠错码和多粒子系统的GHZ佯谬。
在前一个课题中,我们由高维图态出发,构造出了相应的纠错码。首先我们介绍了带权重图的coding clique的概念,并且证明其是如何与稳定子码和非稳定子码的构造相关的。通过此构造方法,我们再通过计算机搜索得出很多码。最后我们依据这些搜索出的码来解析构造出4类奇数维图态的达量子Singleton界的稳定子码和一类非稳定子码。
在后一个课题中,我们主要研究了高维图态的GHZ佯谬和多个测量方向的GHZ佯谬。首先我们简单回顾了量子非定域性和互文的概念和由来。然后我们由一类特殊的图—GHZ图出发,研究了图态的任意多个(大于等于3)粒子的GHZ佯谬。其次,我们从一个图的GHZ佯谬出发,导出了一个关于每个观测者有两个d-测量结果的可观测量的Bell不等式,其最大违背可由相应的图态得出。此外还发现了一个与具体态无关的一个可检验量子互文的Kochen-Specker不等式。最后我们研究了多粒子情况下多个测量方向的GHZ佯谬,并且也给出了一个实验上可以检验的Bell不等式。
Quantum theory is essential for us to understand the behaviors of the micro-scopic scales. The earliest version of it has been formulated in the first decade of last century, but in nowadays many interesting topics in it remain unsettled, such as nonlocality of many-partite systems. Moreover, quantum mechanics led us to reconsider the scope of information processing and computation and its funda-mental principles which brought many new words or new meanings to our lifestyle such as'quantum information','qubit','entanglement'et.al.
In this thesis, we mainly focus on some branches of two topics:nonbina-ry graphical error-correcting codes in quantum error-correcting codes and multi-partite Greenberger-Horne-Zeilinger(GHZ) paradoxes in quantum nonlocality the-ory.
In our first topic, we construct quantum error-correcting codes from nonbina-ry graph states. First, we introduce the concept of a coding clique for a weighted graph and show how it is related to the construction of both stabilizer and non-additive QECCs. Then we present some codes found via computer searches. At last we construct analytically four families of optimal stabilizer codes that satu-rate the quantum Singleton bound for any odd dimension as well as a family of nonadditive codes.
In our second topic, we study GHZ paradoxes from qudit graph states and multi-setting GHZ paradoxes. First, we review the concepts of quantum nonlocal-ity and quantum contextuality. Then we study GHZ paradoxes for an arbitrary number (greater than3) of particles with the help of qudit graph states on a spe-cial kind of graphs, called GHZ graphs. Furthermore, based on the GHZ paradox arising from a GHZ graph, we derive a Bell inequality with two d-outcome observ-ables for each observer, whose maximal violation attained by the corresponding graph state, and a Kochen-Specker inequality testing the quantum contextuali-ty in a state-independent fashion. In the end we investigate multi-settings GHZ paradoxes for multi-particles and in qubit case we give an experimental testable Bell inequality.
本论文主要是关于量子纠错码和量子非定域性这两个方向相关一些细节理论的研究：高维图态纠错码和多粒子系统的GHZ佯谬。
在前一个课题中,我们由高维图态出发,构造出了相应的纠错码。首先我们介绍了带权重图的coding clique的概念,并且证明其是如何与稳定子码和非稳定子码的构造相关的。通过此构造方法,我们再通过计算机搜索得出很多码。最后我们依据这些搜索出的码来解析构造出4类奇数维图态的达量子Singleton界的稳定子码和一类非稳定子码。
在后一个课题中,我们主要研究了高维图态的GHZ佯谬和多个测量方向的GHZ佯谬。首先我们简单回顾了量子非定域性和互文的概念和由来。然后我们由一类特殊的图—GHZ图出发,研究了图态的任意多个(大于等于3)粒子的GHZ佯谬。其次,我们从一个图的GHZ佯谬出发,导出了一个关于每个观测者有两个d-测量结果的可观测量的Bell不等式,其最大违背可由相应的图态得出。此外还发现了一个与具体态无关的一个可检验量子互文的Kochen-Specker不等式。最后我们研究了多粒子情况下多个测量方向的GHZ佯谬,并且也给出了一个实验上可以检验的Bell不等式。
Quantum theory is essential for us to understand the behaviors of the micro-scopic scales. The earliest version of it has been formulated in the first decade of last century, but in nowadays many interesting topics in it remain unsettled, such as nonlocality of many-partite systems. Moreover, quantum mechanics led us to reconsider the scope of information processing and computation and its funda-mental principles which brought many new words or new meanings to our lifestyle such as'quantum information','qubit','entanglement'et.al.
In this thesis, we mainly focus on some branches of two topics:nonbina-ry graphical error-correcting codes in quantum error-correcting codes and multi-partite Greenberger-Horne-Zeilinger(GHZ) paradoxes in quantum nonlocality the-ory.
In our first topic, we construct quantum error-correcting codes from nonbina-ry graph states. First, we introduce the concept of a coding clique for a weighted graph and show how it is related to the construction of both stabilizer and non-additive QECCs. Then we present some codes found via computer searches. At last we construct analytically four families of optimal stabilizer codes that satu-rate the quantum Singleton bound for any odd dimension as well as a family of nonadditive codes.
In our second topic, we study GHZ paradoxes from qudit graph states and multi-setting GHZ paradoxes. First, we review the concepts of quantum nonlocal-ity and quantum contextuality. Then we study GHZ paradoxes for an arbitrary number (greater than3) of particles with the help of qudit graph states on a spe-cial kind of graphs, called GHZ graphs. Furthermore, based on the GHZ paradox arising from a GHZ graph, we derive a Bell inequality with two d-outcome observ-ables for each observer, whose maximal violation attained by the corresponding graph state, and a Kochen-Specker inequality testing the quantum contextuali-ty in a state-independent fashion. In the end we investigate multi-settings GHZ paradoxes for multi-particles and in qubit case we give an experimental testable Bell inequality.
引文
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[41]D. L. Zhou, B. Zeng, Z. Xu, and C. P. Sun, Phys. Rev. A 68,062303 (2003).
[42]M. Hein, W. Dur, J. Eisert, R. Raussendorf, M. Van den Nest, and H. J.Briegel, arXive:quant-ph/0602096.
[43]S. Y. Looi, L. Yu, V. Gheorghiu, R. B. Griffiths, arXiv:0712.1979 [quant-ph].
[44]S. Niskanen and P.R.J. Ostergard, cliquer User's Guide Version 1.0, (Com-munications Laboratory, Helsinki University of Technology, Espoo, Finland, Tech. Rep. T 48,2003).
[45]A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev.47,777 (1935).
[46]J. S. Bell, Physics 1,195 (1964).
[47]Clauser J F,Horne et al, Phys. Rev. Lett.23,880(1969).
[48]D.M. Greenberger, M.A. Home, and A. Zeilinger, in Bell's Theorem, Quan-tum Theory, and Conceptions of the Universe, edited by M. Kafatos (Kluw-er, Dordrecht,1989), p.69; arXiv:0712.0921.
[49]D.M. Greenberger, M.A. Horne, A. Shimony, and A. Zeilinger, Am. J. Phys. 58,1131 (1990).
[50]N.D. Mermin, Phys. Rev. Lett.65,3373 (1990).
[51]N.D. Mermin, Rev. Mod. Phys.65,803 (1993).
[52]S. Kochen and E.P. Specker, J. Math. Mech.17,59 (1967).
[53]A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett.47,460 (1981); R. Lapkiewicz et al., Nature (London) 474,490 (2011); C. Zu, et al., Phys. Rev. Lett.109,150401 (2012).
[54]A. Cabello, Phys. Rev. Lett.101,210401 (2008).
[55]S. Yu and C.H. Oh, Phys. Rev. Lett.108,030402 (2012).
[56]D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett.82,1345 (1999); J.-W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, and A. Zeilinger, Nature(London) 403,515 (2000).
[57]R. Cleve and H. Buhrman, Phys. Rev. A 56,1201 (1997)
[58]M. Zukowski et al., Acta Phys. Pol. A 93,187 (1998)
[59]S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, New J. Phys.8,75 (2006).
[60]A. Cabello, Phys. Rev. A 63,022104 (2001).
[61]A. Cabello et al., Phys. Lett. A 212,183 (1996).
[62]A. A. Klyachko, M. A. Can, S. Binicioglu, and A. S. Shumovsky, Phys. Rev. Lett.101,20403(2008).
[63]C. Pagonis, M.L.G. Redhead, and R.K. Clifton, Phys. Lett. A 155,441 (1991).
[64]N.J. Cerf, S. Massar, and S. Pironio, Phys. Rev. Lett.89,080402 (2002).
[65]J. Lee, S.-W. Lee, and M.S. Kim, Phys. Rev. A 73,032316 (2006).
[66]D. Kaszlikowski and M. Zukowski, Phys. Rev. A 66,042107 (2002).
[67]D.P. DiVincenzo and A. Peres, Phys. Rev. A 55,4089 (1997).
[68]D. Hu, W. Tang, M. Zhao, Q. Chen, S. Yu, and C.H. Oh, Phys. Rev. A 78, 012306 (2008).
[69]W. Son, J. Lee, and M.S. Kim, Phys. Rev. Lett.96,060406 (2006).
[70]Weidong Tang, Sixia Yu, and C.H. Oh, arXiv:1206.2718.
[2]P.W. Shor, Phys. Rev. A 52, R2493 (1995).
[3]C.H. Bennettt, D.P. DiVincenco, J.A. Smolin, and W.K. Wootters, Phys. Rev. A 54,3824 (1996).
[4]E. Knill and R. Laflamme, Phys. Rev. A 55,900 (1997).
[5]A.Steane, Phys. Rev. Lett.77,793(1996).
[6]S. Glancy, E. Knill, and H. M. Vasconcelos, Phys. Rev. A 74,032319 (2006).
[7]A. Ambainis and D. Gottesman, IEEE Trans. Inf. Theory.52,748 (2006).
[8]A.Yu. Kitaev, Ann. Phys. (N.Y.) 303,2 (2003).
[9]X.-G. Wen, Adv. Phys.44,405 (1995).
[10]X.-G. Wen and Q. Niu, Phys. Rev. B 41,9377 (1990).
[11]E. Knill, R. Laflamme, and W. H. Zurek, Science 279,342 (1998); D. Gottes-man, Phys. Rev. A 57,127 (1998).
[12]C.H. Bennett and G. Brassard, Proceedings of IEEE International Con-ference on Computers, Systems, and Signal Processing,175 (1984); A. K. Ekert, Phys. Rev. Lett.67,661 (1991).
[13]D. Gottesman, Phys. Rev. A 54,1862 (1996).
[14]A.R. Calderbank, E.M. Rains, P.W. Shor, and N.J.A. Sloane, Phys. Rev. Lett.78,405 (1997).
[15]A.R. Calderbank, E.M. Rains, P.W. Shor, and N.J.A. Sloane, IEEE Trans. Inf. Theory 44,1369 (1998).
[16]Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quan-tum Information, Cambridge University Press (2000).
[17]John. Preskill Notes on Quantum Computation, Online available at http://www.theory.caltech.edu/-preskill/ph219/index.html#lecture.
[18]E. Knill, Los Alamos National Laboratory Report LAUR-96-2717,1996.
[19]E. Rains, IEEE Trans. Inf. Theory 45,1827 (1999).
[20]A. Ashikhmin and E. Knill, IEEE Trans. Inf. Theory 47,3065 (2001).
[21]M. Grassl and T. Beth, arXive:quant-ph/0312164v1.
[22]A. Ketkar, A. Klappenecker, S. Kumar and P. K. Sarvepalli, IEEE Trans. Inf. Theory,52 4892 (2006); arXiv:quant-ph/0508070.
[23]H.F. Chau, Physical Review A,56, R1 (1997); H.F. Chau, Physical Review A,55, R839 (1997).
[24]K. Feng, IEEE Trans. Inf. Theory 48,2384 (2002).
[25]A. R. Calderbank and P.W. Shor, Phys. Rev. A 54,1098 (1996); A.M. Steane, Phys. Rev. A 54,4741 (1996).
[26]H. Bombin and M. A. Martin-Delgado, Phys. Rev. Lett.97,180501 (2006).
[27]T. Brun, I. Devetak, and M.-H. Hsieh, Science 314,436 (2006).
[28]E.M. Rains, R.H. Hardin, P.W. Shor, and J.A. Sloane, Phys. Rev. Lett,79, 953(1997).
[29]E. Rains, IEEE Trans. Inf. Theory 45,266 (1999).
[30]J.A. Smolin, G. Smith and S. Wehner, arXiv:quant-ph/0701065.
[31]S. Yu, Q. Chen, C.H. Lai, and C.H. Oh, Phys. Rev. Lett 101,090501 (2008).
[32]S. Yu, Q. Chen, and C.H. Oh, arXiv:0709.1780 [quant-ph].
[33]A. Cross, G. Smith, J.A. Smolin, and B. Zeng, arXiv:quant-ph/0708.1021.
[34]M. Hein, J. Eisert, and H. J. Briegel, Phys. Rev. A 69,062311 (2004).
[35]D. Schlingemann and R. F. Werner, Phys. Rev. A,65,012308(2001).
[36]R. Raussendorf and H. J. Briegel, Phys. Rev. Lett.86,5188 (2001).
[37]P. Walther, et.al., Nature 434,169 (2005); C.Y. Lu, et.al., Nature Physics 3,91 (2007).
[38]M. Grassl, A. Klappenecker, and M. Rotteler, IEEE Int. Symp. Inform. Theory Proceedings,45 (2002); arXiv:quant-ph/0703112vl.
[39]D. Schlingemann, Quant. Inform. Comput.2,307 (2002); arXive:quant-ph/01111080.
[40]M. Bahramgiri and S. Beigi, arXiv:quant-ph/0610267.
[41]D. L. Zhou, B. Zeng, Z. Xu, and C. P. Sun, Phys. Rev. A 68,062303 (2003).
[42]M. Hein, W. Dur, J. Eisert, R. Raussendorf, M. Van den Nest, and H. J.Briegel, arXive:quant-ph/0602096.
[43]S. Y. Looi, L. Yu, V. Gheorghiu, R. B. Griffiths, arXiv:0712.1979 [quant-ph].
[44]S. Niskanen and P.R.J. Ostergard, cliquer User's Guide Version 1.0, (Com-munications Laboratory, Helsinki University of Technology, Espoo, Finland, Tech. Rep. T 48,2003).
[45]A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev.47,777 (1935).
[46]J. S. Bell, Physics 1,195 (1964).
[47]Clauser J F,Horne et al, Phys. Rev. Lett.23,880(1969).
[48]D.M. Greenberger, M.A. Home, and A. Zeilinger, in Bell's Theorem, Quan-tum Theory, and Conceptions of the Universe, edited by M. Kafatos (Kluw-er, Dordrecht,1989), p.69; arXiv:0712.0921.
[49]D.M. Greenberger, M.A. Horne, A. Shimony, and A. Zeilinger, Am. J. Phys. 58,1131 (1990).
[50]N.D. Mermin, Phys. Rev. Lett.65,3373 (1990).
[51]N.D. Mermin, Rev. Mod. Phys.65,803 (1993).
[52]S. Kochen and E.P. Specker, J. Math. Mech.17,59 (1967).
[53]A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett.47,460 (1981); R. Lapkiewicz et al., Nature (London) 474,490 (2011); C. Zu, et al., Phys. Rev. Lett.109,150401 (2012).
[54]A. Cabello, Phys. Rev. Lett.101,210401 (2008).
[55]S. Yu and C.H. Oh, Phys. Rev. Lett.108,030402 (2012).
[56]D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett.82,1345 (1999); J.-W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, and A. Zeilinger, Nature(London) 403,515 (2000).
[57]R. Cleve and H. Buhrman, Phys. Rev. A 56,1201 (1997)
[58]M. Zukowski et al., Acta Phys. Pol. A 93,187 (1998)
[59]S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, New J. Phys.8,75 (2006).
[60]A. Cabello, Phys. Rev. A 63,022104 (2001).
[61]A. Cabello et al., Phys. Lett. A 212,183 (1996).
[62]A. A. Klyachko, M. A. Can, S. Binicioglu, and A. S. Shumovsky, Phys. Rev. Lett.101,20403(2008).
[63]C. Pagonis, M.L.G. Redhead, and R.K. Clifton, Phys. Lett. A 155,441 (1991).
[64]N.J. Cerf, S. Massar, and S. Pironio, Phys. Rev. Lett.89,080402 (2002).
[65]J. Lee, S.-W. Lee, and M.S. Kim, Phys. Rev. A 73,032316 (2006).
[66]D. Kaszlikowski and M. Zukowski, Phys. Rev. A 66,042107 (2002).
[67]D.P. DiVincenzo and A. Peres, Phys. Rev. A 55,4089 (1997).
[68]D. Hu, W. Tang, M. Zhao, Q. Chen, S. Yu, and C.H. Oh, Phys. Rev. A 78, 012306 (2008).
[69]W. Son, J. Lee, and M.S. Kim, Phys. Rev. Lett.96,060406 (2006).
[70]Weidong Tang, Sixia Yu, and C.H. Oh, arXiv:1206.2718.