联合幺正操作及其在量子通信中的应用
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摘要
量子信息是现代科学技术发展中出现的一门至关重要的新兴学科,它包括量子计算、量子测量、量子密码术以及量子通信等,在现代通信技术的发展、仪器设备的改进以及国防科技等方面有着巨大的潜在作用。在量子信息学中,量子纠缠是个不可忽视的重要物理资源,可以说,量子纠缠贯穿着整个量子信息学领域。自从量子信息学诞生以来,诸多学者一直在探索怎样增强和传递量子系统之间的纠缠等问题。而对两体系或是多体系的联合量子操作可实现量子纠缠。另外,借助于量子操作,可使系统之间的纠缠得到转移,从而使部分系统之间的纠缠增强。在量子通信过程中,我们往往要根据实际的需要制备具有一定纠缠度的纠缠态或是通过量子操作实现系统间的纠缠度增加到某一特定值。故此,研究量子操作的纠缠特性就具有非常重要的理论和实际意义。
在量子通信过程中,人们常用控制非(C-NOT)操作实现两体之间的纠缠。而在实际操作中,由于环境噪声、仪器设备精密性等因素所造成的消相干影响,量子通道常为混合态通道。此时,常用的C-NOT操作是否是实现相应通信目的的最优操作尚未可知。故此,在本论文中我们将以联合幺正操作可以实现量子纠缠为主线,探讨一般联合幺正操作算符的纠缠特性。根据一般联合幺正算符的算符纠缠度量方式的多样性,比较各种算符纠缠度量方式的优缺点,并最终选用特定度量方式对两量子比特一般联合幺正算符的算符纠缠进行分析。通过分析两量子比特一般联合幺正算符的纠缠属性,探索两量子比特一般联合幺正算符的相关属性(如施密特数等)与其算符纠缠之间的关系,以达到对两量子比特幺正算符的纠缠特性的更深层次理解。至此,我们将对两量子比特一般联合幺正算符的算符纠缠进行系统地总结和归纳。最后,根据以上研究结果,探讨两量子比特一般联合幺正算符在不同量子通信过程中(量子纠缠交换、量子态纯化、量子隐形传态等)的实际应用,并研究量子通道容量、两量子比特一般联合幺正操作的算符纠缠、输出结果保真度这三者之间的匹配关系以及最优化选择方法。本文的主要成果如下：
(1)给出了两量子比特一般幺正算符的算符纠缠度量通式；分析了一般幺正算符的相关属性；基于算符纠缠度量通式的提出,给出了一般幺正算符的相关属性及其算符纠缠之间的具体联系。如,一般幺正操作算符的施密特数与其算符纠缠之间的联系。
(2)在量子隐形传态过程中,研究了联合幺正操作的算符纠缠、通道容量以及输出态保真度这三者之间的匹配关系；在量子纠缠交换过程中,研究了联合幺正操作对交换后粒子对纠缠度的影响,并找到了纠缠交换前后纠缠粒子对的纠缠度之间的关系；在量子态纯化过程中,采用一般联合幺正操作来探讨幺正操作对非局域纠缠粒子对的纯化结果和效率的影响,并找出了联合幺正操作的算符纠缠能力、量子通道的纠缠度以及输出态纠缠度之间的匹配关系,根据所得的匹配关系,得出了最优化选择途径。
Quantum information is a crucial emerging science in the development of modern science and technology. Quantum information includes Quantum computation, Quantum measurement, Quantum cryptography, Quantum communication and so on. It has enormous potentials in the modern communication technology, devices designing and National defense. In Quantum information science, Quantum entanglement is an important physical resource. We can say that quantum entanglement exists in all of the branches in quantum information science. After the emergence of quantum information science, many scholars have been exploring the solution for transmitting and enhancing the entanglement of quantum systems. They found that quantum entanglement can be realized by applying joint unitary operation on quantum bipartite system or quantum multi-system. Entanglement can be transferred with the help of joint unitary operations, and the entanglement of the left entangled pairs can be enhanced after post-selection measurements. In quantum communication process, to prepare some systems in some specific entanglement and to enhance the entanglement of systems to a specific value by joint quantum unitary operation are in urgent need in practice. Therefore, the study on the entangling property of joint quantum operation is of great theoretical and practical significance.
In quantum communication processes, ones usually take the controlled-NOT(C-NOT) operation to generate entanglement of bipartite systems. But in the practical situation, the quantum channel are usually mixed entangled state because of the decoherence effect, such as the noisy environment, device precision etc. So we can't be sure whether the C-NOT operation is the optimal operation for the corresponding communication process. Therefore, in this thesis, with the fact that the joint unitary operation can generate quantum entanglement in mind, we discuss the operator entanglement of the general joint unitary operation firstly. And then we will compare several kinds of operator entanglement measures for joint unitary operators and chose one of them to measure the operator entanglement of the general joint unitary operator in bipartite system. By studying the relationship between some of the intrinsic attributes of the general joint unitary operation and the operator entanglement of it, we can make further conjectures on the entanglement property of join unitary operator in bipartite system. At last, we will apply the above theory to the general join unitary operators in quantum communication processes (such as Quantum entanglement swapping, Quantum state purification, Quantum teleportation and so on) and discuss the matching relation between the quantum channel, the operator entanglement of bipartite general joint unitary operation and the fidelity of the output state of the process and thus an optimal scheme can be designed. Our main results are as follows:
(1) The general formula of operator entanglement for the general bipartite joint unitary operator has been derived in this thesis, and we further, studied several intrinsic attributes of the general joint unitary operator. In light of the general formula and the intrinsic attributes of the joint unitary operator, we found the exact relationship between some of the intrinsic attributes of the general joint unitary operation and its operator entanglement.
(2) In quantum teleportation process, we studied and got the exact matching relation among the operator entanglement of the joint unitary operator the channel entanglement and the fidelity of the output states for the process; In quantum entanglement swapping process, the exact relationship among the operator entanglement of the joint unitary operation in the intermediate location and the entanglements of the final swapped state and the initial state has also been derived; In quantum state purification process, we took the general bipartite joint unitary operation for the process rather than the perfect C-NOT operation in the standard purification process, found its effect on the purification efficiency and the entanglement of the final state and got the exact matching relation among the operator entanglement of the join unitary operation the quantum channel entanglement and the entanglement of the output state. In light of this matching relation, we can design the corresponding optimal scheme for the quantum state purification process.
在量子通信过程中,人们常用控制非(C-NOT)操作实现两体之间的纠缠。而在实际操作中,由于环境噪声、仪器设备精密性等因素所造成的消相干影响,量子通道常为混合态通道。此时,常用的C-NOT操作是否是实现相应通信目的的最优操作尚未可知。故此,在本论文中我们将以联合幺正操作可以实现量子纠缠为主线,探讨一般联合幺正操作算符的纠缠特性。根据一般联合幺正算符的算符纠缠度量方式的多样性,比较各种算符纠缠度量方式的优缺点,并最终选用特定度量方式对两量子比特一般联合幺正算符的算符纠缠进行分析。通过分析两量子比特一般联合幺正算符的纠缠属性,探索两量子比特一般联合幺正算符的相关属性(如施密特数等)与其算符纠缠之间的关系,以达到对两量子比特幺正算符的纠缠特性的更深层次理解。至此,我们将对两量子比特一般联合幺正算符的算符纠缠进行系统地总结和归纳。最后,根据以上研究结果,探讨两量子比特一般联合幺正算符在不同量子通信过程中(量子纠缠交换、量子态纯化、量子隐形传态等)的实际应用,并研究量子通道容量、两量子比特一般联合幺正操作的算符纠缠、输出结果保真度这三者之间的匹配关系以及最优化选择方法。本文的主要成果如下：
(1)给出了两量子比特一般幺正算符的算符纠缠度量通式；分析了一般幺正算符的相关属性；基于算符纠缠度量通式的提出,给出了一般幺正算符的相关属性及其算符纠缠之间的具体联系。如,一般幺正操作算符的施密特数与其算符纠缠之间的联系。
(2)在量子隐形传态过程中,研究了联合幺正操作的算符纠缠、通道容量以及输出态保真度这三者之间的匹配关系；在量子纠缠交换过程中,研究了联合幺正操作对交换后粒子对纠缠度的影响,并找到了纠缠交换前后纠缠粒子对的纠缠度之间的关系；在量子态纯化过程中,采用一般联合幺正操作来探讨幺正操作对非局域纠缠粒子对的纯化结果和效率的影响,并找出了联合幺正操作的算符纠缠能力、量子通道的纠缠度以及输出态纠缠度之间的匹配关系,根据所得的匹配关系,得出了最优化选择途径。
Quantum information is a crucial emerging science in the development of modern science and technology. Quantum information includes Quantum computation, Quantum measurement, Quantum cryptography, Quantum communication and so on. It has enormous potentials in the modern communication technology, devices designing and National defense. In Quantum information science, Quantum entanglement is an important physical resource. We can say that quantum entanglement exists in all of the branches in quantum information science. After the emergence of quantum information science, many scholars have been exploring the solution for transmitting and enhancing the entanglement of quantum systems. They found that quantum entanglement can be realized by applying joint unitary operation on quantum bipartite system or quantum multi-system. Entanglement can be transferred with the help of joint unitary operations, and the entanglement of the left entangled pairs can be enhanced after post-selection measurements. In quantum communication process, to prepare some systems in some specific entanglement and to enhance the entanglement of systems to a specific value by joint quantum unitary operation are in urgent need in practice. Therefore, the study on the entangling property of joint quantum operation is of great theoretical and practical significance.
In quantum communication processes, ones usually take the controlled-NOT(C-NOT) operation to generate entanglement of bipartite systems. But in the practical situation, the quantum channel are usually mixed entangled state because of the decoherence effect, such as the noisy environment, device precision etc. So we can't be sure whether the C-NOT operation is the optimal operation for the corresponding communication process. Therefore, in this thesis, with the fact that the joint unitary operation can generate quantum entanglement in mind, we discuss the operator entanglement of the general joint unitary operation firstly. And then we will compare several kinds of operator entanglement measures for joint unitary operators and chose one of them to measure the operator entanglement of the general joint unitary operator in bipartite system. By studying the relationship between some of the intrinsic attributes of the general joint unitary operation and the operator entanglement of it, we can make further conjectures on the entanglement property of join unitary operator in bipartite system. At last, we will apply the above theory to the general join unitary operators in quantum communication processes (such as Quantum entanglement swapping, Quantum state purification, Quantum teleportation and so on) and discuss the matching relation between the quantum channel, the operator entanglement of bipartite general joint unitary operation and the fidelity of the output state of the process and thus an optimal scheme can be designed. Our main results are as follows:
(1) The general formula of operator entanglement for the general bipartite joint unitary operator has been derived in this thesis, and we further, studied several intrinsic attributes of the general joint unitary operator. In light of the general formula and the intrinsic attributes of the joint unitary operator, we found the exact relationship between some of the intrinsic attributes of the general joint unitary operation and its operator entanglement.
(2) In quantum teleportation process, we studied and got the exact matching relation among the operator entanglement of the joint unitary operator the channel entanglement and the fidelity of the output states for the process; In quantum entanglement swapping process, the exact relationship among the operator entanglement of the joint unitary operation in the intermediate location and the entanglements of the final swapped state and the initial state has also been derived; In quantum state purification process, we took the general bipartite joint unitary operation for the process rather than the perfect C-NOT operation in the standard purification process, found its effect on the purification efficiency and the entanglement of the final state and got the exact matching relation among the operator entanglement of the join unitary operation the quantum channel entanglement and the entanglement of the output state. In light of this matching relation, we can design the corresponding optimal scheme for the quantum state purification process.
引文
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4 M. Ikram, S.-Y. Zhu, M. S. Zubairy, Quantum teleportation of an entangled state [J], Phys. Rev. A 62,022307 (2000).
5 J. Lee, H. Min, S. D. Oh, Multipartite entanglement for entanglement teleportation [J], Phys. Rev. A 66,052318 (2002).
6 X.-F. Yin, Y.-M. Liu, Z.-Y. Zhang, W. Zhang, Z.-J. Zhang, Perfect teleportation of an arbitrary three-qubit state with the highly entangled six-qubit genuine state [J], Sci China Ser G-Phys Mech Astron,53 (11):2059-2063(2010).
7 M. Zukowski, A. Zeilinger, M. A. Home, A. K. Ekert, "Event-Ready-Detectors" Bell Experiment via Entanglement Swapping [J], Phys. Rev. Lett.71, 4287-4290(1993).
8 S. Bose, Purification via entanglement swapping and conserved entanglement [J], Phys. Rev. A 60,194-197(1999).
9 B. Kraus and J. I. Cirac, Optimal creation of entanglement using a two-qubit gate [J], Phys. Rev. A 63,062309(2001).
10 B.-L. Hu, Y.-M. Di, Entanglement Capacity of Two-Qubit Unitary Operator with the Help of Auxiliary System [J], Commun. Theor. Phys.47(2007).
11 J. Modlawska, A. Grudka, Increasing Singlet fraction with entanglement swapping [J], Phys. Rev. A 78,032321(2008).
12 S. Bose, V. Vedral, P. L. Knight, Multiparticle generalization of entanglement swapping [J], Phys. Rev. A 57,822-829(1998).
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16 P. Zanardi, C. Zalka, and L. Faoro, Entangling power of quantum evolutions [J]. Phys. Rev. A 62,030301(2000).
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21 X.-G. Wang, P. Zanardi, Quantum entanglement of unitary operators on bipartite systems[J], Phys. Rev. A 66,044303(2002).
22 M. A. Ballester, Estimation of unitary quantum operations[J], Phys. Rev. A 69, 022303(2004).
23 M. W. Coffey, R. Deiotte, Relation of operator schmidt decomposition and CNOT complexity [J], Quantum Inf Process 7,117-124(2008).
24 S. Balakrishnan, R. Sankaranarayanan, Entangling power and local invariants of two qubit gates[J], Phys. Rev. A 82,034301(2010).
25 Y.-Z. Zheng, P. Ye, G.-C. Guo, Probabilistic implementation of non-local CNOT operation and entanglement purification[J], CHIN. PHYS. LETT.21,9 (2004).
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27 A. T. Rezakhani, Characterization of two-qubit perfect entanglers[J], Phys. Rev. A 70,052313(2004).
28 R. F. Werner, Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model [J]. Phys. Rev. A,1989,40:4277-4281.
29 P. Zanardi, Entanglement of quantum evolutions[J], Phys. Rev. A 63,040304 (R) (2001).
30 P. Zanardi, C. Zalka, and L. Faoro, Entangling power of quantum evolutions[J], Phys. Rev. A 62,030301 (R) (2000).
31 M. A. Nielsen, C. M. Dawson, J. L. Dodd, A. Gilchrist, D. Mortimer. T. J. Osborne. M. J. Bremner, A. W. Harrow, A. Hines, Quantum dynamics as a physical resource[J], Phys. Rev. A 67,052301 (2003).
32 M.-Y. Ye, D. Sun, Y.-S. Zhang, and G.-C. Guo, Entanglement-changing power of two-qubit unitary operations[J], Phys. Rev. A 70,022326 (2004).
33 P. Zanardi, Entanglement of quantum evolutions[J], Phys. Rev. A 63,040304(R) (2001).
34 S. Balakrishnan and R. Sankaranarayanan, Measures of operator entanglement of two-qubit gates[J], Phys. Rev. A 83,062320 (2011).
35 J.-S. Jin, C.-S. Yu, P Pei, and H.-S. Song, Quantum nondemolition measurement of the Werner state[J], Phys. Rev. A 82,042112 (2010).
36 M.-J. Zhao, Z.-G. Li, S.-M. Fei, Z.-X.Wang, X. L.-Jost, Faithful teleportation with arbitrary pure or mixed resource states[J], J. Phys. A:Math. Theor.44 (2011) 215302 (9pp).
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40 W. Son, J. Lee, M. S. Kim, and Y.-J. Park, Conclusive teleportation of a d-dimensional unknown state [J], Phys. Rev. A 64,064304 (2001).
41 C. D. Franco, D. Ballester, Teleportation protocol with non-ideal conditional local operations [J], Phys. Lett. A 374,3164-3169 (2010).
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48 C. H. Bennett, G. Brassarf, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels[J]. Phys. Rev. Lett.,1996,76:722-725.
2 S. Adhikar, A. S. Msjumdar, Quantum teleportation using non-orthogonal entangled channels [J], arXiv:1002.2047v1 (2010).
3 P. Agrawal, A. K. Pati, Probabilistic quantum teleportation [J], Phys. Lett. A 305, 12-17(2002).
4 M. Ikram, S.-Y. Zhu, M. S. Zubairy, Quantum teleportation of an entangled state [J], Phys. Rev. A 62,022307 (2000).
5 J. Lee, H. Min, S. D. Oh, Multipartite entanglement for entanglement teleportation [J], Phys. Rev. A 66,052318 (2002).
6 X.-F. Yin, Y.-M. Liu, Z.-Y. Zhang, W. Zhang, Z.-J. Zhang, Perfect teleportation of an arbitrary three-qubit state with the highly entangled six-qubit genuine state [J], Sci China Ser G-Phys Mech Astron,53 (11):2059-2063(2010).
7 M. Zukowski, A. Zeilinger, M. A. Home, A. K. Ekert, "Event-Ready-Detectors" Bell Experiment via Entanglement Swapping [J], Phys. Rev. Lett.71, 4287-4290(1993).
8 S. Bose, Purification via entanglement swapping and conserved entanglement [J], Phys. Rev. A 60,194-197(1999).
9 B. Kraus and J. I. Cirac, Optimal creation of entanglement using a two-qubit gate [J], Phys. Rev. A 63,062309(2001).
10 B.-L. Hu, Y.-M. Di, Entanglement Capacity of Two-Qubit Unitary Operator with the Help of Auxiliary System [J], Commun. Theor. Phys.47(2007).
11 J. Modlawska, A. Grudka, Increasing Singlet fraction with entanglement swapping [J], Phys. Rev. A 78,032321(2008).
12 S. Bose, V. Vedral, P. L. Knight, Multiparticle generalization of entanglement swapping [J], Phys. Rev. A 57,822-829(1998).
13 A. Kent, Entangled mixed states and local purification [J], Phys. Rev. Lett.81, 2839-2841 (1998).
14 D. Salart, O. Landry, N. Sangouard, N. Gisin, H. Herrmann, B. Sanguinetti, C. Simon, W. Sohler, R. T. Thew, A. Thomas, and H. Zbinden, Purification of Single-Photon Entanglement [J], Phys. Rev. Lett.104.180504(2010).
15 N. Sangouard, C. Simon, T. Coudreau, N. Gisin, Purification of single-photon entanglement with linear optics [J], Phys. Rev. A 78,050301(R) (2008).
16 P. Zanardi, C. Zalka, and L. Faoro, Entangling power of quantum evolutions [J]. Phys. Rev. A 62,030301(2000).
17 W. K. Wotters, Entanglement of formation of an arbitrary state of two qubits [J], Phys. Rev. Lett.80.2245-2248(1998).
18 M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information[M]. Cambridge:Cambridge University Press,2000:20,510-514.
19 M. Horodecki, P. Horodecki, and R. Horodecki, General Teleportation channel Singlet fraction and quasidistillation[J], Phys. Rev. A 60,1888-1898 (1999).
20 X.-G. Wang, B. C.Sanders, D. W. Berry, Entangling power and operator entanglement in qudit systems[J], Phys. Rev. A 67,042323 (2003).
21 X.-G. Wang, P. Zanardi, Quantum entanglement of unitary operators on bipartite systems[J], Phys. Rev. A 66,044303(2002).
22 M. A. Ballester, Estimation of unitary quantum operations[J], Phys. Rev. A 69, 022303(2004).
23 M. W. Coffey, R. Deiotte, Relation of operator schmidt decomposition and CNOT complexity [J], Quantum Inf Process 7,117-124(2008).
24 S. Balakrishnan, R. Sankaranarayanan, Entangling power and local invariants of two qubit gates[J], Phys. Rev. A 82,034301(2010).
25 Y.-Z. Zheng, P. Ye, G.-C. Guo, Probabilistic implementation of non-local CNOT operation and entanglement purification[J], CHIN. PHYS. LETT.21,9 (2004).
26 J. Zhang, J. Vala, S. Sastry, K. B. Whalay, Geometric theory of nonlocal two qubit operations[J], Phys. Rev. A 67,042313(2003).
27 A. T. Rezakhani, Characterization of two-qubit perfect entanglers[J], Phys. Rev. A 70,052313(2004).
28 R. F. Werner, Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model [J]. Phys. Rev. A,1989,40:4277-4281.
29 P. Zanardi, Entanglement of quantum evolutions[J], Phys. Rev. A 63,040304 (R) (2001).
30 P. Zanardi, C. Zalka, and L. Faoro, Entangling power of quantum evolutions[J], Phys. Rev. A 62,030301 (R) (2000).
31 M. A. Nielsen, C. M. Dawson, J. L. Dodd, A. Gilchrist, D. Mortimer. T. J. Osborne. M. J. Bremner, A. W. Harrow, A. Hines, Quantum dynamics as a physical resource[J], Phys. Rev. A 67,052301 (2003).
32 M.-Y. Ye, D. Sun, Y.-S. Zhang, and G.-C. Guo, Entanglement-changing power of two-qubit unitary operations[J], Phys. Rev. A 70,022326 (2004).
33 P. Zanardi, Entanglement of quantum evolutions[J], Phys. Rev. A 63,040304(R) (2001).
34 S. Balakrishnan and R. Sankaranarayanan, Measures of operator entanglement of two-qubit gates[J], Phys. Rev. A 83,062320 (2011).
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