量子信道理论中的若干问题
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摘要
量子信息论，作为一门迅速崛起的新兴学科，主要研究使用量子物理体系来进行信息的传输。量子信息论充分的利用了量子力学中的基本原理(如量子态叠加原理)和基本概念(如量子纠缠)来实现信息的处理。虽然目前量子信息论仍处在实验和理论物理学家的原创性研究阶段，但它为量子论的实际应用提供了一个全新的视点和生长点，对它的深入研究必将拓宽和深化量子论本身。与以前应用量子力学完全不同的是，在量子信息论中人们利用的是量子态本身，其任务是量子态的存储、操纵、传输和读出。我们可以谨慎的预言，量子信息论的发展很可能会导致一个新的量子技术时代。
最初的一些量子信息处理方案都是针对离散变量(如自旋和极化)的量子体系(即量子比特)提出的。近几年，连续变量(如动量和位置)的量子信息处理方案引起了广泛的关注。连续变量体系的teleportation、纠缠交换、量子克隆、量子计算、量子纠错、纠缠纯化等被相应的提出；我们还提出了新的用连续变量和离散变量纠缠实现量子信息处理的方案。经过数十年的发展，量子光学已经是一门非常成熟而又充满活力的科学。它为检验量子力学的一些基本问题提供了必不可少的精密手段。连续变量的量子信息处理的一个突出特点是：它可以在量子光学实验中利用线性光学元件(如相移及分束器)操纵压缩态来实现；线性光学元件易于实现较高效率和精度的量子操作。因此，量子光学为各种连续变量量子信息方案提供了可行的手段。但在Bell不等式的实验检验中，人们大多使用那些具有离散变量的量子系统。运用非简并光学参数放大过程，人们可以产生双模压缩真空态，从而实现连续量子变量(如位置和动量)的Einstein-Podolsky-Rosen佯谬。在此基础上，连续量子变量变量系统的量子纠缠和非局域性及它们之间的关系就成了极大的理论兴趣之所在。
本文的主要工作有如下四个方面：
1．使用主方程对量子信道进行了讨论。给出了有损耗玻色信道的信道容量和保真度的具体表达式。
2．使用算子理论，给出连续变量teleportation量子信道的信道容量和保真度的具体表达式，并展开了相关的讨论。
3．研究了使用相对熵来区分离散量子信道和连续变量量子信道的方法。
4．研究了各种各样的高斯信道的保真度的变化规律。
Quantum information theory, as a new rapidly emerging subject, its aim is to study the transmission of information in quantum physical systems. Quantum information theory fully takes advantages of the basic principles of quantum mechanics (quantum states overlapping principles) and basic concepts (quantum entanglement, for example)to carry out the processing of information. Although at current stages, quantum information theory still remains the original research of experimentalists and theorists, it provides a new angle and growing point for the practical applications of quantum theory. And the extensive researches will broaden and deepen quantum theory itself. Completely different from quantum mechanics, in quantum information theory what people exploit is quantum states themselves, and the task is to preserve、 manipulate、 transmitting and reading out the quantum states. We can prudently declare that the developments of quantum information theory could lead to a brand new quantum technology era.
The initial quantum information processing schemes are only aimed at discrete variables (spins and polarizations, for example ) quantum systems (viz. qubits) For recent years, continuous variables (positions and momentums) quantum information processing proposals arouse wide concern. Continuous variables teleportation、 entanglement swapping 、 quantum cloning、 quantum computation、 quantum error correction、 entanglement purification have been put forth; we also come up with new quantum information schemes via continuous variable and discrete variable entanglement. With decades of development, quantum optics has become an mature and vigorous science. It provides some necessary sophisticated methods to verify certain fundamental problems in quantum mechanics. The outstanding characteristic of continuous variable quantum information processing is that: it can exploit linear optical devices (phase shifting and beam splitters, for example) to realize squeezed states; linear optical devices are easy to perform quantum operations with higher efficiency and precision. Hence, quantum optics provide applicable methods for various continuous variables quantum information schemes. However, in the experiments that testify Bell inequalities, people generally use discrete variable quantum systems. Applying non-degenerating optical parameter amplification, people can produce bimodal squeezed vacuum states, so as to realize the continuous variables (positions and momentums, for example) Einstein-Podolsky-Rosen paradox. On this basis, continuous variable quantum entanglement and nonlocality and their relations become great theoretical concern.
This thesis contains four aspects of work:
1. We use master equations to study quantum bosonic channels. And we give the exact form of the channel capacity and fidelity of noisy lossy bosonic channels.
2. Using operators theory, we give the continuous variable teleportation qunatum channel its channel capacity and fidelity , and we have relevant discussions.
3. We use relative entropy to distinguish discrete variable and continuous variable quantum channels.
4. We study the mechanism of the fidelity of various Gaussian channels.
最初的一些量子信息处理方案都是针对离散变量(如自旋和极化)的量子体系(即量子比特)提出的。近几年，连续变量(如动量和位置)的量子信息处理方案引起了广泛的关注。连续变量体系的teleportation、纠缠交换、量子克隆、量子计算、量子纠错、纠缠纯化等被相应的提出；我们还提出了新的用连续变量和离散变量纠缠实现量子信息处理的方案。经过数十年的发展，量子光学已经是一门非常成熟而又充满活力的科学。它为检验量子力学的一些基本问题提供了必不可少的精密手段。连续变量的量子信息处理的一个突出特点是：它可以在量子光学实验中利用线性光学元件(如相移及分束器)操纵压缩态来实现；线性光学元件易于实现较高效率和精度的量子操作。因此，量子光学为各种连续变量量子信息方案提供了可行的手段。但在Bell不等式的实验检验中，人们大多使用那些具有离散变量的量子系统。运用非简并光学参数放大过程，人们可以产生双模压缩真空态，从而实现连续量子变量(如位置和动量)的Einstein-Podolsky-Rosen佯谬。在此基础上，连续量子变量变量系统的量子纠缠和非局域性及它们之间的关系就成了极大的理论兴趣之所在。
本文的主要工作有如下四个方面：
1．使用主方程对量子信道进行了讨论。给出了有损耗玻色信道的信道容量和保真度的具体表达式。
2．使用算子理论，给出连续变量teleportation量子信道的信道容量和保真度的具体表达式，并展开了相关的讨论。
3．研究了使用相对熵来区分离散量子信道和连续变量量子信道的方法。
4．研究了各种各样的高斯信道的保真度的变化规律。
Quantum information theory, as a new rapidly emerging subject, its aim is to study the transmission of information in quantum physical systems. Quantum information theory fully takes advantages of the basic principles of quantum mechanics (quantum states overlapping principles) and basic concepts (quantum entanglement, for example)to carry out the processing of information. Although at current stages, quantum information theory still remains the original research of experimentalists and theorists, it provides a new angle and growing point for the practical applications of quantum theory. And the extensive researches will broaden and deepen quantum theory itself. Completely different from quantum mechanics, in quantum information theory what people exploit is quantum states themselves, and the task is to preserve、 manipulate、 transmitting and reading out the quantum states. We can prudently declare that the developments of quantum information theory could lead to a brand new quantum technology era.
The initial quantum information processing schemes are only aimed at discrete variables (spins and polarizations, for example ) quantum systems (viz. qubits) For recent years, continuous variables (positions and momentums) quantum information processing proposals arouse wide concern. Continuous variables teleportation、 entanglement swapping 、 quantum cloning、 quantum computation、 quantum error correction、 entanglement purification have been put forth; we also come up with new quantum information schemes via continuous variable and discrete variable entanglement. With decades of development, quantum optics has become an mature and vigorous science. It provides some necessary sophisticated methods to verify certain fundamental problems in quantum mechanics. The outstanding characteristic of continuous variable quantum information processing is that: it can exploit linear optical devices (phase shifting and beam splitters, for example) to realize squeezed states; linear optical devices are easy to perform quantum operations with higher efficiency and precision. Hence, quantum optics provide applicable methods for various continuous variables quantum information schemes. However, in the experiments that testify Bell inequalities, people generally use discrete variable quantum systems. Applying non-degenerating optical parameter amplification, people can produce bimodal squeezed vacuum states, so as to realize the continuous variables (positions and momentums, for example) Einstein-Podolsky-Rosen paradox. On this basis, continuous variable quantum entanglement and nonlocality and their relations become great theoretical concern.
This thesis contains four aspects of work:
1. We use master equations to study quantum bosonic channels. And we give the exact form of the channel capacity and fidelity of noisy lossy bosonic channels.
2. Using operators theory, we give the continuous variable teleportation qunatum channel its channel capacity and fidelity , and we have relevant discussions.
3. We use relative entropy to distinguish discrete variable and continuous variable quantum channels.
4. We study the mechanism of the fidelity of various Gaussian channels.
引文
[1] Yongde Zhang, Principles of Quantum Information Physics(2006) Science Press.
[2] Yongde Zhang, Quantum Mechanics(2002) Science Press.
[3] S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77,(2005) 513.
[4] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information(2000) Cambridge University Press, Cambridge.
[5] S. L. Braunstein and A. K. Pati, Quantum Information Theory with Continuous Variables(2001) Springer.
[6] C. H. Bennett and P. W. Shor, IEEE Trans. Inf. Theory 44(1998) 2724; A. S. Holevo, e-print quant-ph/9809023.
[7] C. M. Caves and P. D. Drummond, Rev. Mod. Phys. 66(1994) 481.
[8] A. S. Holevo and R. F. Werner, Phys. Rev. A 63(2001) 032312.
[9] V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H. Shapiro, and H. P. Yuen, Phys. Rev. Lett. 92(2004) 027902; V. Giovannetti and S. Mancini, Phys. Rev. A 71(2005) 062304.
[10] Chiara Macchiavello, G. Massimo Palma, and S. Virmani, Phys. Rev. A 69(2004) 010303(R); Nicolas J. Cerf, Julien Clavareau, Chiara Macchiavello and Jeremie Roland, ibid. 72(2005) 042330; V. Giovannetti, S. Lloyd, L. Maccone, J. H. Shapiro, and B. J. Yen, ibid. 70(2004) 022328.
[11] A. S. Holevo, M. Sohma, and O. Hirota, Phys. Rev. A 59(1999) 1820.
[12] W. H. Zurek, Phys. Today 44(10) (1991) 36.
[13] E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, K-Kupsch, I. -O. Stamatescu, Decoherence and the Appearance of a Classical World in Quantum Theory(2003) Springer.
[14] Sonja Daffer, Krzysztof Wodkiewicz, and John K. McIver, Phys. Rev. A 67(2003) 062312; Sonja Daffer, Krzysztof Wodkiewicz, James D. Cresser, and John K. McIver, ibid, 70(2004) 010304.
[15] John Preskill, http://www.theory.caltech.edu/-preskill/ph229.
[16] C. W. Gardiner and P. Zoller, Quantum Noise(2000) Springer.
[17] A. S. Holevo, IEEE Trans. Inf. Theory 44(1998) 269; P. Hausladen, R. Jozsa, B. Schumacher, M. Westmoreland, and W. K. Wootters, Phys. Rev. A 54(1996) 1869; B. Schumacher and M. D. Westmoreland, ibid. 56(1997) 131.
[18] Marian O. Scully and M. Suhail Zubairy, Quantum Optics(1997) Cambridge.
[19] D. F. Walls and G. J. Milburn, Quantum Optics(1994) Springer.
[20] H. X. Lu, J. Yang, Y. D. Zhang and Z. B. Chen, Phys. Rev. A 67(2003) 024101.
[21] N. A. Peters, T. -C. Wei, and P. G. Kwiat, Phys. Rev. A 70(2004) 052309.
[22] J. Fiurasek, Phys. Rev. A 70(2004) 032308.
[23] P. Zanardi and D. A. Lidar, Phys. Rev. A 70(2004) 012315.
[24] S. Oh, S. Lee, and H. -w. Lee, Phys. Rev. A 66(2002) 022316.
[25] Bennett C. H., Brassard G., Crepeau C., Jozsa R., Peres A., and Wootters W. K., Phys. Rev. Lett. 70, 1895(1993).
[26] Bowen G. and Bose S., Phys. Rev. Lett. 87, 267901(2001).
[27] Vaidman L., Phys. Rev. A 49, 1473(1994).
[28] Bouwmeester D., Pan J-W, Mattle K., Eibl M., Weinfurter H., Zeilinger A., Nature 390, 575(1997).
[29] Furusawa A., Sφrensen J. L., Braunstein S. L., Fuchs C. A., Kimble H. J., and Polzik E. S, Science 282, 706(1998); Zhang T. C., Goh K. W., Chou C. W., Lodahl P., and Kimble H. J., Phys. Rev. A 67, 033802(2003).
[30] Thomas M. C. and Thomas J. A., Elements of Information Theory, Wiley-Interscience(1991).
[31] Braunstein S. L. and Kimble H. J., Phys. Rev. Lett. 80, 869(1998).
[32] Ban M., Sasaki M. and Takeoka M., J. Phys. A: Math. Gen. 35, L401(2002).
[33] Takeoka M., Ban M. and Sasaki M., J. Opt. B: Quantum Semiclass. Opt.4, 114(2002).
[34] Ban M., J. Opt. B: Quantum Semiclass. Opt. 6, 224(2004).
[35] Pirandola S., Mancini S., and Vitali D., Phys. Rev. A 71, 042326(2005).
[36] Li Y., Zhang J., Zhang J-X and Zhang T-C, Chinese Physics, Vol.15, No.8, 1766(2006).
[37] Li Y., Zhang T-C, Zhang J-X and Xie C-D, Chinese Physics, Vol.12, No.8, 861(2003).
[38] Bennett C. H. and Shor P. W., IEEE Trans. Inf. Theory 44, 2724(1998); Holevo A. S., Tamagawa Univ. Res. Rev. No.4(1998), quant-ph/9809023.
[39] Qin T., Zhao M-S and Zhang Y-D, quant-ph/0512068.
[40] Ban M., J. Opt. B: Quantum Semiclass. Opt. 2, 786(2000).
[41] Braunstein S. L. and Kimble H. J., Phys. Rev. A 61, 042302(2000).
[42] Ban M., J. Phys. A: Math. Gen. 37, L385(2004).
[43] W. K. Wootters, Phys. Rev. D 23, 357(1981); S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439(1994).
[44] A. Gilchrist, N. K. Langford, and M. A. Nielson, e-print quant-ph/0408063; V. P. Belavkin, G. M. D'Ariano, and M. Raginsky, J. Math. Phys. 46, 062106(2005).
[45] Nicolas J. Cerf, Phys. Rev. Lett. 84, 4497(2000).
[46] C. King, e-print quant-ph/0204172.
[47] Chiara Macchiavello and G. Massimo Palma, Phys. Rev. A 65, 050301(R) (2002).
[48] Massimiliano and F. Sacchi, Phys. Rev. A 72, 014305(2005); Massimiliano F. Sacchi, ibid., A 71, 062340(2005).
[49] Nilanjana Datta and Alexander S. Holevo, e-print quant-ph/0510145.
[50] V. Vedral, Rev. Mod. Phys. 74, 197(2002).
[51] K. Kraus, States, Effects and Operations: Fundamental Notions of Quantum Theory (Springer-Verlag, Berlin, 1983)
[52] Xiang-Bin Wang, L C Kwek and C H Oh, J. Phys. A: Math. Gen. 33, 4925(2000).
[53] Carlton M. Caves, arXiv: quant-ph/0409063.
[54] H. P. Yuen and M. Ozawa, Phys. Rev. Lett. 70, 363(1992).
[55] M. J. W. Hall, Phys. Rev. A 50, 3295(1994).
[56] Tao Qin, Meisheng Zhao and Yongde Zhang, arXiv: quant-ph/0605171.
[57] V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H. Shapiro, and H. P. Yuen, Phys. Rev. Lett. 92, 027902(2004).
[58] Tao Qin, Meisheng Zhao and Yongde Zhang, Physics Letters A 355, 308(2006).
[2] Yongde Zhang, Quantum Mechanics(2002) Science Press.
[3] S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77,(2005) 513.
[4] M. Nielsen and I. Chuang, Quantum Computation and Quantum Information(2000) Cambridge University Press, Cambridge.
[5] S. L. Braunstein and A. K. Pati, Quantum Information Theory with Continuous Variables(2001) Springer.
[6] C. H. Bennett and P. W. Shor, IEEE Trans. Inf. Theory 44(1998) 2724; A. S. Holevo, e-print quant-ph/9809023.
[7] C. M. Caves and P. D. Drummond, Rev. Mod. Phys. 66(1994) 481.
[8] A. S. Holevo and R. F. Werner, Phys. Rev. A 63(2001) 032312.
[9] V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H. Shapiro, and H. P. Yuen, Phys. Rev. Lett. 92(2004) 027902; V. Giovannetti and S. Mancini, Phys. Rev. A 71(2005) 062304.
[10] Chiara Macchiavello, G. Massimo Palma, and S. Virmani, Phys. Rev. A 69(2004) 010303(R); Nicolas J. Cerf, Julien Clavareau, Chiara Macchiavello and Jeremie Roland, ibid. 72(2005) 042330; V. Giovannetti, S. Lloyd, L. Maccone, J. H. Shapiro, and B. J. Yen, ibid. 70(2004) 022328.
[11] A. S. Holevo, M. Sohma, and O. Hirota, Phys. Rev. A 59(1999) 1820.
[12] W. H. Zurek, Phys. Today 44(10) (1991) 36.
[13] E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, K-Kupsch, I. -O. Stamatescu, Decoherence and the Appearance of a Classical World in Quantum Theory(2003) Springer.
[14] Sonja Daffer, Krzysztof Wodkiewicz, and John K. McIver, Phys. Rev. A 67(2003) 062312; Sonja Daffer, Krzysztof Wodkiewicz, James D. Cresser, and John K. McIver, ibid, 70(2004) 010304.
[15] John Preskill, http://www.theory.caltech.edu/-preskill/ph229.
[16] C. W. Gardiner and P. Zoller, Quantum Noise(2000) Springer.
[17] A. S. Holevo, IEEE Trans. Inf. Theory 44(1998) 269; P. Hausladen, R. Jozsa, B. Schumacher, M. Westmoreland, and W. K. Wootters, Phys. Rev. A 54(1996) 1869; B. Schumacher and M. D. Westmoreland, ibid. 56(1997) 131.
[18] Marian O. Scully and M. Suhail Zubairy, Quantum Optics(1997) Cambridge.
[19] D. F. Walls and G. J. Milburn, Quantum Optics(1994) Springer.
[20] H. X. Lu, J. Yang, Y. D. Zhang and Z. B. Chen, Phys. Rev. A 67(2003) 024101.
[21] N. A. Peters, T. -C. Wei, and P. G. Kwiat, Phys. Rev. A 70(2004) 052309.
[22] J. Fiurasek, Phys. Rev. A 70(2004) 032308.
[23] P. Zanardi and D. A. Lidar, Phys. Rev. A 70(2004) 012315.
[24] S. Oh, S. Lee, and H. -w. Lee, Phys. Rev. A 66(2002) 022316.
[25] Bennett C. H., Brassard G., Crepeau C., Jozsa R., Peres A., and Wootters W. K., Phys. Rev. Lett. 70, 1895(1993).
[26] Bowen G. and Bose S., Phys. Rev. Lett. 87, 267901(2001).
[27] Vaidman L., Phys. Rev. A 49, 1473(1994).
[28] Bouwmeester D., Pan J-W, Mattle K., Eibl M., Weinfurter H., Zeilinger A., Nature 390, 575(1997).
[29] Furusawa A., Sφrensen J. L., Braunstein S. L., Fuchs C. A., Kimble H. J., and Polzik E. S, Science 282, 706(1998); Zhang T. C., Goh K. W., Chou C. W., Lodahl P., and Kimble H. J., Phys. Rev. A 67, 033802(2003).
[30] Thomas M. C. and Thomas J. A., Elements of Information Theory, Wiley-Interscience(1991).
[31] Braunstein S. L. and Kimble H. J., Phys. Rev. Lett. 80, 869(1998).
[32] Ban M., Sasaki M. and Takeoka M., J. Phys. A: Math. Gen. 35, L401(2002).
[33] Takeoka M., Ban M. and Sasaki M., J. Opt. B: Quantum Semiclass. Opt.4, 114(2002).
[34] Ban M., J. Opt. B: Quantum Semiclass. Opt. 6, 224(2004).
[35] Pirandola S., Mancini S., and Vitali D., Phys. Rev. A 71, 042326(2005).
[36] Li Y., Zhang J., Zhang J-X and Zhang T-C, Chinese Physics, Vol.15, No.8, 1766(2006).
[37] Li Y., Zhang T-C, Zhang J-X and Xie C-D, Chinese Physics, Vol.12, No.8, 861(2003).
[38] Bennett C. H. and Shor P. W., IEEE Trans. Inf. Theory 44, 2724(1998); Holevo A. S., Tamagawa Univ. Res. Rev. No.4(1998), quant-ph/9809023.
[39] Qin T., Zhao M-S and Zhang Y-D, quant-ph/0512068.
[40] Ban M., J. Opt. B: Quantum Semiclass. Opt. 2, 786(2000).
[41] Braunstein S. L. and Kimble H. J., Phys. Rev. A 61, 042302(2000).
[42] Ban M., J. Phys. A: Math. Gen. 37, L385(2004).
[43] W. K. Wootters, Phys. Rev. D 23, 357(1981); S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439(1994).
[44] A. Gilchrist, N. K. Langford, and M. A. Nielson, e-print quant-ph/0408063; V. P. Belavkin, G. M. D'Ariano, and M. Raginsky, J. Math. Phys. 46, 062106(2005).
[45] Nicolas J. Cerf, Phys. Rev. Lett. 84, 4497(2000).
[46] C. King, e-print quant-ph/0204172.
[47] Chiara Macchiavello and G. Massimo Palma, Phys. Rev. A 65, 050301(R) (2002).
[48] Massimiliano and F. Sacchi, Phys. Rev. A 72, 014305(2005); Massimiliano F. Sacchi, ibid., A 71, 062340(2005).
[49] Nilanjana Datta and Alexander S. Holevo, e-print quant-ph/0510145.
[50] V. Vedral, Rev. Mod. Phys. 74, 197(2002).
[51] K. Kraus, States, Effects and Operations: Fundamental Notions of Quantum Theory (Springer-Verlag, Berlin, 1983)
[52] Xiang-Bin Wang, L C Kwek and C H Oh, J. Phys. A: Math. Gen. 33, 4925(2000).
[53] Carlton M. Caves, arXiv: quant-ph/0409063.
[54] H. P. Yuen and M. Ozawa, Phys. Rev. Lett. 70, 363(1992).
[55] M. J. W. Hall, Phys. Rev. A 50, 3295(1994).
[56] Tao Qin, Meisheng Zhao and Yongde Zhang, arXiv: quant-ph/0605171.
[57] V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H. Shapiro, and H. P. Yuen, Phys. Rev. Lett. 92, 027902(2004).
[58] Tao Qin, Meisheng Zhao and Yongde Zhang, Physics Letters A 355, 308(2006).