量子纠错码理论若干问题研究
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摘要
量子计算技术因其强大的计算能力,近十几年来,引起了人们极大的兴趣。然而,要使量子计算机成为现实,一个核心问题就是克服由消相干带来的量子噪声。近年来发展起来的量子纠错编码技术能够比较有效的解决这一难题。这就是本论文研究工作的目的。
本论文就量子纠错码理论中的若干问题进行了研究。利用量子码与经典码之间的联系,借助经典常数循环码,构造出了量子常数循环码,并找到了一些新的量子Hamming码;借助经典generalized Reed-Solomon码,构造出了量子generalized Reed-Solomon码,这是一类量子最大距离可分(MDS)码,并在此基础上,提出了量子MDS码的统一框架;提出了量子码级联结构,并具体构造出了一类基于级联结构的量子渐近好码;借助经典Justesen码的思想,构造出了量子Justesen码,这是第一次利用非量子好码具体构造出量子渐近好码。
Quantum computer has interested people greatly for its remarkable computation capacity. However, to make quantum computer practical, it is necessary to find the effective method to win through decoherence. Quantum error-correcting code is one of good methods to get over decoherence.
This dissertation focuses on the study of several problems in the theory of quantum error-correcting codes. By the connection between quantum codes and classical codes, quantum constacyclic codes are constructed based on classical constacyclic codes. In the meantime, some new quantum Hamming codes are found. A class of quantum MDS codes, quantum generalized Reed-Solomon codes, is derived from classical generalized Reed-Solomon codes. Moreover, a unified framework of quantum MDS codes is given. Quantum code concatenation structure is presented, based on which a family of asymptotically good quantum codes is constructed. From classical Justesen codes, quantum Justesen codes are obtained with the property that this is the first time that good quantum codes are derived from bad quantum codes.
本论文就量子纠错码理论中的若干问题进行了研究。利用量子码与经典码之间的联系,借助经典常数循环码,构造出了量子常数循环码,并找到了一些新的量子Hamming码;借助经典generalized Reed-Solomon码,构造出了量子generalized Reed-Solomon码,这是一类量子最大距离可分(MDS)码,并在此基础上,提出了量子MDS码的统一框架;提出了量子码级联结构,并具体构造出了一类基于级联结构的量子渐近好码;借助经典Justesen码的思想,构造出了量子Justesen码,这是第一次利用非量子好码具体构造出量子渐近好码。
Quantum computer has interested people greatly for its remarkable computation capacity. However, to make quantum computer practical, it is necessary to find the effective method to win through decoherence. Quantum error-correcting code is one of good methods to get over decoherence.
This dissertation focuses on the study of several problems in the theory of quantum error-correcting codes. By the connection between quantum codes and classical codes, quantum constacyclic codes are constructed based on classical constacyclic codes. In the meantime, some new quantum Hamming codes are found. A class of quantum MDS codes, quantum generalized Reed-Solomon codes, is derived from classical generalized Reed-Solomon codes. Moreover, a unified framework of quantum MDS codes is given. Quantum code concatenation structure is presented, based on which a family of asymptotically good quantum codes is constructed. From classical Justesen codes, quantum Justesen codes are obtained with the property that this is the first time that good quantum codes are derived from bad quantum codes.
引文
[1] Shor P. and Laflamme R., Quantum analog of the MacWilliams identities for classical coding theory, Phys. Rev. Lett., 1997, 78(8), 1600-1602.
[2] Ashikhmin A., Remarks on bounds for quantum codes, quant-ph/9705037, 1997.
[3] Ashikhmin A. and Litsyn S., Upper bounds on the size of quantum codes, IEEE Trans. Inf. Theory, 1999, 45(4), 1206-1215.
[4] Rains E., Quantum weight enumerators, IEEE Trans. Inf. Theory, 1998, 44 (4), 1388-1394.
[5] Rains E., Monotonicity of the quantum linear programming bound, IEEE Trans. Inf. Theory, 1999, 45(7), 2489-2492.
[6] Rains E., Quantum shadow enumerators, IEEE Trans. Inf. Theory, 1999, 45(7), 2361-2366.
[7] Matsmoto R., Improvement of Ashikhmin-Litsyn-Tsfasman bound for quantum codes, IEEE Trans. Inf. Theory, 2002, 48, 2122-2124.
[8] Bennet C., Bessette G., Brassard S. and Wiesner S., Quantum cryptograph, Advance in cryptograph: Proceedings of Crypto. 82, Plenum Press, New York, 1982, 267-275.
[9] Bennet C. and Brassard G., Quantum cryptography: public key distribution and coin tossing, Proceedings of IEEE International Conference on Computers, Systems & Signal Processing, Bangalore, India, December 10-12, 1984, 175-179.
[10] Bennet C., Brassard G. and Robert J., Privacy amplification by public discussion, SIAM J. Comp., 1988, 7, 210-229.
[11] Deutch D., Quantum communication thwarts the eavesdroppers, New Scientist, 1989, 25-26.
[12] Ekert A., Quantum cryptography based on Bell theorem, Phys. Rev. Lett., 1991, 67, 661-663.
[13] Bennet C., Quantum cryptography using any two non-orthogonal states, Phys. Rev. Lett., 1992, 68, 3121-3124.
[14] Bennet C., Charles H., Brassard G., etc, Quantum cryptography, Scientist American, October 1992, 50-57.
[15] Ekert A. and Palma G., Quantum cryptography with interferometer quantum entanglement, J. Mod. Opt., 1994, 41(12), 2413-2423.
[16] Bennett C., Brassard G., Crepeau C., etc, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett., 1993, 70(13), 1895-1899.
[17] Ekert A., Hutmer B., and Palma G., Eavesdropping on quantum cryptographical systems, Phys. Rev. A, 1994, 50(2), 1047-1056.
[18] Bennet C. and Phoenix A., Information-theoretic limits to quantum cryptography, Phys. Rev. A, 1993, 48(1), R5-R8.
[19] Brassard G., A bibliography of quantum cryptography, Sigact News, 1993, 24(3), 16-20.
[20] Ekert A. and Palma G., Quantum cryptography with the interferometer quantum entanglement, J. Mod. Opt. 1992, 41(12), 2413-2423.
[21] Schumacher B. and Westmoreland M., Quantum privacy and quantum coherence, Phys. Rev. Lett., 1998, 80(25), 5695-5697.
[22] Townsend P., Rarity J., and Tapster P., Enhanced single photon fringe visibility in a 10km long prototype quantum cryptography channel, Electron. Lett., 1993, 29, 1291-1293.
[23] Townsend P., Secure key distribution system based on quantum cryptography, Electron. Lett., 1994, 30, 809-811.
[24] Townsend P., Rarity J., and Tapster P., Single photon interference in a 10km long optical fiber interferometer, Electron. Lett., 1995, 19, 634-639.
[25] Marand C. and Townsend P., Quantum key distribution over distances as long as 30km, Optics Lett., 1995, 20, 1695-1697.
[26] Muller A., Breguet J., and Gisin N., Experimental demonstration of quantum cryptography using polarized photons in optical fiber over more than 1km, Euro. Phys. Lett., 1993, 23, 383-388.
[27] Muller A., Zbinden H., and Gisin N., Underwater quantum coding, Nature, 1995, 378, 449-449.
[28] Muller A., Zbinden H., and Gisin N., Quantum cryptography over 23km in installed under-lake telecom fiber, Euro. Phys. Lett., 1995, 33, 335-339.
[29] Muller A., Herzog T., Huttner B., etc., Plug and play systems for quantum cryptography, Appl. Phys. Lett., 1997, 70, 793-795.
[30] Hughes R., Morgan R., and Peterson C., Quantum key distribution over a 48km optical fiber network, Los Alamos report, LA-UR-99-1593, quantum-ph/9904038.
[31] Hughes R., Morgan R., and Peterson C., Quantum key distribution over a 48km optical fiber network, Journal of Modern Optics, 2000, 47, 533-547.
[32] Reid M., Quantum cryptography with a predetermined key using continuous variable EPR correlations, Phys. Rev. A, 2000, 62, 062308.
[33] Pereira S., Ou Z., and Kimble H., Quantum communications with correlated non-classical states, Phys. Rev. A, 2000, 62, 042311.
[34] Kurtsiefer C., Zarda P., Halder M., etc., A step towards global key distribution, Nature, 2002, 419-420.
[35] Stucki D., Gisin N., Guinnard O., Ribordy G., and Zbinden H., Quantum key distribution over 67km with a plug and play system, New Journal of Physics, 2002, 4, 411-418.
[36] Kosaka H., Tomita A., Namba Y., etc., Single-photon interference experiment over 100km forquantum cryptography system using balanced gated-mode photon detector, Electron. Lett., 2003, 39, 1199-1201.
[37] Gobby C., Yuan Z., and Shields A., Quantum key distribution over 122km of standard telecom fiber, Appl. Phys. Lett., 2004, 84, 3762-3764.
[38] Gordon K., Fernandez V., Townsend P., and Butler Q., A short wavelength GigaHertz clocked fiber-optic quantum key distribution system, IEEE Journal of quantum electronics, 2004, 40(7), 900-908.
[39] Ralph T., Continuous variable quantum cryptography, Phys. Rev. A, 1999, 61, 010303.
[40] Ralph T., Security of continuous variable quantum cryptography, Phys. Rev. A, 2000, 62, 062306.
[41] Hillery M., Quantum cryptography with squeeze states, Phys. Rev. A, 2000, 62, 022309.
[42] Braunstein S. and Kimble H., Teleportation of continuous quantum variables, Phys. Rev. Lett., 1998, 80, 869-872.
[43] Braunstein S., Error correction for continuous quantum variable, Phys. Rev. Lett., 1998, 80, 4084-4087.
[44] Lloyd S. and Slotine J., Analog quantum error correction, Phys. Rev. Lett., 1998, 80, 4088-4091.
[45] Braunstein S. and Kimble H., Dense coding for continuous quantum variable, Phys. Rev. A, 2000, 61, 042302.
[46] Zeng G. and Zhang W., Identity verification in quantum cryptography, Phys. Rev. A, 2000, 61, 032303.
[47] Ljunggren D., Bourennane M., and Karlsson A., Authority-based user authentication in quantum key distribution, Phys. Rev. A, 2000, 62, 022305.
[48] Curty M. and Santos D., Quantum authentication of classical messages, Phys. Rev. A, 2001, 64, 062309.
[49] Dusek M., Haderka O., and Hendrych M., Quantum identification system, Phys. Rev. A, 1999, 60, 149-156.
[50] Mihara T., Quantum identification schemes with entanglement, Phys. Rev. A, 2002, 65, 052306.
[51] Dam W., Comment on quantum identification schemes with entanglement, Phys. Rev. A, 2003, 68, 026301.
[52] Li X. and Banrum H., Quantum authentication using entangled states, Int. J. Found. Com. Sci., 2004, 15, 609.
[53] Garcia P., Femandez E., Perez E., etc., New encoding schemes for quantum authentication, Quantum Inf.&Comp., 2005, 5(1), 1-12.
[54] Winter A., Quantum and classical message protect identification via quantum channels, Quantum Inf.&Comp., 2004, 4(6-7), 563-578.
[55] Cleve R., Gottesman D., and Lo H., How to share a quantum secret, Phys. Rev. Lett., 1999, 65, 648-651.
[56] Rains E., Quantum codes of minimum distance two, IEEE Trans. Inf. Theory, 1999, 45, 266-271.
[57] Guo Y. and Zeng G., Quantum BCH codes based on spectral techniques, Communications in Theoretical Physics, 2006, 45(2), 283-286.
[58] Guo Y. and Zeng G., Quantum convolution codes based on the stabilizer of linear codes, IEEE APCC2005, Perth, Western Australia, October 2005.
[59] Guo Y. and Zeng G., Message authentication scheme based on quantum error-correction codes, First international conference on scalable system, Hong-Kong, May29 2006.
[60] Duan L. and Guo G., Preserving coherence in quantum computation by pairing quantum bits. Phys. Rev. Lett., 1997, 79(10), 1953-1956.
[61] Duan L. and Guo G., etc., Reducing decoherence in quantum-computer memory with all quantum bits coupling to the same environment. Phys. Rev. A, 1998, 57(2), 737-741.
[62] Duan L. and Guo G., Scheme for reducing collective decoherence in quantum memory, Phys. Lett. A, 1998, 43(5-6), 265-269.
[63] Duan L. and Guo G., Prevention of dissipation with two particles, Phys. Rev. A, 1998, 57(4), 2399-2402.
[64] Duan L. and Guo G., A probabilistic cloning machine for replicating two non-orthogonal states, Phys. Lett. A, 1998, 43(5), 261-264.
[65] Duan L. and Guo G., Probabilistic cloning and identification of linearly independent quantum states, Phys. Rev. Lett., 1998, 80(22), 4999-5002.
[66] Marcos C. and David J., Quantum authentication of classical messages, Phys. Rev. A, 2001, 64, 062309.
[67] Lo H. and Chau H., Unconditional security of quantum key distribution over arbitrarily long distance, Science, 1999, 283, 2050-2056.
[68] Hwang W., Matsumoto K., Imai H., Kim J., and Lee H., Shor-Preskill-type security proof for concatenated BB84 quantum-key-distribution protocol, Phys. Rev. A, 2003, 67, 024302.
[69] Wootters W. and Zurek W., A single quantum cannot be cloned, Nature, 1982, 299, 802-803.
[70] Chau H., Quantum convolutional error-correcting codes, Phys. Rev. A, 1998, 58, 905-909.
[71] Gottesman D. and Preskill J., Secure quantum key distribution using squeezed states, Phys. Rev. A, 2001, 63, 022309.
[72] Bourennane M., Karlsson A., and Bjork G., Quantum key distribution using multilevelencoding, Phys. Rev. A, 2001, 64, 012306.
[73] Bostrom K. and Felbinger T., Deterministic secure direct communication using entanglement, Phys. Rev. Lett., 2001, 89(18), 187902.
[74] Wojcik A., Comment on quantum dense key distribution, Phys. Rev. A, 2005, 71, 016301.
[75] Briegel H., Dur W., VanEnk S., Cirac J., and Zoller P., Quantum repeaters: the role of imperfect local operations in quantum communication, Phys. Rev. Lett., 1998, 81, 5932.
[76] Buhrman H., Quantum fingerprinting, Phys. Rev. Lett., 2001, 87, 167902.
[77] Cai Q., The Ping-Pong protocol can be attacked without eavesdropping, Phys. Rev. Lett., 2003, 91, 109801.
[78] Cai Q. and Li B., Improving the capacity of Bostrom-Felbinger protocol, Phys. Rev. A, 2004, 69, 054301.
[79] Deng F. and Long G., Secure direct communication with a quantum one-time pad, Phys. Rev. A, 2004, 69, 052319.
[80] Kye W., Kim C., etc., Quantum key distribution with Blind Polarization Bases, Phys. Rev. Lett., 2005, 95, 040501.
[81] Zhang Q., Wang X., Chen Y., etc., Comment on quantum key distribution with blind polarization bases, Phys. Rev. Lett., 2006, 96, 078901.
[82] Lucamarini M. and Mancini S., Secure deterministic communication without entanglement, Phys. Rev. Lett., 2005, 94, 140501.
[83] Schumacher B., Quantum coding, Phys. Rev. A, 1995, 51, 2738-2747.
[84] Shor P., Scheme for reducing decoherence in computer memory, Phys. Rev. A, 1995, 52, 2493-2496.
[85] Steane A., Error correcting codes in quantum theory, Phys. Rev. Lett., 1996, 77, 793-797.
[86] Steane A., Multiple particle interference and quantum error correction, Proc. R. Soc. London A, 1996, 452, 2551-2576.
[87] Calderbank A. and Shor P., Good quantum error correcting code exist, Phys. Rev. A, 1996, 54, 1098-1103.
[88] Bennet C., Divincenro D., Smolin J. and Wooters W., Mixed state entanglement and quantum error correction, Phys. Rev. A, 1996, 54, 3824-3826.
[89] Laflamme R., Miquel C., Paz P. and Zurek W., Perfect quantum error correction code, Phys. Rev. Lett., 1996, 77, 198-202.
[90] Gottesman D., Class of quantum error-correcting code saturating the quantum Hamming bound, Phys. Rev. A, 1996, 54, 1862-1868.
[91] Gottesman D., Stabilizer codes and quantum error correction, Ph.D. thesis, California Institute of Technology, Pasadena, CA, 1997.
[92] Calderbank A., Rains E., Shor P. and Sloane N., Quantum error correction and orthogonal geometry, Phys. Rev. Lett., 1997, 78, 405-408.
[93] Calderbank A., Rains E., Shor P. and Sloane N., Quantum error correction via codes over GF(4), IEEE Trans. Inf. Theory, 1998, 44(4), 1369-1387.
[94] Ekert A. and Macchiavello C., Error correction in quantum communication, Phys. Rev. Lett., 1996, 77, 2585-2589.
[95] Knill E. and Laflamme R., Theory of quantum error-correcting codes, Phys. Rev. A, 1997, 55, 900-908.
[96] Grassl M. and Beth T., Codes for the quantum erasure channel, Phys. Rev. A, 1997, 56, 33-38.
[97] Rains E., Hardin R., Shor P. and Sloane N., Nonadditive quantum code, Phys. Rev. Lett., 1997, 79(5), 953-954.
[98] Leung D., Nielson M., Chuang I. and Yamamoto Y., Approximate quantum error correction can lead to better codes, Phys. Rev. A, 1997, 56, 2567-2573.
[99] Zanardi P and Rasetti M., Noiseless quantum, Phys. Rev. Lett., 1998, 79(17), 3306-3309.
[100] Zanardi P., Stabilizing quantum information, quant-ph/9910016, 1999.
[101] Lidar D., Chuang I. and Whaley K., Decoherence-free subspaces for quantum computation, Phys. Rev. Lett., 1998, 81(12), 2594-2597.
[102] Bacon D., Kempe J., Lidar D. and Whaley K., Universal fault tolerant computation on decoherence-free subspaces, quant-ph/9909058, 1999.
[103] Lidar D., Bacon D. and Whaley K., Concatenating decoherence free subspaces for error correction, Phys. Rev. Lett., 1999, 82(22), 4556-4559.
[104] Knill E., Laflamme R. and Viola L., Theory of quantum error correction for general noise, quant-ph/9908066, 1999.
[105] Steane A., Enlargement of Calderbank-Shor-Steane quantum codes, IEEE Trans. Inf. Theory, 1999, 45, 2492-2495.
[106] Cohen G., Encheva S. and Litsyn S., On binary construction of quantum codes, IEEE Trans. Inf. Theory, 1999, 45, 2495-2498.
[107] Steane A., Quantum Reed-Muller codes, IEEE Trans. Inf. Theory, 1999, 45, 1701-1703.
[108] Chau H., Good quantum convolution error-correction codes and their decoding algorithm exist, Phys. Rev. A, 1999, 60, 1966-1974.
[109] Vatan F., Roychowdhury V. and Anantram M., Spatially correlated qubit errors and burst-correcting quantum codes, IEEE Trans. Inf. Theory, 1999, 45(5), 1703-1708.
[110] Ashikhim A., Litsyn S. and Tsfasman M., Asymptotically good quantum codes, IEEE Trans. Inf. Theory, 2001, 47, 1206-1215.
[111] Chen H., Some good quantum error-correction codes from algebric geometric codes, IEEETrans. Inf. Theory, 2001, 47, 2059-2061.
[112] Chen H., Ling S. and Xing C., Asymptotically good quantum codes exceeding the Ashikhmin-Litsyn-Tsfasman bound, IEEE Trans. Inf. Theory, 2001, 47, 2057-2058.
[113] Chen H., Ling S. and Xing C., Quantum codes from concatenated algebric geometric codes, Preprint, 2001.
[114] Li R. and Li X., Quantum codes constructed from binary cyclic codes, International Journal of Quantum information, 2004, 2, 265-272.
[115] Li R. and Li X., Binary construction of quantum codes of minimum distance three and four, IEEE Trans. Inf. Theory, 2004, 50, 1331-1335.
[116] Gottesman D., Fault-tolerant quantum computation with higher dimensional system, quant-ph/9802007, 1998.
[117] Rains E., Nonbinary quantum codes, IEEE Trans. Inf. Theory, 1999, 45(7), 2489-2492.
[118] Klappenecker A., Beyond Stabilizer Codes I: Nice Error Bases, IEEE Trans. Inf. Theory, 2002, 48(8), 2392-2395.
[119] Grassl M., Geiselmann W. and Beth T., Quantum Reed-Solomon codes, AAECC 1999, quant-ph/9910059, 1999.
[120] Grassl M. and Beth T., Quantum BCH codes, 1999, quant-ph/9910060, 1999.
[121] Rains E., Nonbinary quantum codes, IEEE Trans. Inf. Theory, 1999, 45, 1827-1832.
[122] Aharonov D. and Ben-Or M., Fault-tolerant quantum computation with constant error rate, quant-ph/9906129, 1999.
[123] Bierbrauer J. and Edel Y., Quantum twisted codes, J. Comb. Designs, 2000, 8, 174-178.
[124] Ashikhim A. and Knill E., Nonbinary stabilizer codes, IEEE Trans. Inf. Theory, 2001, 47, 3065-3072.
[125] Schlingemann D. and Werner R., Quantum error-correcting codes associated with graphs, Phys. Rev. A, 2002, 65, 012308.
[126] Tonchev V., Varshamov-Gilbert bound for a class of formally self-dual codes and related quantum code, IEEE Trans. Inf. Theory, 2002, 48, 975-977.
[127] Feng K., Quantum codes [[6,2,3]]p [[7,3,3]]p exist, IEEE Trans. Inf. Theory, 2002, 48, 2384-2391.
[128] Feng K. and Ma Z., A finite Gilbert-Varshamov bound for pure stabilizer quantum codes, IEEE Trans. Inf. Theory, 2004, 50, 1369-1387.
[129] MacKay D. J. C., Mitchison G. and McFadden P. L., Sparse graph codes for quantum error- correction, IEEE Trans. Inf. Theory, 2004, 50, 2315-2330.
[130] Kribs D. W., Laflamme R. and Poulin D., Unified and generalized approach to quantum error correction, Phys. Rev. Lett., 2005, 94, 180501.
[131] Forney G. D., Grassl M. and Guha S., Convolutional and tail-biting quantum error- correcting codes, IEEE Trans. Inf. Theory, 2007, 53, 865-880.
[132] Kschischang F. and Pasupathy S., Some ternary and quaternary codes and associated sphere packings, IEEE Trans. Inf. Theory, 1992, 38(2), 227-246.
[133] MacWilliams F. J. and Sloane N. J. A., The Theory of Error-Correcting Codes, New York: North-Holland, 1977, p310.
[2] Ashikhmin A., Remarks on bounds for quantum codes, quant-ph/9705037, 1997.
[3] Ashikhmin A. and Litsyn S., Upper bounds on the size of quantum codes, IEEE Trans. Inf. Theory, 1999, 45(4), 1206-1215.
[4] Rains E., Quantum weight enumerators, IEEE Trans. Inf. Theory, 1998, 44 (4), 1388-1394.
[5] Rains E., Monotonicity of the quantum linear programming bound, IEEE Trans. Inf. Theory, 1999, 45(7), 2489-2492.
[6] Rains E., Quantum shadow enumerators, IEEE Trans. Inf. Theory, 1999, 45(7), 2361-2366.
[7] Matsmoto R., Improvement of Ashikhmin-Litsyn-Tsfasman bound for quantum codes, IEEE Trans. Inf. Theory, 2002, 48, 2122-2124.
[8] Bennet C., Bessette G., Brassard S. and Wiesner S., Quantum cryptograph, Advance in cryptograph: Proceedings of Crypto. 82, Plenum Press, New York, 1982, 267-275.
[9] Bennet C. and Brassard G., Quantum cryptography: public key distribution and coin tossing, Proceedings of IEEE International Conference on Computers, Systems & Signal Processing, Bangalore, India, December 10-12, 1984, 175-179.
[10] Bennet C., Brassard G. and Robert J., Privacy amplification by public discussion, SIAM J. Comp., 1988, 7, 210-229.
[11] Deutch D., Quantum communication thwarts the eavesdroppers, New Scientist, 1989, 25-26.
[12] Ekert A., Quantum cryptography based on Bell theorem, Phys. Rev. Lett., 1991, 67, 661-663.
[13] Bennet C., Quantum cryptography using any two non-orthogonal states, Phys. Rev. Lett., 1992, 68, 3121-3124.
[14] Bennet C., Charles H., Brassard G., etc, Quantum cryptography, Scientist American, October 1992, 50-57.
[15] Ekert A. and Palma G., Quantum cryptography with interferometer quantum entanglement, J. Mod. Opt., 1994, 41(12), 2413-2423.
[16] Bennett C., Brassard G., Crepeau C., etc, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett., 1993, 70(13), 1895-1899.
[17] Ekert A., Hutmer B., and Palma G., Eavesdropping on quantum cryptographical systems, Phys. Rev. A, 1994, 50(2), 1047-1056.
[18] Bennet C. and Phoenix A., Information-theoretic limits to quantum cryptography, Phys. Rev. A, 1993, 48(1), R5-R8.
[19] Brassard G., A bibliography of quantum cryptography, Sigact News, 1993, 24(3), 16-20.
[20] Ekert A. and Palma G., Quantum cryptography with the interferometer quantum entanglement, J. Mod. Opt. 1992, 41(12), 2413-2423.
[21] Schumacher B. and Westmoreland M., Quantum privacy and quantum coherence, Phys. Rev. Lett., 1998, 80(25), 5695-5697.
[22] Townsend P., Rarity J., and Tapster P., Enhanced single photon fringe visibility in a 10km long prototype quantum cryptography channel, Electron. Lett., 1993, 29, 1291-1293.
[23] Townsend P., Secure key distribution system based on quantum cryptography, Electron. Lett., 1994, 30, 809-811.
[24] Townsend P., Rarity J., and Tapster P., Single photon interference in a 10km long optical fiber interferometer, Electron. Lett., 1995, 19, 634-639.
[25] Marand C. and Townsend P., Quantum key distribution over distances as long as 30km, Optics Lett., 1995, 20, 1695-1697.
[26] Muller A., Breguet J., and Gisin N., Experimental demonstration of quantum cryptography using polarized photons in optical fiber over more than 1km, Euro. Phys. Lett., 1993, 23, 383-388.
[27] Muller A., Zbinden H., and Gisin N., Underwater quantum coding, Nature, 1995, 378, 449-449.
[28] Muller A., Zbinden H., and Gisin N., Quantum cryptography over 23km in installed under-lake telecom fiber, Euro. Phys. Lett., 1995, 33, 335-339.
[29] Muller A., Herzog T., Huttner B., etc., Plug and play systems for quantum cryptography, Appl. Phys. Lett., 1997, 70, 793-795.
[30] Hughes R., Morgan R., and Peterson C., Quantum key distribution over a 48km optical fiber network, Los Alamos report, LA-UR-99-1593, quantum-ph/9904038.
[31] Hughes R., Morgan R., and Peterson C., Quantum key distribution over a 48km optical fiber network, Journal of Modern Optics, 2000, 47, 533-547.
[32] Reid M., Quantum cryptography with a predetermined key using continuous variable EPR correlations, Phys. Rev. A, 2000, 62, 062308.
[33] Pereira S., Ou Z., and Kimble H., Quantum communications with correlated non-classical states, Phys. Rev. A, 2000, 62, 042311.
[34] Kurtsiefer C., Zarda P., Halder M., etc., A step towards global key distribution, Nature, 2002, 419-420.
[35] Stucki D., Gisin N., Guinnard O., Ribordy G., and Zbinden H., Quantum key distribution over 67km with a plug and play system, New Journal of Physics, 2002, 4, 411-418.
[36] Kosaka H., Tomita A., Namba Y., etc., Single-photon interference experiment over 100km forquantum cryptography system using balanced gated-mode photon detector, Electron. Lett., 2003, 39, 1199-1201.
[37] Gobby C., Yuan Z., and Shields A., Quantum key distribution over 122km of standard telecom fiber, Appl. Phys. Lett., 2004, 84, 3762-3764.
[38] Gordon K., Fernandez V., Townsend P., and Butler Q., A short wavelength GigaHertz clocked fiber-optic quantum key distribution system, IEEE Journal of quantum electronics, 2004, 40(7), 900-908.
[39] Ralph T., Continuous variable quantum cryptography, Phys. Rev. A, 1999, 61, 010303.
[40] Ralph T., Security of continuous variable quantum cryptography, Phys. Rev. A, 2000, 62, 062306.
[41] Hillery M., Quantum cryptography with squeeze states, Phys. Rev. A, 2000, 62, 022309.
[42] Braunstein S. and Kimble H., Teleportation of continuous quantum variables, Phys. Rev. Lett., 1998, 80, 869-872.
[43] Braunstein S., Error correction for continuous quantum variable, Phys. Rev. Lett., 1998, 80, 4084-4087.
[44] Lloyd S. and Slotine J., Analog quantum error correction, Phys. Rev. Lett., 1998, 80, 4088-4091.
[45] Braunstein S. and Kimble H., Dense coding for continuous quantum variable, Phys. Rev. A, 2000, 61, 042302.
[46] Zeng G. and Zhang W., Identity verification in quantum cryptography, Phys. Rev. A, 2000, 61, 032303.
[47] Ljunggren D., Bourennane M., and Karlsson A., Authority-based user authentication in quantum key distribution, Phys. Rev. A, 2000, 62, 022305.
[48] Curty M. and Santos D., Quantum authentication of classical messages, Phys. Rev. A, 2001, 64, 062309.
[49] Dusek M., Haderka O., and Hendrych M., Quantum identification system, Phys. Rev. A, 1999, 60, 149-156.
[50] Mihara T., Quantum identification schemes with entanglement, Phys. Rev. A, 2002, 65, 052306.
[51] Dam W., Comment on quantum identification schemes with entanglement, Phys. Rev. A, 2003, 68, 026301.
[52] Li X. and Banrum H., Quantum authentication using entangled states, Int. J. Found. Com. Sci., 2004, 15, 609.
[53] Garcia P., Femandez E., Perez E., etc., New encoding schemes for quantum authentication, Quantum Inf.&Comp., 2005, 5(1), 1-12.
[54] Winter A., Quantum and classical message protect identification via quantum channels, Quantum Inf.&Comp., 2004, 4(6-7), 563-578.
[55] Cleve R., Gottesman D., and Lo H., How to share a quantum secret, Phys. Rev. Lett., 1999, 65, 648-651.
[56] Rains E., Quantum codes of minimum distance two, IEEE Trans. Inf. Theory, 1999, 45, 266-271.
[57] Guo Y. and Zeng G., Quantum BCH codes based on spectral techniques, Communications in Theoretical Physics, 2006, 45(2), 283-286.
[58] Guo Y. and Zeng G., Quantum convolution codes based on the stabilizer of linear codes, IEEE APCC2005, Perth, Western Australia, October 2005.
[59] Guo Y. and Zeng G., Message authentication scheme based on quantum error-correction codes, First international conference on scalable system, Hong-Kong, May29 2006.
[60] Duan L. and Guo G., Preserving coherence in quantum computation by pairing quantum bits. Phys. Rev. Lett., 1997, 79(10), 1953-1956.
[61] Duan L. and Guo G., etc., Reducing decoherence in quantum-computer memory with all quantum bits coupling to the same environment. Phys. Rev. A, 1998, 57(2), 737-741.
[62] Duan L. and Guo G., Scheme for reducing collective decoherence in quantum memory, Phys. Lett. A, 1998, 43(5-6), 265-269.
[63] Duan L. and Guo G., Prevention of dissipation with two particles, Phys. Rev. A, 1998, 57(4), 2399-2402.
[64] Duan L. and Guo G., A probabilistic cloning machine for replicating two non-orthogonal states, Phys. Lett. A, 1998, 43(5), 261-264.
[65] Duan L. and Guo G., Probabilistic cloning and identification of linearly independent quantum states, Phys. Rev. Lett., 1998, 80(22), 4999-5002.
[66] Marcos C. and David J., Quantum authentication of classical messages, Phys. Rev. A, 2001, 64, 062309.
[67] Lo H. and Chau H., Unconditional security of quantum key distribution over arbitrarily long distance, Science, 1999, 283, 2050-2056.
[68] Hwang W., Matsumoto K., Imai H., Kim J., and Lee H., Shor-Preskill-type security proof for concatenated BB84 quantum-key-distribution protocol, Phys. Rev. A, 2003, 67, 024302.
[69] Wootters W. and Zurek W., A single quantum cannot be cloned, Nature, 1982, 299, 802-803.
[70] Chau H., Quantum convolutional error-correcting codes, Phys. Rev. A, 1998, 58, 905-909.
[71] Gottesman D. and Preskill J., Secure quantum key distribution using squeezed states, Phys. Rev. A, 2001, 63, 022309.
[72] Bourennane M., Karlsson A., and Bjork G., Quantum key distribution using multilevelencoding, Phys. Rev. A, 2001, 64, 012306.
[73] Bostrom K. and Felbinger T., Deterministic secure direct communication using entanglement, Phys. Rev. Lett., 2001, 89(18), 187902.
[74] Wojcik A., Comment on quantum dense key distribution, Phys. Rev. A, 2005, 71, 016301.
[75] Briegel H., Dur W., VanEnk S., Cirac J., and Zoller P., Quantum repeaters: the role of imperfect local operations in quantum communication, Phys. Rev. Lett., 1998, 81, 5932.
[76] Buhrman H., Quantum fingerprinting, Phys. Rev. Lett., 2001, 87, 167902.
[77] Cai Q., The Ping-Pong protocol can be attacked without eavesdropping, Phys. Rev. Lett., 2003, 91, 109801.
[78] Cai Q. and Li B., Improving the capacity of Bostrom-Felbinger protocol, Phys. Rev. A, 2004, 69, 054301.
[79] Deng F. and Long G., Secure direct communication with a quantum one-time pad, Phys. Rev. A, 2004, 69, 052319.
[80] Kye W., Kim C., etc., Quantum key distribution with Blind Polarization Bases, Phys. Rev. Lett., 2005, 95, 040501.
[81] Zhang Q., Wang X., Chen Y., etc., Comment on quantum key distribution with blind polarization bases, Phys. Rev. Lett., 2006, 96, 078901.
[82] Lucamarini M. and Mancini S., Secure deterministic communication without entanglement, Phys. Rev. Lett., 2005, 94, 140501.
[83] Schumacher B., Quantum coding, Phys. Rev. A, 1995, 51, 2738-2747.
[84] Shor P., Scheme for reducing decoherence in computer memory, Phys. Rev. A, 1995, 52, 2493-2496.
[85] Steane A., Error correcting codes in quantum theory, Phys. Rev. Lett., 1996, 77, 793-797.
[86] Steane A., Multiple particle interference and quantum error correction, Proc. R. Soc. London A, 1996, 452, 2551-2576.
[87] Calderbank A. and Shor P., Good quantum error correcting code exist, Phys. Rev. A, 1996, 54, 1098-1103.
[88] Bennet C., Divincenro D., Smolin J. and Wooters W., Mixed state entanglement and quantum error correction, Phys. Rev. A, 1996, 54, 3824-3826.
[89] Laflamme R., Miquel C., Paz P. and Zurek W., Perfect quantum error correction code, Phys. Rev. Lett., 1996, 77, 198-202.
[90] Gottesman D., Class of quantum error-correcting code saturating the quantum Hamming bound, Phys. Rev. A, 1996, 54, 1862-1868.
[91] Gottesman D., Stabilizer codes and quantum error correction, Ph.D. thesis, California Institute of Technology, Pasadena, CA, 1997.
[92] Calderbank A., Rains E., Shor P. and Sloane N., Quantum error correction and orthogonal geometry, Phys. Rev. Lett., 1997, 78, 405-408.
[93] Calderbank A., Rains E., Shor P. and Sloane N., Quantum error correction via codes over GF(4), IEEE Trans. Inf. Theory, 1998, 44(4), 1369-1387.
[94] Ekert A. and Macchiavello C., Error correction in quantum communication, Phys. Rev. Lett., 1996, 77, 2585-2589.
[95] Knill E. and Laflamme R., Theory of quantum error-correcting codes, Phys. Rev. A, 1997, 55, 900-908.
[96] Grassl M. and Beth T., Codes for the quantum erasure channel, Phys. Rev. A, 1997, 56, 33-38.
[97] Rains E., Hardin R., Shor P. and Sloane N., Nonadditive quantum code, Phys. Rev. Lett., 1997, 79(5), 953-954.
[98] Leung D., Nielson M., Chuang I. and Yamamoto Y., Approximate quantum error correction can lead to better codes, Phys. Rev. A, 1997, 56, 2567-2573.
[99] Zanardi P and Rasetti M., Noiseless quantum, Phys. Rev. Lett., 1998, 79(17), 3306-3309.
[100] Zanardi P., Stabilizing quantum information, quant-ph/9910016, 1999.
[101] Lidar D., Chuang I. and Whaley K., Decoherence-free subspaces for quantum computation, Phys. Rev. Lett., 1998, 81(12), 2594-2597.
[102] Bacon D., Kempe J., Lidar D. and Whaley K., Universal fault tolerant computation on decoherence-free subspaces, quant-ph/9909058, 1999.
[103] Lidar D., Bacon D. and Whaley K., Concatenating decoherence free subspaces for error correction, Phys. Rev. Lett., 1999, 82(22), 4556-4559.
[104] Knill E., Laflamme R. and Viola L., Theory of quantum error correction for general noise, quant-ph/9908066, 1999.
[105] Steane A., Enlargement of Calderbank-Shor-Steane quantum codes, IEEE Trans. Inf. Theory, 1999, 45, 2492-2495.
[106] Cohen G., Encheva S. and Litsyn S., On binary construction of quantum codes, IEEE Trans. Inf. Theory, 1999, 45, 2495-2498.
[107] Steane A., Quantum Reed-Muller codes, IEEE Trans. Inf. Theory, 1999, 45, 1701-1703.
[108] Chau H., Good quantum convolution error-correction codes and their decoding algorithm exist, Phys. Rev. A, 1999, 60, 1966-1974.
[109] Vatan F., Roychowdhury V. and Anantram M., Spatially correlated qubit errors and burst-correcting quantum codes, IEEE Trans. Inf. Theory, 1999, 45(5), 1703-1708.
[110] Ashikhim A., Litsyn S. and Tsfasman M., Asymptotically good quantum codes, IEEE Trans. Inf. Theory, 2001, 47, 1206-1215.
[111] Chen H., Some good quantum error-correction codes from algebric geometric codes, IEEETrans. Inf. Theory, 2001, 47, 2059-2061.
[112] Chen H., Ling S. and Xing C., Asymptotically good quantum codes exceeding the Ashikhmin-Litsyn-Tsfasman bound, IEEE Trans. Inf. Theory, 2001, 47, 2057-2058.
[113] Chen H., Ling S. and Xing C., Quantum codes from concatenated algebric geometric codes, Preprint, 2001.
[114] Li R. and Li X., Quantum codes constructed from binary cyclic codes, International Journal of Quantum information, 2004, 2, 265-272.
[115] Li R. and Li X., Binary construction of quantum codes of minimum distance three and four, IEEE Trans. Inf. Theory, 2004, 50, 1331-1335.
[116] Gottesman D., Fault-tolerant quantum computation with higher dimensional system, quant-ph/9802007, 1998.
[117] Rains E., Nonbinary quantum codes, IEEE Trans. Inf. Theory, 1999, 45(7), 2489-2492.
[118] Klappenecker A., Beyond Stabilizer Codes I: Nice Error Bases, IEEE Trans. Inf. Theory, 2002, 48(8), 2392-2395.
[119] Grassl M., Geiselmann W. and Beth T., Quantum Reed-Solomon codes, AAECC 1999, quant-ph/9910059, 1999.
[120] Grassl M. and Beth T., Quantum BCH codes, 1999, quant-ph/9910060, 1999.
[121] Rains E., Nonbinary quantum codes, IEEE Trans. Inf. Theory, 1999, 45, 1827-1832.
[122] Aharonov D. and Ben-Or M., Fault-tolerant quantum computation with constant error rate, quant-ph/9906129, 1999.
[123] Bierbrauer J. and Edel Y., Quantum twisted codes, J. Comb. Designs, 2000, 8, 174-178.
[124] Ashikhim A. and Knill E., Nonbinary stabilizer codes, IEEE Trans. Inf. Theory, 2001, 47, 3065-3072.
[125] Schlingemann D. and Werner R., Quantum error-correcting codes associated with graphs, Phys. Rev. A, 2002, 65, 012308.
[126] Tonchev V., Varshamov-Gilbert bound for a class of formally self-dual codes and related quantum code, IEEE Trans. Inf. Theory, 2002, 48, 975-977.
[127] Feng K., Quantum codes [[6,2,3]]p [[7,3,3]]p exist, IEEE Trans. Inf. Theory, 2002, 48, 2384-2391.
[128] Feng K. and Ma Z., A finite Gilbert-Varshamov bound for pure stabilizer quantum codes, IEEE Trans. Inf. Theory, 2004, 50, 1369-1387.
[129] MacKay D. J. C., Mitchison G. and McFadden P. L., Sparse graph codes for quantum error- correction, IEEE Trans. Inf. Theory, 2004, 50, 2315-2330.
[130] Kribs D. W., Laflamme R. and Poulin D., Unified and generalized approach to quantum error correction, Phys. Rev. Lett., 2005, 94, 180501.
[131] Forney G. D., Grassl M. and Guha S., Convolutional and tail-biting quantum error- correcting codes, IEEE Trans. Inf. Theory, 2007, 53, 865-880.
[132] Kschischang F. and Pasupathy S., Some ternary and quaternary codes and associated sphere packings, IEEE Trans. Inf. Theory, 1992, 38(2), 227-246.
[133] MacWilliams F. J. and Sloane N. J. A., The Theory of Error-Correcting Codes, New York: North-Holland, 1977, p310.