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On Some Questions of C. Ampadu Associated with the Quantum Random Walk

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DOI: 10.4236/am.2014.519291    2,924 Downloads   3,561 Views  
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ABSTRACT

We review (not exhaustively) the quantum random walk on the line in various settings, and propose some questions that we believe have not been tackled in the literature. In a sense, this article invites the readers (beginner, intermediate, or advanced), to explore the beautiful area of quantum random walks.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Ampadu, C. (2014) On Some Questions of C. Ampadu Associated with the Quantum Random Walk. Applied Mathematics, 5, 3040-3066. doi: 10.4236/am.2014.519291.

References

[1] Aharonov, Y., Davidovich, L. and Zagury, N. (1993) Quantum Random Walks. Physical Review A, 48, 1687.
http://dx.doi.org/10.1103/PhysRevA.48.1687
[2] Hughes, B.D. (1995) Volume 1: Random Walks and Random Environments. In: Random Walks, Oxford University Press, Oxford.
[3] Ambainis, A., Bach, E., Nayak, A., Vishwanath, A. and Watrous, J. (2001) One Dimensional Quantum Walks. Proceedings of the 33rd Annual ACM Symposium on the Theory of Computing, Heraklion, 6-8 July 2001, 50.
[4] Nayak, A. and Vishwanath, A. (2000) Quantum Walk on the Line. arXiv: Quant-ph/0010117.
[5] Childs, A.M., Farhi, E. and Gutmann, S. (2002) An Example of the Difference between Quantum and Classical Random Walks. Quantum Information Processing, 1, 35-43.
http://dx.doi.org/10.1023/A:1019609420309
[6] Fahri, E. and Gutmann, S. (1998) Quantum Computation and Decision Trees. Physical Review A, 58, 915.
[7] Meyer, D. (1996) From Quantum Cellular Automata to Quantum Lattice Gases. Journal of Statistical Physics, 85, 551-574.
http://dx.doi.org/10.1007/BF02199356
[8] Strauch, F.W. (2006) Connecting the Discrete- and Continuous-Time Quantum Walks. Physical Review A, 74, Article ID: 030301.
http://dx.doi.org/10.1103/PhysRevA.74.030301
[9] Chandrashekar, C.M. (2008) Generic Quantum Walk Using a Coin-Embedded Shift Operator. Physical Review A, 78, Article ID: 052309.
http://dx.doi.org/10.1103/PhysRevA.78.052309
[10] Childs, A.M. (2010) On the Relationship between Continuous- and Discrete-Time Quantum Walk. Communications in Mathematical Physics, 294, 581-603.
http://dx.doi.org/10.1007/s00220-009-0930-1
[11] Aharonov, D., et al. (2001) Quantum Walks on Graphs. Proceedings of the 33rd Annual ACM Symposium on the Theory of Computing, Heraklion, 6-8 July 2001, 50.
[12] Bach, E., Coppersmith, S., Goldschen, M.P., Joynt, R. and Watrous, J. (2004) One Dimensional Quantum Walks with Absorbing Boundaries. Journal of Computer and System Sciences, 69, 562-592.
http://dx.doi.org/10.1016/j.jcss.2004.03.005
[13] Dür, W., Raussendorf, R., Kendon, V.M. and Briegel, H.-J. (2002) Quantum Random Walks in Optical Lattices. Physical Review A, 66, Article ID: 052139.
http://dx.doi.org/10.1103/PhysRevA.66.052319
[14] Kempe, J. (2005) Quantum Random Walks Hit Exponentially Faster. Probability Theory and Related Fields, 133, 215-235.
http://dx.doi.org/10.1007/s00440-004-0423-2
[15] Konno, N. (2005) A New Type of Limit Theorems for the One-Dimensional Quantum Random Walk. Journal of the Mathematical Society of Japan, 57, 1179-1195.
http://dx.doi.org/10.2969/jmsj/1150287309
[16] Konno, N., Namiki, T. and Soshi, T. (2004) Symmetry of Distribution for the One-Dimensional Hadamard Walk. Interdisciplinary Information Sciences, 10, 11-22.
http://dx.doi.org/10.4036/iis.2004.11
[17] Mackay, T.D., Bartlett, S.D., Stephenson, L.T. and Sanders, B.C. (2002) Quantum Walks in Higher Dimensions. Journal of Physics A: Mathematical and General, 35, 2745-2753.
http://dx.doi.org/10.1088/0305-4470/35/12/304
[18] Moore, C. and Russell, A. (2002) Quantum Walks on the Hypercubes, Randomization and Approximation Techniques in Computer Science. Lecture Notes in Computer Science, 2483, 164-178.
[19] Travaglione, B.C. and Milburn, G.J. (2002) Implementing the Quantum Random Walk. Physical Review A, 65, Article ID: 032310.
http://dx.doi.org/10.1103/PhysRevA.65.032310
[20] Yamasaki, T., Kobayashi, H. and Imai, H. (2003) Analysis of Absorbing Times of Quantum Walks. Physical Review A, 68, Article ID: 012302.
http://dx.doi.org/10.1103/PhysRevA.68.012302
[21] Aharonov, D., Ambainis, A., Kempe, J. and Vazirani, U. (2001) Quantum Walks on Graphs. Proceedings of the 33rd STOC, New York, 50-59.
[22] Shenvi, N., Kempe, J. and Whaley, K. (2003) Quantum Random-Walk Search Algorithm. Physical Review A, 67, Article ID: 052307.
http://dx.doi.org/10.1103/PhysRevA.67.052307
[23] Ambainis, A. (2007) Quantum Walk Algorithm for Element Distinctness. SIAM Journal on Computing, 37, 210-239.
http://dx.doi.org/10.1137/S0097539705447311
[24] Childs, A. and Goldstone, J. (2004) Spatial Search by Quantum Walk. Physical Review A, 70, Article ID: 022314.
http://dx.doi.org/10.1103/PhysRevA.70.022314
[25] Kendon, V. (2006) A Random Walk Approach to Quantum Algorithms. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 364, 3407-3422.
http://dx.doi.org/10.1098/rsta.2006.1901
[26] Gabris, A., Kiss, T. and Jex, I. (2007) Scattering Quantum Random-Walk Search with Errors. Physical Review A, 76, Article ID: 062315.
http://dx.doi.org/10.1103/PhysRevA.76.062315
[27] Magniez, F., Nayak, A., Roland, J. and Santha, M. (2007) Search via Quantum Walk. Proceedings of the 33rd STOC, New York, 575-584.
[28] Reitzner, D., Hillery, M., Feldman, E. and Buzek, V. (2009) Quantum Searches on Highly Symmetric Graphs. Physical Review A, 79, Article ID: 012323.
http://dx.doi.org/10.1103/PhysRevA.79.012323
[29] Potocek, V., Gabris, A., Kiss, T. and Jex, I. (2009) Optimized Quantum Random-Walk Search Algorithms on the Hypercube. Physical Review A, 79, Article ID: 012325.
http://dx.doi.org/10.1103/PhysRevA.79.012325
[30] Hein, B. and Tanner, G. (2009) Quantum Search Algorithms on the Hypercube. Journal of Physics A: Mathematical and Theoretical, 42, Article ID: 085303.
http://dx.doi.org/10.1088/1751-8113/42/8/085303
[31] Hein, B. and Tanner, G. (2010) Quantum Search Algorithms on a Regular Lattice. Physical Review A, 82, Article ID: 012326.
http://dx.doi.org/10.1103/PhysRevA.82.012326
[32] Travaglione, B.C. and Milburn, G.J. (2002) Implementing the Quantum Random Walk. Physical Review A, 65, Article ID: 032310.
http://dx.doi.org/10.1103/PhysRevA.65.032310
[33] Dur, W., Raussendorf, R., Kendon, V. and Briegel, H.-J. (2002) Quantum Walks in Optical Lattices. Physical Review A, 66, Article ID: 052319.
http://dx.doi.org/10.1103/PhysRevA.66.052319
[34] Du, J., Li, H., Xu, X., Shi, M., Wu, J., Zhou, X. and Han, R. (2003) Experimental Implementation of Quantum Random-Walk Algorithm. Physical Review A, 67, Article ID: 042316.
http://dx.doi.org/10.1103/PhysRevA.67.042316
[35] Ryan, C.A., Laforest, M., Boileau, J.C. and Laflamme, R. (2005) Experimental Implementation of a Discrete-Time Quantum Random Walk on an NMR Quantum-Information Processor. Physical Review A, 72, Article ID: 062317.
http://dx.doi.org/10.1103/PhysRevA.72.062317
[36] Xue, P., Sanders, B.C., Blais, A. and Lalumiere, K. (2008) Quantum Walks on Circles in Phase Space via Superconducting Circuit Quantum Electrodynamics. Physical Review A, 78, Article ID: 042334.
http://dx.doi.org/10.1103/PhysRevA.78.042334
[37] Witthaut, D. (2010) Quantum Walks and Quantum Simulations with Bloch-Oscillating Spinor Atoms. Physical Review A, 82, Article ID: 033602.
http://dx.doi.org/10.1103/PhysRevA.82.033602
[38] Schreiber, A., Cassemiro, K.N., Potocek, V., Gabris, A., Mosley, P.J., Andersson, E., Jex, I. and Silberhorn, C. (2010) Photons Walking the Line: A Quantum Walk with Adjustable Coin Operations. Physical Review Letters, 104, Article ID: 050502.
http://dx.doi.org/10.1103/PhysRevLett.104.050502
[39] Xue, P., Sanders, B.C. and Leibfried, D. (2009) Quantum Walk on a Line for a Trapped Ion. Physical Review Letters, 103, Article ID: 183602.
http://dx.doi.org/10.1103/PhysRevLett.103.183602
[40] Zhao, Z., Du, J., Li, H., Yang, T., Chen, Z.-B. and Pan, J.-W. (2002) Implement Quantum Random Walks with Linear Optics Elements. arXiv: Quant-ph/0212149.
[41] Konno, N. (2002) Quantum Random Walks in One Dimension. Quantum Information Processing, 1, 345-354.
http://dx.doi.org/10.1023/A:1023413713008
[42] Venagas-Andraca, S.E., Ball, J., Burnett, K. and Bose, S. (2005) Quantum Walks with Entangle Coins. New Journal of Physics, 7, 221.
http://dx.doi.org/10.1088/1367-2630/7/1/221
[43] Bednarska, M., Grudka, A., Kurzynski, P., Luczak, T. and Wojcik, A. (2003) Quantum Walks on Cycles. Physics Letters A, 317, 21-25.
http://dx.doi.org/10.1016/j.physleta.2003.08.023
[44] Omar, Y., Paunkovic, N., Sheridian, L. and Bose, S. (2006) Quantum Walk on a Line with Two Entangled Particles. Physical Review A, 74, Article ID: 042304.
http://dx.doi.org/10.1103/PhysRevA.74.042304
[45] Oliveira, A., Portugal, R. and Donangelo, R. (2006) Decoherence in Two Dimensional Quantum Walks. Physical Review A, 74, Article ID: 012312.
[46] Watabe, K., Kobayashi, N., Katori, M. and Konno, N. (2008) Limit Distributions of Two Dimensional Quantum Walks. Physical Review A, 77, Article ID: 062331.
http://dx.doi.org/10.1103/PhysRevA.77.062331
[47] Adamczak, W., Andrew, K., Bergen, L., Ethier, D., Hernberg, P., Lin, J. and Tamon, C. (2007) Non-Uniform Mixing of Quantum Walk on Cycles. International Journal of Quantum Information, 5, 781.
http://dx.doi.org/10.1142/S0219749907003195
[48] Mackay, T.D., Bartlett, S., Stephenson, L. and Sanders, B. (2002) Quantum Walks in Higher Dimensions. Journal of Physics A, 35, 2745-2753.
http://dx.doi.org/10.1088/0305-4470/35/12/304
[49] Carneiro, I., Loo, M., Xu, X., Girerd, M., Kendon, V. and Knight, P. (2005) Entanglement in Coined Quantum Walks on Regular Graphs. New Journal of Physics, 7, 156.
http://dx.doi.org/10.1088/1367-2630/7/1/156
[50] Romanelli, A., Sicardi-Schifino, A.C., Siri, R., Abal, G., Auyuanet, A. and Donangelo, R. (2004) Quantum Random Walk on the Line as Markovian Process. Physica A: Statistical Mechanics and Its Applications, 338, 395-405.
http://dx.doi.org/10.1016/j.physa.2004.02.061
[51] Venegas-Andraca, S. (2012) Quantum Walks: A Comprehensive Review. Quantum Information Processing, 11, 1015-1106.
http://dx.doi.org/10.1007/s11128-012-0432-5
[52] Shikano, Y. (2013) From Discrete Time Quantum Walk to Continuous Time Quantum Walk in Limit Distribution. Journal of Computational and Theoretical Nanoscience, 10, 1558-1570.
http://dx.doi.org/10.1166/jctn.2013.3097
[53] Kempe, J. (2003) Quantum Random Walks: An Introductory Overview. Contemporary Physics, 44, 307-327.
http://dx.doi.org/10.1080/00107151031000110776
[54] Kendon, V. (2007) Decoherence in Quantum Walks: A Review. Mathematical Structures in Computer Science, 17, 1169-1220.
[55] Venegas-Andracas, S.E. (2008) Quantum Walks for Computer Scientists. Morgan and Claypool Publishers.
http://dx.doi.org/10.2200/S00144ED1V01Y200808QMC001
[56] Konno, N. (2008) Quantum Walks. In: Franz, U. and Schurmann, M., Eds., Quantum Potential Theory, Springer-Verlag, Heidelberg, 309-452.
[57] Konno, N. (2005) A Path Integral Approach for Disordered Quantum Walks in One Dimensions. Fluctuation and Noise Letters, 5, No. 4.
http://dx.doi.org/10.1142/S0219477505002987
[58] Mackay, T.D., Bartlett, S.D., Stephanson, L.T. and Sanders, B.C. (2002) Quantum Walks in Higher Dimensions. Journal of Physics A: Mathematical and General, 35, 2745-2753.
[59] Schijven, P, Kohlberger, J., Blumen, A. and Muelken, O. (2011) Transport Efficiency in Topologically Disordered Networks with Environmentally Induced Diffusion. Journal of Physics A: Mathematical and Theoretical, 45, Article ID: 215003.
[60] Lavicka, H., Potocek, V., Kiss, T., Lutz, E. and Jex, I. (2011) Quantum Walk with Jumps. European Physical Journal D, 64, 119-129.
http://dx.doi.org/10.1140/epjd/e2011-20138-8
[61] Abasto, D.F., Mohseni, M., Lloyd, S. and Zanardi, P. (2012) Exciton Diffusion Length in Complex Quantum Systems: The Effects of Disorder and Environmental Fluctuations on Symmetry-Enhanced Supertransfer. Philosophical Transactions of the Royal Society A, 370, 3750-3770.
http://dx.doi.org/10.1098/rsta.2011.0213
[62] Obuse, H. and Kawakami, N. (2011) Topological Phases and Delocalization of Quantum Walks in Random Environments. Physical Review B, 84, Article ID: 195139.
http://dx.doi.org/10.1103/PhysRevB.84.195139
[63] Chandrashekar, C.M. (2011) Quantum Walk through Lattice with Temporal, Spatial and Fluctuating Disordered Operations. arXiv:1103.2704.
[64] Wootton, J. and Pachos, J. (2011) Bringing Order through Disorder: Localization of Errors in Topological Quantum Memories. Physical Review Letters, 107, Article ID: 030503.
http://dx.doi.org/10.1103/PhysRevLett.107.030503
[65] Schreiber, A., Cassemiro, K.N., Potocek, V., Gábris, A., Jex, I. and Silberhorn, C. (2011) Decoherence and Disorder in Quantum Walks: From Ballistic Spread to Localization. Physical Review Letters, 106, Article ID: 180403.
http://dx.doi.org/10.1103/PhysRevLett.106.180403
[66] Ahlbrecht, A., Scholz, V.B. and Werner, A.H. (2011) Disordered Quantum Walks in One Lattice Dimension. Journal of Mathematical Physics, 52, Article ID: 102201.
http://dx.doi.org/10.1063/1.3643768
[67] Chandrashekar, C.M. (2011) Disordered Quantum Walk-Induced Localization of a Bose-Einstein Condensate. Physical Review A, 83, Article ID: 022320.
http://dx.doi.org/10.1103/PhysRevA.83.022320
[68] Leung, G., Knott, P., Bailey, J. and Kendon, V. (2010) Coined Quantum Walks on Percolation Graphs. New Journal of Physics, 12, Article ID: 123018.
http://dx.doi.org/10.1088/1367-2630/12/12/123018
[69] Monthus, C. and Garel, T. (2009) An Eigenvalue Method to Compute the Largest Relaxation Time of Disordered Systems. Journal of Statistical Mechanics: Theory and Experiment, 2009, 12017.
http://dx.doi.org/10.1088/1742-5468/2009/12/P12017
[70] Yin, Y., Katsanos, D.E. and Evangelou, S.N. (2008) Quantum Walks on a Random Environment. Physical Review A, 77, Article ID: 022302.
http://dx.doi.org/10.1103/PhysRevA.77.022302
[71] Mülken, O., Bierbaum, V. and Blumen, A. (2007) Localization of Coherent Exciton Transport in Phase Space. Physical Review E, 75, Article ID: 031121.
http://dx.doi.org/10.1103/PhysRevE.75.031121
[72] Iglói, F., Karevski, D. and Rieger, H. (1998) Comparative Study of the Critical Behavior in One-Dimensional Random and Aperiodic Environments. The European Physical Journal B: Condensed Matter and Complex Systems, 5, 613-625.
http://dx.doi.org/10.1007/s100510050486
[73] Godsil, C. and Severini, S. (2010) Control by Quantum Dynamics on Graphs. Physical Review A, 81, Article ID: 052316.
http://dx.doi.org/10.1103/PhysRevA.81.052316
[74] Machida, T. (2013) Limit Distribution with a Combination of Density Functions for a 2-State Quantum Walk. Journal of Computational and Theoretical Nanoscience, 10, 1571-1578.
http://dx.doi.org/10.1166/jctn.2013.3090
[75] Shikano, Y. and Katsura, H. (2010) Localization and Fractality in Inhomogeneous Quantum Walks with Self-Duality. Physical Review E, 82, Article ID: 031122.
http://dx.doi.org/10.1103/PhysRevE.82.031122
[76] Linden, N. and Sharam, J. (2009) Inhomogeneous Quantum Walks. Physical Review A, 80, Article ID: 052327.
http://dx.doi.org/10.1103/PhysRevA.80.052327
[77] Konno, N., ?uczak, T. and Segawa, E. (2013) Limit Measures of Inhomogeneous Discrete-Time Quantum Walks in One Dimension. Quantum Information Processing, 12, 33-53.
http://dx.doi.org/10.1007/s11128-011-0353-8
[78] Konno, N. (2009) One-Dimensional Discrete-Time Quantum Walks on Random Environments. Quantum Information Processing, 8, 387-399.
[79] Konno, N. (2010) Localization of an Inhomogeneous Discrete-Time Quantum Walk on the Line. Quantum Information Processing, 9, 405-418.
[80] Ampadu, C. (2012) On an Inhomogeneous Quantum Walk. Unpublished.
[81] Villagra, M., Nakanishi, M., Yamashita, S. and Nakashima, Y. (2012) Quantum Walk on the Line with Phase Parameters. IEICE Transactions on Information and Systems, E95.D, 722-730.
[82] Ampadu, C. (2011) The Parametrized Grover Walk on the Line. Unpublished.
[83] Ampadu, C. (2012) Asymptotic Entanglement in the Parametrized Hadamard Walk. International Journal of Quantum Information, 10, Article ID: 1250066.
http://dx.doi.org/10.1142/S0219749912500669
[84] Ampadu, C. (2012) Limit Theorem for the Parametrized Grover Walk on the Line. Proceedings of AIP Conference, Vaxjo, 11-14 June 2012, 343.
[85] Ampadu, C. (2012) On a Parametrized Quantum Walk in Random Environments. Unpublished.
[86] McGettrick, M. (2010) One Dimensional Quantum Walks with Memory. Quantum Information & Computation, 10, 509-524.
[87] Konno, N. and Machida, T. (2010) Limit Theorems for Quantum Walks with Memory. Quantum Information and Computation, 10, 1004-1017.
[88] Inui, N., Konno, N. and Segawa, E. (2005) One-Dimensional Three-State Quantum Walk. Physical Review E, 72, Article ID: 056112.
http://dx.doi.org/10.1103/PhysRevE.72.056112
[89] Brun, T.A., Carteret, H.A. and Ambainis, A. (2003) Quantum Walks Driven by Many Coins. Physical Review A, 67, Article ID: 052317.
http://dx.doi.org/10.1103/PhysRevA.67.052317
[90] Venegas-Andreca, S.E., Ball, J.L., Burnett, K. and Bose, S. (2005) Quantum Walks with Entangled Coins. New Journal of Physics, 7, 221.
[91] Segawa, E. and Konno, N. (2008) Limit Theorems for Quantum Walks Driven by Many Coins. International Journal of Quantum Information, 6, 1231-1243.
[92] Ampadu, C. (2011) Limit Theorems for the Grover Walk without Memory. arXiv:1108.4149.
[93] Brun, T.A., Carteret, H.A. and Ambainis, A. (2003) Quantum Random Walks with Decoherent Coins. Physical Review A, 67, Article ID: 032304.
http://dx.doi.org/10.1103/PhysRevA.67.032304
[94] Liu, C. and Pentulante, N. (2011) Asymptotic Evolution of Quantum Walks on the N Cycle Subject to Decoherence on Both the Coin and Position Degrees of Freedom. Physical Review A, 84, Article ID: 012317.
http://dx.doi.org/10.1103/PhysRevA.84.012317
[95] Schreiber, A., Cassemiro, K.N., Potocek, V., Gábris, A., Jex, I. and Silberhorn, C. (2011) Decoherence and Disorder in Quantum Walks: From Ballistic Spread to Localization. Physical Review Letters, 106, Article ID: 180403.
http://dx.doi.org/10.1103/PhysRevLett.106.180403
[96] Romanelli, A. and Hernandez, G. (2011) Quantum Walks: Decoherence and Coin Flipping Games. Physica A: Statistical Mechanics and Its Applications, 390, 1209-1220.
[97] Srikanth, R., Banerjee, S. and Chandrashekar, C.M. (2010) Quantumness in Decoherent Quantum Walk Using Measurement-Induced Disturbance. Physical Review A, 81, Article ID: 062123.
http://dx.doi.org/10.1103/PhysRevA.81.062123
[98] Broome, M.A., Fedrizzi, A., Lanyon, B.P., Kassal, I., Aspuru-Guzik, A. and White, A.G. (2010) Discrete Single-Photon Quantum Walks with Tunable Decoherence. Physical Review Letters, 104, Article ID: 153602.
http://dx.doi.org/10.1103/PhysRevLett.104.153602
[99] Annabestani, M., Akhtarshenas, S.J. and Abolhassani, M.R. (2010) Tunneling Effects in a One-Dimensional Quantum Walk. arXiv: 1004.4352.
[100] Gönülol, M., Aydiner, E. and Müstecapl?oglu, Ö.E. (2009) Decoherence in Two-Dimensional Quantum Random Walks with Traps. Physical Review A, 80, Article ID: 022336.
http://dx.doi.org/10.1103/PhysRevA.80.022336
[101] Liu, C. and Petulante, N. (2010) Quantum Random Walks on the N Cycle Subject to Decoherence on the Coin Degree of Freedom. Physical Review E, 81, Article ID: 031113.
http://dx.doi.org/10.1103/PhysRevE.81.031113
[102] Xue, P., Sanders, B.C., Blais, A. and Lalumière, K. (2008) Quantum Walks on Circles in Phase Space via Superconducting Circuit Quantum Electrodynamics. Physical Review A, 78, Article ID: 042334.
http://dx.doi.org/10.1103/PhysRevA.78.042334
[103] Abal, G., Donangelo, R., Severo, F. and Siri, R. (2007) Decoherent Quantum Walks Driven by a Generic Coin Operation. Physica A, 387, 335-345.
http://dx.doi.org/10.1016/j.physa.2007.08.058
[104] Maloyer, O. and Kendon, V. (2007) Decoherence vs. Entanglement in Coined Quantum Walks. New Journal of Physics, 9, 87.
http://dx.doi.org/10.1088/1367-2630/9/4/087
[105] Kosík, J., Buzek, V. and Hillery, M. (2006) Quantum Walks with Random Phase Shifts. Physical Review A, 74, Article ID: 022310.
http://dx.doi.org/10.1103/PhysRevA.74.022310?
[106] Salimi, S. and Radgohar, R. (2006) Mixing and Decoherence in Continuous-Time Quantum Walks on Cycles. Quantum Information and Computation, 6, 263-276.
[107] Solenov, D. and Fedichkin, L. (2006) Non-Unitary Quantum Walks on Hyper-Cycles. Physical Review A, 73, Article ID: 012308.
http://dx.doi.org/10.1103/PhysRevA.73.012308
[108] Ryan, C.A., Laforest, M., Boileau, J.C. and Laflamme, R. (2005) Experimental Implementation of Discrete Time Quantum Random Walk on an NMR Quantum Information Processor. Physical Review A, 72, Article ID: 062317.
http://dx.doi.org/10.1103/PhysRevA.72.062317
[109] Algaic, G. and Russell, A. (2005) Decoherence in Quantum Walks on the Hypercube. Physical Review A, 72, Article ID: 062304.
http://dx.doi.org/10.1103/PhysRevA.72.062304
[110] Brun, T.A., Carteret, H.A. and Ambainis, A. (2003) The Quantum to Classical Transition for Random Walks. Physical Review Letters, 91, Article ID: 130602.
http://dx.doi.org/10.1103/PhysRevLett.91.130602
[111] Lopez, C.C. and Paz, J.P. (2003) Phase-Space Approach to the Study of Decoherence in Quantum Walks. Physical Review A, 68, Article ID: 052305.
http://dx.doi.org/10.1103/PhysRevA.68.052305
[112] Dür, W., Raussendorf, R., Kendon, V.M. and Briegel, H.-J. (2002) Quantum Random Walks in Optical Lattices. Physical Review A, 66, Article ID: 052319.
http://dx.doi.org/10.1103/PhysRevA.66.052319
[113] Solenov, D. and Fedichkin, L. (2006) Continuous-Time Quantum Walks on a Cycle Graph. Physical Review A, 73, Article ID: 012313.
http://dx.doi.org/10.1103/PhysRevA.73.012313
[114] Yin, Y., Katsanos, D.E. and Evangelou, S.N. (2008) Quantum Walks on a Random Environment. Physical Review A, 77, Article ID: 022302.
http://dx.doi.org/10.1103/PhysRevA.77.022302
[115] Annabestani, M., Akhtarshenas, S.J. and Abolhassani, M.R. (2010) Decoherence in a One-Dimensional Quantum Walk. Physical Review A, 81, Article ID: 032321.
http://dx.doi.org/10.1103/PhysRevA.81.032321
[116] Romanelli, A., Siri, R., Abal, G., Auyuanet, A. and Donangelo, R. (2005) Decoherence in the Quantum Walk on the Line. Physica A: Statistical Mechanics and Applications, 347, 137-152.
[117] Romanelli, A., Schifino, A.C.S., Abal, G., Siri, R. and Donangelo, R. (2003) Markovian Behavior and Constrained Maximization of the Entropy in Chaotic Quantum Systems. Physics Letters A, 313, 325-329.
[118] Romanelli, A., Sicardi-Schifino, A.C., Siri, R., Abal, G., Auyuanet, A. and Donangelo, R. (2004) Quantum Random Walk on the Line as a Markovian Process. Physica A: Statistical Mechanics and Applications, 338, 395-405.
[119] Ampadu, C. (2012) Brun-Type Formalism for Decoherence in Two Dimensional Quantum Walks. Communications in Theoretical Physics, 57, 41-55.
[120] Fan, S.M., Feng, Z.Y., Xiong, S. and Yang, W.-S. (2011) Convergence of Quantum Random Walks with Decoherence. Physical Review A, 84, Article ID: 042317.
http://dx.doi.org/10.1103/PhysRevA.84.042317
[121] Correa, L.A., Valido, A.A. and Alonso, D. (2012) Asymptotic Discord and Entanglement of Non-Resonant Harmonic Oscillators in an Equilibrium Environment. Physical Review A, 86, Article ID: 012110.
http://dx.doi.org/10.1103/PhysRevA.86.012110
[122] Liu, C. (2012) Asymptotic Distributions of Quantum Walks on the Line with Two Entangled Coin. Quantum Information Processing, 11, 1193-1205.
http://dx.doi.org/10.1007/s11128-012-0361-3
[123] Jakóbczyk, L., Olkiewicz, R. and Zaba, M. (2011) Asymptotic Entanglement of Two Atoms in Squeezed Light Field. Physical Review A, 83, Article ID: 062322.
http://dx.doi.org/10.1103/PhysRevA.83.062322
[124] Benatti, F. (2011) Asymptotic Entanglement and Lindblad Dynamics: A Perturbative Approach. Journal of Physics A: Mathematical and Theoretical, 44, Article ID: 155303.
http://dx.doi.org/10.1088/1751-8113/44/15/155303
[125] Salimi, S. and Yosefjani, R. (2012) Asymptotic Entanglement in 1D Quantum Walks with a Time-Dependent Coined. International Journal of Modern Physics B, 26, Article ID: 1250112.
http://dx.doi.org/10.1142/S0217979212501123
[126] Benatti, F. (2011) Three Qubits in a Symmetric Environment: Dissipatively Generated Asymptotic Entanglement. Annals of Physics, 326, 740-753.
http://dx.doi.org/10.1016/j.aop.2010.09.006
[127] Leung, D., Mancinska, L., Matthews, W., Ozols, M. and Roy, A. (2012) Entanglement Can Increase Asymptotic Rates of Zero-Error Classical Communication over Classical Channels. Communications in Mathematical Physics, 311, 97-111.
http://dx.doi.org/10.1007/s00220-012-1451-x
[128] Drumond, R.C., Souza, L.A.M. and Cunha, M.T. (2010) Asymptotic Entanglement Dynamics Phase Diagrams for Two Electromagnetic Field Modes in a Cavity. Physical Review A, 82, Article ID: 042302.
http://dx.doi.org/10.1103/PhysRevA.82.042302
[129] Gütschow, J., Uphoff, S., Werner, R.F. and Zimborás, Z. (2010) Time Asymptotics and Entanglement Generation of Clifford Quantum Cellular Automata. Journal of Mathematical Physics, 51, Article ID: 015203.
http://dx.doi.org/10.1063/1.3278513
[130] Annabestani, M., Abolhasani, M.R. and Abal, G. (2010) Asymptotic Entanglement in a Two-Dimensional Quantum Walk. Journal of Physics A: Mathematical and Theoretical, 43, Article ID: 075301.
http://dx.doi.org/10.1088/1751-8113/43/7/075301
[131] Drumond, R. and Cunha, M. (2009) Asymptotic Entanglement Dynamics and Geometry of Quantum States. Journal of Physics A: Mathematical and Theoretical, 42, Article ID: 285308.
http://dx.doi.org/10.1088/1751-8113/42/28/285308
[132] Pan, Q. and Jing, J. (2008) Hawking Radiation, Entanglement and Teleportation in Background of an Asymptotically Flat Static Black Hole. Physical Review D, 78, Article ID: 065015.
http://dx.doi.org/10.1103/PhysRevD.78.065015
[133] Isar, A. (2008) Asymptotic Entanglement in Open Quantum Systems. International Journal of Infectious Diseases, 6, 689-694.
http://dx.doi.org/10.1142/S0219749908003967
[134] Isar, A. (2007) Decoherence and Asymptotic Entanglement in Open Quantum Dynamics. Journal of Russian Laser Research, 28, 439-452.
http://dx.doi.org/10.1007/s10946-007-0033-4
[135] Abal, G., et al. (2006) Asymptotic Entanglement in the Discrete-Time Quantum Walk. Annals of the 1st Workshop on Quantum Computation and Information, Universidade Católica de Pelotas, 189-200.
[136] Bowen, G. and Datta, N. (2008) Asymptotic Entanglement Manipulation of Bipartite Pure States. IEEE Transaction on Information Theory, 54, 3677-3686.
http://dx.doi.org/10.1109/TIT.2008.926377
[137] Benatti, F. and Floreanini, R. (2006) Asymptotic Entanglement of Two Independent Systems in a Common Bath. International Journal of Quantum Information, 4, 395.
http://dx.doi.org/10.1142/S0219749906001864
[138] Hostens, E., Dehaene, J. and De Moor, B. (2006) Asymptotic Adaptive Bipartite Entanglement Distillation Protocol. Physical Review A, 73, Article ID: 062337.
http://dx.doi.org/10.1103/PhysRevA.73.062337
[139] Ishizaka, S. and Plenio, M.B. (2005) Multi-Particle Entanglement under Asymptotic Positive Partial Transpose Preserving Operations. Physical Review A, 72, Article ID: 042325.
http://dx.doi.org/10.1103/PhysRevA.72.042325
[140] Duan, R.Y., Feng, Y. and Ying, M.S. (2005) Entanglement-Assisted Transformation Is Asymptotically Equivalent to Multiple-Copy Transformation. Physical Review A, 72, Article ID: 024306.
http://dx.doi.org/10.1103/PhysRevA.72.024306
[141] Childs, A.M., Leung, D.W., Verstraete, F. and Vidal, G. (2003) Asymptotic Entanglement Capacity of the Ising and Anisotropic Heisenberg Interactions. Quantum Information and Computation, 3, 97-105.
[142] Horodecki, M., Sen(De), A. and Sen, U. (2003) Rates of Asymptotic Entanglement Transformations for Bipartite Mixed States: Maximally Entangled States Are Not Special. Physical Review A, 67, Article ID: 062314.
http://dx.doi.org/10.1103/PhysRevA.67.062314
[143] Audenaert, K., De Moor, B., Vollbrecht, K.G.H. and Werner, R.F. (2002) Asymptotic Relative Entropy of Entanglement for Orthogonally Invariant States. Physical Review A, 66, Article ID: 032310.
http://dx.doi.org/10.1103/PhysRevA.66.032310
[144] Vidal, G. (2002) On the Continuity of Asymptotic Measures of Entanglement. arXiv:quant-ph/0203107.
[145] Hwang, W.-Y. and Matsumoto, K. (2002) Entanglement Measures with Asymptotic Weak-Monotonicity as Lower (Upper) Bound for the Entanglement of Cost (Distillation). Physics Letters A, 300, 581-585.
[146] Audenaert, K. (2001) The Asymptotic Relative Entropy of Entanglement. Physical Review Letters, 87, Article ID: 217902.
http://dx.doi.org/10.1103/PhysRevLett.87.217902
[147] Vidal, G. and Cirac, J. (2001) Irreversibility in Asymptotic Manipulations of Entanglement. Physical Review Letters, 86, 5803-5806.
[148] Hayden, P.M., Horodecki, M. and Terhal, B.M. (2001) The Asymptotic Entanglement Cost of Preparing a Quantum State. Journal of Physics A: Mathematical and General, 34, 6891-6898.
http://dx.doi.org/10.1088/0305-4470/34/35/314
[149] Bennett, C.H., Popescu, S., Rohrlich, D., Smolin, J.A. and Thapliyal, A.V. (2000) Exact and Asymptotic Measures of Multipartite Pure State Entanglement. Physical Review A, 63, Article ID: 012307.
http://dx.doi.org/10.1103/PhysRevA.63.012307
[150] Machida, T. (2013) Limit Theorems for the Interference Terms of Discrete-Time Quantum Walks on the Line. Quantum Information and Computation, 13, 661-671.
[151] Wootters, W. (1998) Entanglement of Formation of an Arbitrary State of Two Quibits. Physical Review Letters, 80, 2245-2248.
http://dx.doi.org/10.1103/PhysRevLett.80.2245
[152] Rungta, P., Buzek, V., Caves, C.M., Hillery, M. and Milburn, G.J. (2001) Universal State Inversion and Concurrence in Arbitrary Dimensions. Physical Review A, 64, Article ID: 042315.
http://dx.doi.org/10.1103/PhysRevA.64.042315
[153] Kuang, L.-M. and Zhou, L. (2003) Generation of Atom-Photon Entangled States in Atomic Bose-Einstein Condensate via Electromagnetically Induced Transparency. Physical Review A, 68, Article ID: 043606.
http://dx.doi.org/10.1103/PhysRevA.68.043606
[154] Uhlmann, A. (2000) Fidelity and Concurrence of Conjugated States. Physical Review A, 62, Article ID: 032307.
http://dx.doi.org/10.1103/PhysRevA.62.032307
[155] Peres, A. (1996) Separability Criterion for Density Matrices. Physical Review Letters, 77, 1413-1415.
http://dx.doi.org/10.1103/PhysRevLett.77.1413
[156] Horodeckia, M., Horodeckib, P. and Horodecki, R. (1996) Separability of Mixed States: Necessary and Sufficient Conditions. Physical Review A, 223, 1-8.
http://dx.doi.org/10.1016/S0375-9601(96)00706-2
[157] Vidal, G. and Werner, R.F. (2002) Computable Measure of Entanglement. Physical Review A, 65, Article ID: 032314.
http://dx.doi.org/10.1103/PhysRevA.65.032314
[158] Coffman, V., Kundu, J. and Wootters, W.K. (2000) Distributed entanglement. Physical Review A, 61, Article ID: 052306.
http://dx.doi.org/10.1103/PhysRevA.61.052306
[159] Wong, A. and Christensen, N. (2001) Potential Multiparticle Entanglement. Physical Review A, 63, Article ID: 044301.
http://dx.doi.org/10.1103/PhysRevA.63.044301
[160] Osborne, T.J. and Verstraete, F. (2006) General Monogamy Inequality for Bipartite Qubit Entanglement. Physical Review Letters, 96, Article ID: 220503.
http://dx.doi.org/10.1103/PhysRevLett.96.220503
[161] Liu, C.B. and Petulante, N. (2010) On the von Neumann Entropy of Certain Quantum Walks Subject to Decoherence. Mathematical Structures in Computer Science, 20, 1099-1115.
[162] Abal, G., Siri, R., Romanelli, A. and Donangelo, R. (2006) Quantum Walk on the Line: Entanglement and Non-Local Conditions. Physical Review A, 73, Article ID: 042302.
http://dx.doi.org/10.1103/PhysRevA.73.042302
[163] Ide, Y., Konno, N. and Machida, T. (2011) Entanglement for Discrete-Time Quantum Walks on the Line. Quantum Information and Computation, 11, 855-866.
[164] Ampadu, C. (2011) Von Neumann Entanglement and Decoherence in Two Dimensional Quantum Walks. arXiv: 1110.0681.
[165] Ampadu, C. (2012) On the von Neumann and Shannon Entropies for Quantum Walks on Z^2. International Journal of Quantum Information, 10, Article ID: 1250020.
http://dx.doi.org/10.1142/S0219749912500207
[166] Grimmett, G., Janson, S. and Scudo, P. (2004) Weak Limits for Quantum Random Walks. Physical Review E, 69, Article ID: 026119.
http://dx.doi.org/10.1103/PhysRevE.69.026119
[167] Konno, N. (2005) A New Type of Limit Theorems for the One Dimensional Quantum Random Walk. Journal of the Mathematical Society of Japan, 57, 1179-1195.
http://dx.doi.org/10.2969/jmsj/1150287309
[168] Konno, N. (2002) Quantum Random Walks in One Dimension. Quantum Information Processing, 1, 345-354.
http://dx.doi.org/10.1023/A:1023413713008
[169] Ampadu, C. (2013) Interference Phenomena in the Continuous-Time Quantum Random Walk on Z. Unpublished.
[170] Ampadu, C. (2012) Limit Theorems for Decoherent Two Dimensional Quantum Walks. Quantum Information Processing, 11, 1921-1929.
http://dx.doi.org/10.1007/s11128-011-0349-4
[171] Venegas-Andraca, S.E., Ball, J.L., Burnett, K. and Bose, S. (2005) Quantum Walks with Entangled Coins. New Journal of Physics, 7, 221.
[172] Liu, C. and Pentulante, N. (2009) One-Dimensional Quantum Random Walk with Two Entangled Coins. Physical Review A, 79, Article ID: 032312.
http://dx.doi.org/10.1103/PhysRevA.79.032312

  
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