JQIS  Vol.2 No.2 , June 2012
Entanglement Generation in Spatially Separated Systems Using Quantum Walk
Abstract: We present a scheme for generating entanglement between two spatially separated systems from the spatial entanglement generated by the interference effect during the evolution of a single-particle quantum walk. Any two systems which can interact with the spatial modes entangled during the walk evolution can be entangled using this scheme. A notable feature is the ability to control the quantum walk dynamics and its localization at desired pair lattice sites irrespective of separation distance resulting in a substantial control and improvement in the entanglement output. Implementation schemes to entangle spatially separated atoms using quantum walk on a single atom is also presented.
Cite this paper: C. Chandrashekar, S. Goyal and S. Banerjee, "Entanglement Generation in Spatially Separated Systems Using Quantum Walk," Journal of Quantum Information Science, Vol. 2 No. 2, 2012, pp. 15-22. doi: 10.4236/jqis.2012.22004.

[1]   R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, “Quantum Entanglement,” Reviews of Modern Physics, Vol. 81, No. 2, 2009, pp. 865-942. doi:10.1103/RevModPhys.81.865

[2]   T. Brougham, V. Kostak, I. Jex, E. Andersson and T. Kiss, “Entanglement Preparation Using Symmetric Multiports,” The European Physical Journal D, Vol. 61, No. 1, 2011, pp. 231-236. doi:10.1140/epjd/e2010-10337-2.

[3]   C. Cabrillo, J. I. Cirac, P. Garcia-Fernandez and P. Zoller, “Creation of Entangled States of Distant Atoms by Interference,” Physical Review A, Vol. 59, No. 2, 1999, pp. 1025-1033. doi:10.1103/PhysRevA.59.1025

[4]   M. B. Plenio, S. F. Huelga, A. Beige and P. L. Knight, “Cavity-Loss-Induced Generation of Entangled Atoms,” Physical Review A, Vol. 59, No. 3, 1999, pp. 2468-2475. doi:10.1103/PhysRevA.59.2468

[5]   L. Duan, M. D. Lukin, J. I. Cirac and P. Zoller, “Long-Distance Quantum Communication with Atomic Ensembles and Linear Optics,” Nature (London), Vol. 414, 2001, pp. 413-418. doi:10.1038/35106500

[6]   S. Bose and D. Home, “Generic Entanglement Generation, Quantum Statistics, and Complementarity,” Physical Review Letters, Vol. 88, 2002, Article ID: 050401. doi:10.1103/PhysRevLett.88.050401

[7]   A. K. Ekert, “Quantum Cryptography Based on Bell’s Theorem,” Physical Review Letters, Vol. 67, No. 6, 1991, pp. 661-633. doi:10.1103/PhysRevLett.67.661

[8]   C. H. Bennett, “Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels,” Physical Review Letters, Vol. 70, No. 13, 1993, pp. 1895-1899. doi:10.1103/PhysRevLett.70.1895

[9]   S. J. van Enk, “Single-Particle Entanglement,” Physical Review A, Vol. 72, No. 6, 2005, Article ID: 064306. doi:10.1103/PhysRevA.72.064306

[10]   S. Goyal and C. M. Chandrashekar, “Spatial Entanglement Using a Quantum Walk on a Many-Body System,” Journal of Physics A: Mathematical and Theoretical, Vol. 43, No. 23, 2010, Article ID: 235303. doi:10.1088/1751-8113/43/23/235303

[11]   D. A. Meyer, “From Quantum Cellular Automata to Quantum Lattice Gases,” Journal of Statistical Physics, Vol. 85, No. 5-6, 1996, pp. 551-574. doi:10.1007/BF02199356

[12]   C. M. Chandrashekar, R. Srikanth and R. Laflamme, “Optimizing the Discrete Time Quantum Walk Using a SU(2) Coin,” Physical Review A, Vol. 77, No. 3, 2008, Article ID: 032326. doi:10.1103/PhysRevA.77.032326

[13]   J. Du, H. Li, X. Xu, M. Shi, J. Wu, X. Zhou and R. Han, “Experimental Implementation of the Quantum Random-Walk Algorithm,” Physical Review A, Vol. 67, No. 4, 2003, Article ID: 042316. doi:10.1103/PhysRevA.67.042316

[14]   C. A. Ryan, M. Laforest, J. C. Boileau and R. Laflamme, “Experimental Implementation of a Discrete-Time Quantum Random Walk on an NMR Quantum-Information Processor,” Physical Review A, Vol. 72, No. 6, 2005, Article ID: 062317. doi:10.1103/PhysRevA.72.062317

[15]   D. Lu, J. Zhu, P. Zou, X. Peng, Y. Yu, S. Zhang, Q. Chen and J. Du, “Experimental Implementation of a Quantum Random-Walk Search Algorithm Using Strongly Dipolar Coupled Spins,” Physical Review A, Vol. 81, No. 2, 2010, Article ID: 022308. doi:10.1103/PhysRevA.81.022308

[16]   D. Bouwmeester, I. Marzoli, G. P. Karman, W. Schleich and J. P. Woerdman, “Optical Galton Board,” Physical Review A, Vol. 61, No. 1, 1999, Article ID: 013410. doi:10.1103/PhysRevA.61.013410

[17]   B. Do, M. L. Stohler, S. Balasubramanian, D. S. Elliott, C. Eash, E. Fischbach, M. A. Fischbach, A. Mills and B. Zwickl, “Experimental Realization of a Quantum Quincunx by Use of Linear Optical Elements,” Journal of the Optical Society of America B, Vol. 22, No. 2, 2005, pp. 499-504. doi:10.1364/JOSAB.22.000499

[18]   H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti and Y. Silberberg, “Realization of Quantum Walks with Negligible Decoherence in Waveguide Lattices,” Physical Review Letters, Vol. 100, No. 17, 2008, Article ID: 170506. doi:10.1103/PhysRevLett.100.170506

[19]   H. Schmitz, R. Matjeschk, Ch. Schneider, J. Glueckert, M. Enderlein, T. Huber and T. Schaetz, “Quantum Walk of a Trapped Ion in Phase Space,” Physical Review Letters, Vol. 103, No. 9, 2009, Article ID: 090504. doi:10.1103/PhysRevLett.103.090504

[20]   F. Zahringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt and C. F. Roos, “Realization of a Quantum Walk with One and Two Trapped Ions,” Physical Review Letters, Vol. 104, No. 10, 2010, Article ID: 100503. doi:10.1103/PhysRevLett.104.100503

[21]   K. Karski, L. Foster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede and A. Widera, “Quantum Walk in Position Space with Single Optically Trapped Atoms,” Science, Vol. 325, No. 5937, 2009, pp. 174-177.

[22]   A. Schreiber, K. N. Cassemiro, V. Potocek, A. Gabris, P. Mosley, E. Andersson, I. Jex and Ch. Silberhorn, “Photons Walking the Line: A Quantum Walk with Adjustable Coin Operations,” Physical Review Letters, Vol. 104, No. 5, 2010, Article ID: 050502. doi:10.1103/PhysRevLett.104.050502

[23]   M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-Guzik and A. G. White, “Discrete Single-Photon Quantum Walks with Tunable Decoherence,” Physical Review Letters, Vol. 104, No. 15, 2010, Article ID: 153602. doi:10.1103/PhysRevLett.104.153602

[24]   W. Dur, R. Raussendorf, V. M. Kendon and H. J. Briegel, “Quantum Walks in Optical Lattices,” Physical Review A, Vol. 66, No. 5, 2002, Article ID: 052319. doi:10.1103/PhysRevA.66.052319

[25]   K. Eckert, J. Mompart, G. Birkl and M. Lewenstein, “One- and Two-Dimensional Quantum Walks in Arrays of Optical Traps,” Physical Review A, Vol. 72, No. 1, 2005, Article ID: 012327. doi:10.1103/PhysRevA.72.012327

[26]   C. M. Chandrashekar, “Implementing the One-Dimensional Quantum (Hadamard) Walk Using a Bose-Einstein Condensate,” Physical Review A, Vol. 74, No. 3, 2006, Article ID: 032307. doi:10.1103/PhysRevA.74.032307

[27]   Y. Aharonov, L. Davidovich and N. Zagury, “Quantum Random Walks,” Physical Review A, Vol. 48, No. 2, 1993, pp. 1687-1690. doi:10.1103/PhysRevA.48.1687

[28]   A. Ambainis, E. Bach, A. Nayak, A. Vishwanath and J. Watrous, “One Dimensional Quantum Walks with Absorbing Boundaries,” Journal of Computer and System Sciences, Vol. 69, No. 4, 2001, pp. 562-592.

[29]   A. Peres, “Separability Criterion for Density Matrices,” Physical Review Letters, Vol. 77, No. 8, 1996, pp. 1413-1415. doi:10.1103/PhysRevLett.77.1413

[30]   H. J. Lee, W. Namgung and D. Ahna, “Entanglement Generates Entanglement: Entanglement Transfer by Interaction,” Physical Review A, Vol. 338, No. 3-5, 2005, pp. 192-196. doi:10.1016/j.physleta.2005.03.010

[31]   E. Lieb, T. Schultz and D. Mattis, “Two Soluble Models of an Antiferromagnetic Chain,” Annals of Physics, Vol. 16, No. 3, 1961, pp. 407-466. doi:10.1016/0003-4916(61)90115-4

[32]   M. Christandl, N. Datta, A. Ekert and Andrew J. Landahl, “Perfect State Transfer in Quantum Spin Networks,” Physical Review Letters, Vol. 92, No. 18, 2004, Article ID: 187902. doi:10.1103/PhysRevLett.92.187902

[33]   W. K. Wootters, “Entanglement of Formation of an Arbitrary State of Two Qubits,” Physical Review Letters, Vol. 80, No. 10, 1998, pp. 2245-2248. doi:10.1103/PhysRevLett.80.2245

[34]   T. Oka, N. Konno, R. Arita and H. Aoki, “Breakdown of an Electric-Field Driven System: A Mapping to a Quantum Walk,” Physical Review Letters, Vol. 94, No. 10, 2005, Article ID: 100602. doi:10.1103/PhysRevLett.94.100602

[35]   N. Konno, “Localization of an Inhomogeneous Discrete-Time Quantum Walk on the Line,” Quantum Information Processing, Vol. 9, No. 3, 2010, pp. 405-418. doi:10.1007/s11128-009-0147-4

[36]   C. M. Chandrashekar, “arXiv: 1001.5326,” 2010.

[37]   A. Joye and M. Merkli, “Dynamical Localization of Quantum Walks in Random Environments,” Journal of Statistical Physics, Vol. 140, No. 6, 2010, pp. 1-29. doi:10.1007/s10955-010-0047-0

[38]   C. M. Chandrashekar, “Disordered-Quantum-Walk-Induced Localization of a Bose-Einstein Condensate,” Physical Review A, Vol. 83, No. 2, 2011, Article ID: 022320. doi:10.1103/PhysRevA.83.022320

[39]   K.-A. Brickman Soderberg, N. Gemelke and C. Chin, “Ultracold Molecules: Vehicles to Scalable Quantum Information Processing,” New Journal of Physics, Vol. 11, 2009, Article ID: 055022. doi:10.1088/1367-2630/11/5/055022

[40]   R. Simon and N. Mukunda, “Minimal Three-Component SU(2) Gadget for Polarization Optics,” Physics Letters A, Vol. 143, No. 4-5, 1990, pp. 165-169.