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On the von Neumann entropy of certain quantum walks subject to decoherence†

Published online by Cambridge University Press:  08 November 2010

CHAOBIN LIU  and
NELSON PETULANTE
CHAOBIN LIU
Affiliation:
Department of Mathematics, Bowie State University, 14000 Jericho Park Road, Bowie, Maryland 20715, U.S.A. Email: [email protected], [email protected]
NELSON PETULANTE
Affiliation:
Department of Mathematics, Bowie State University, 14000 Jericho Park Road, Bowie, Maryland 20715, U.S.A. Email: [email protected], [email protected]
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Abstract

In this paper, we consider a discrete-time quantum walk on the N-cycle governed by the condition that at every time step of the walk, the option persists, with probability p, of exercising a projective measurement on the coin degree of freedom. For a bipartite quantum system of this kind, we prove that the von Neumann entropy of the total density operator converges to its maximum value. Thus, when influenced by decoherence, the mutual information between the two subsystems corresponding to the space of the coin and the space of the walker must eventually diminish to zero. Put plainly, any level of decoherence greater than zero forces the system to become completely ‘disentangled’ eventually.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 20 , Special Issue 6: Quantum Algorithms , December 2010 , pp. 1099 - 1115
Copyright
Copyright © Cambridge University Press 2010

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References

Abal, G., Siri, R., Romanelli, A. and Donangelo, R (2006) Quantum walk on the line: entanglement and non-local initial conditions. Physical Review A 73 042302 (Erratum: Physical Review A 73 069905(E)).CrossRefGoogle Scholar
Aharanov, D., Ambainis, A., Kempe, J. and Vazirani, U. (2001) Quantum walks on graphs. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, ACM 50–59.CrossRefGoogle Scholar
Ambainis, A. (2003) Quantum walks and their algorithmic applications. Int. J. Quantum Information 1 (4)507–518.CrossRefGoogle Scholar
Ambainis, A., Bach, E., Nayak, A., Vishwanath, A and Watrous, J. (2001) One-dimensional quantum walks. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, ACM 37–49.CrossRefGoogle Scholar
Annabestani, M., Abolhasani, M. R. and Abal, G. (2010a) Asymptotic entanglement in a two-dimensional quantum walk. J. Phys. A: Math. Theor. 43 075301.CrossRefGoogle Scholar
Annabestani, M., Akhtarshenas, S. J. and Abolhassani, M. R. (2010b) Decoherence in one-dimensional Quantum Walk. Physical Review A 81 032321.CrossRefGoogle Scholar
Banerjee, S., Srikanth, R., Chandrashekar, C. M. and Rungta, P. (2008) Symmetry-noise interplay in a quantum walk on an n-cycle. Physical Review A 78 052316.CrossRefGoogle Scholar
Brun, T. A., Carteret, H. A. and Ambainis, A. (2003a) The quantum to classical transition for random walks. Physical Review Letters 91 (13)130602.CrossRefGoogle ScholarPubMed
Brun, T. A., Carteret, H. A. and Ambainis, A. (2003b) Quantum random walks with decoherent coins. Physical Review A 67 032304.CrossRefGoogle Scholar
Carneiro, I., Loo, M., Xu, X., Girerd, M., Kendon, V. and Knight, P. L. (2005) Entanglement in coined quantum walks on regular graphs. New J. Phys. 7 156.CrossRefGoogle Scholar
Childs, A. M. (2010) On the relationship between continuous- and discrete-time quantum walk. Communications in Mathematical Physics 294 581–603.CrossRefGoogle Scholar
Childs, A. M., Fahri, E and Gutmann, S. (2002) An example of the difference between quantum and classical random walks. Quant. Inf. Proc. 1 35.CrossRefGoogle Scholar
Du, J., Li, H., Xu, X., Shi, M., Wu, J., Zhou, X. and Han, R. (2003) Experimental implementation of the quantum random-walk algorithm. Physical Review A 67 042316.CrossRefGoogle Scholar
Dur, W., Raussendorf, R., Kendon, V. and Briegel, H. J. (2002) Quantum walks in optical lattices. Physical Review A 66 052319.CrossRefGoogle Scholar
Eckert, K., Mompart, J., Birkl, G. and Lewenstein, M. (2005) One- and two-dimensional quantum walks in arrays of optical traps. Physical Review A 72 012327.CrossRefGoogle Scholar
Fahri, E and Gutmann, S. (1998) Quantum computation and decision trees. Physical Review A 58 915.Google Scholar
Kempe, J. (2003) Quantum random walks – an introductory overview. Contemp. Phys. 44 (4)307–327.CrossRefGoogle Scholar
Kendon, V. (2006) Decoherence in quantum walks – a review. Mathematical Structures in Computer Science 17 1169–1220.Google Scholar
Kendon, V. and Tregenna, B. (2002) Decoherence in a quantum walk on a line. In: Shapiro, J. H. and Hirota, O. (eds.) Quantum Communication, Measurement & Computing (QCMC–02), Rinton Press 463.Google Scholar
Kendon, V. and Tregenna, B. (2003a) Decoherence in discrete quantum walks (available at arXiv:quant-ph/0301182).CrossRefGoogle Scholar
Kendon, V. and Tregenna, B. (2003b) Decoherence is useful in quantum walks. Physical Review A 67 042315.CrossRefGoogle Scholar
Konno, N. (2008) Quantum walks. In: Franz, U. and Schurmann, M. (eds.) Quantum Potential Theory. Springer-Verlag Lecture Notes in Mathematics 1954 309–452.CrossRefGoogle Scholar
Kosik, J., Buzek, V. and Hillery, M. (2006) Quantum walks with random phase shifts. Physical Review A 74 022310.CrossRefGoogle Scholar
Liu, C. and Petulante, N. (2010) Quantum walks on the N-cycle subject to decoherence on the coin degree of freedom. Physical Review E 81 031113.CrossRefGoogle ScholarPubMed
Maloyer, O and Kendon, V. (2007) Decoherence versus entanglement in coined quantum walks. New Journal of Physics 9 87.CrossRefGoogle Scholar
Nayak, A and Vishwanath, A. (2000) Quantum Walk on the Line (available at arXiv:quant-ph/0010117).Google Scholar
Richter, P. (2007) Quantum speedup of classical mixing processes. Physical Review A 76 042306.CrossRefGoogle Scholar
Romanelli, A., Siri, R., Abal, G., Auyuanet, A. and Donangelo, R. (2005) Decoherence in the quantum walk on the line. Physica A 347 137–152.CrossRefGoogle Scholar
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 062317.CrossRefGoogle Scholar
Sanders, B., Bartlett, S., Tregenna, B. and Knight, P. (2003) Quantum quincunx in cavity quantum electrodynamics. Physical Review A 67 042305.Google Scholar
Shikano, Y., Chisaki, K., Segawa, E. and Konno, N. (2010) Emergence of randomness and arrow of time in quantum walks. Physical Review A 81 062129.CrossRefGoogle Scholar
Strauch, F. (2006) Connecting the discrete- and continuous-time quantum walks. Physical Review A 74 030301(R).CrossRefGoogle Scholar
Travaglione, B. C. and Milburn, G. J. (2002) Implementing the quantum random walk. Physical Review A 65 032310.CrossRefGoogle Scholar
van Hoogdalem, K. A. and Blaauboer, M. (2009) Implementation of the quantum-walk step operator in lateral quantum dots. Physical Review B 80 125309.CrossRefGoogle Scholar
Venegas-Andraca, S. E. (2008) Quantum Walks for Computer Scientists, (Synthesis Lectures on Quantum Computing), Morgan and Claypool Publishers.CrossRefGoogle Scholar
Venegas-Andraca, S. E. and Bose, S. (2009) Quantum Walk-based Generation of Entanglement Between Two Walkers (available at quant-ph/0901.3946).Google Scholar
Watrous, J. (2008) Theory of Quantum Information, Lecture notes from Fall 2008, Institute for Quantum Computing, University of Waterloo, Canada.Google Scholar
Zhang, K. (2008) The limiting distribution of decoherent quantum random walks. Physical Review A 77 062302.CrossRefGoogle Scholar
Zou, X., Dong, Y. and Guo, G. (2006) Optical implementation of one-dimensional quantum random walks using orbital angular momentum of a single photon. New J. Phys. 8 81.CrossRefGoogle Scholar
Zurek, W. H. (2003) Decoherence and the transition from quantum to classical – Revisited. In: Duplantier, B., Raimond, J.-M. and Rivasseau, V. (eds.) Decoherence Poincaré Seminar 2005, Progress in Mathematical Physics, Birkhuser Verlag 1–31. (Eprint available at arXiv:quant-ph/0306072.)Google Scholar