Abstract
We study the analogues of irreducibility, period, and communicating classes for open quantum random walks, as defined in (J Stat Phys 147(4):832–852, 2012). We recover results similar to the standard ones for Markov chains, in terms of ergodic behaviour, decomposition into irreducible subsystems, and characterization of invariant states.
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Albeverio, S., Høegh-Krohn, R.: Frobenius theory for positive maps of von Neumann algebras. Comm. Math. Phys. 64(1), 83–94 (1978/79)
Attal, S., Guillotin, N., Sabot, C.: Central limit theorems for open quantum random walks. Ann. Henri Poincaré (to appear) (2015)
Attal S., Petruccione F., Sabot C., Sinayskiy I.: Open quantum random walks. J. Stat. Phys. 147(4), 832–852 (2012)
Baumgartner B., Narnhofer H.: The structures of state space concerning quantum dynamical semigroups. Rev. Math. Phys. 24(2), 1250001 (2012)
Carbone, R., Pautrat, Y.: Homogeneous open quantum random walks a lattice. Arxiv preprint
Davies E.B.: Quantum stochastic processes. II. Comm. Math. Phys. 19, 83–105 (1970)
Enomoto, M., Watatani, Y.: A Perron–Frobenius type theorem for positive linear maps on C*-algebras. Math. Japon. 24(1), 53–63 (1979/80)
Evans D.E., Høegh-Krohn R.: Spectral properties of positive maps on C*-algebras. J. Lond. Math. Soc. 17(2), 345–355 (1978)
Fagnola F., Pellicer R.: Irreducible and periodic positive maps. Commun. Stoch. Anal. 3(3), 407–418 (2009)
Farenick D.R.: Irreducible positive linear maps on operator algebras. Proc. Am. Math. Soc. 124(11), 3381–3390 (1996)
Frigerio, A.: Quantum dynamical semigroups and approach to equilibrium. Lett. Math. Phys. 2(2), 79–87 (1977/78)
Frigerio A., Verri M.: Long-time asymptotic properties of dynamical semigroups on W*-algebras. Math. Z. 180(2), 275–286 (1982)
Gärtner, A., Kümmerer, B.: A Coherent Approach to Recurrence and Transience for Quantum Markov Operators. ArXiv e-prints, Nov. (2012)
Groh U.: The peripheral point spectrum of Schwarz operators on C*-algebras. Math. Z. 176(3), 311–318 (1981)
Kraus, K.: States, effects, and operations. Volume 190 of Lecture Notes in Physics. Springer, Berlin, (1983). In: Böhm, A., Dollard, J.D., Wootters, W.H. (eds.) Fundamental notions of quantum theory, Lecture notes
Kümmerer B., Maassen H.: A pathwise ergodic theorem for quantum trajectories. J. Phys. A 37(49), 11889–11896 (2004)
Marais, A., Sinayskiy, I., Kay, A., Petruccione, F., Ekert, A.: Decoherence-assisted transport in quantum networks. New J. Phys. 15, 013038, 18, (2013)
Nielsen M.A., Chuang I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2000)
Pellegrini C.: Continuous time open quantum random walks and non-Markovian Lindblad master equations. J. Stat. Phys. 154(3), 838–865 (2014)
Reed, M., Simon, B.: Methods of modern mathematical physics. I. In: Functional analysis, 2nd edn. Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers), New York (1980)
Russo B., Dye H.A.: A note on unitary operators in C*-algebras. Duke Math. J. 33, 413–416 (1966)
Schrader, R.: Perron–Frobenius theory for positive maps on trace ideals. In: Mathematical physics in mathematics and physics (Siena, 2000), volume 30 of Fields Inst. Commun., pp. 361–378. Am. Math. Soc., Providence, RI (2001)
Sinayskiy I., Petruccione F.: Efficiency of open quantum walk implementation of dissipative quantum computing algorithms. Quantum Inf. Process. 11(5), 1301–1309 (2012)
Umanità V.: Classification and decomposition of quantum Markov semigroups. Probab. Theory Relat Fields 134(4), 603–623 (2006)
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Communicated by Denis Bernard.
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Carbone, R., Pautrat, Y. Open Quantum Random Walks: Reducibility, Period, Ergodic Properties. Ann. Henri Poincaré 17, 99–135 (2016). https://doi.org/10.1007/s00023-015-0396-y
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DOI: https://doi.org/10.1007/s00023-015-0396-y