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Open quantum random walks: ergodicity, hitting times, gambler’s ruin and potential theory. (English) Zbl 1358.82016

Summary: In this work we study certain aspects of open quantum random walks (OQRWs), a class of quantum channels described by S. Attal et al. [J. Stat. Phys. 147, No. 4, 832–852 (2012; Zbl 1246.82039)]. As a first objective we consider processes which are nonhomogeneous in time, i.e., at each time step, a possibly distinct evolution kernel. Inspired by a spectral technique described by L. Saloff-Coste and J. Zúñiga [Stochastic Processes Appl. 117, No. 8, 961–979 (2007; Zbl 1124.60057)], we define a notion of ergodicity for finite nonhomogeneous quantum Markov chains and describe a criterion for ergodicity of such objects in terms of singular values. As a second objective, and based on a quantum trajectory approach, we study a notion of hitting time for OQRWs and we see that many constructions are variations of well-known classical probability results, with the density matrix degree of freedom on each site giving rise to systems which are seen to be nonclassical. In this way we are able to examine open quantum versions of the gambler’s ruin, birth-and-death chain and a basic theorem on potential theory.

MSC:

82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
37A60 Dynamical aspects of statistical mechanics
15A18 Eigenvalues, singular values, and eigenvectors
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[1] Accardi, L., Koroliuk, D.: Quantum Markov chains: the recurrence problem. Quantum Prob. Relat. Top. VII, 6373 (1991) · Zbl 0929.60094
[2] Accardi, L., Koroliuk, D.: Stopping times for quantum Markov chains. J. Theor. Probab. 5(3), 521-535 (1992) · Zbl 0751.60081 · doi:10.1007/BF01060433
[3] Alicki, R., Fannes, M.: Quantum Dynamical Systems. Oxford University Press, Oxford (2000) · Zbl 1140.81308
[4] Attal, S., Petruccione, F., Sabot, C., Sinayskiy, I.: Open quantum random walks. J. Stat. Phys. 147, 832-852 (2012) · Zbl 1246.82039 · doi:10.1007/s10955-012-0491-0
[5] Attal, S., Guillotin-Plantard, N., Sabot, C.: Central limit theorems for open quantum random walks and quantum measurement records. Ann. Henri Poincaré 16, 15-43 (2015) · Zbl 1319.81056 · doi:10.1007/s00023-014-0319-3
[6] Baraviera, A., Lardizabal, C.F., Lopes, A.O., Terra Cunha, M.: Quantum stochastic processes, quantum iterated function systems and entropy. São Paulo Journ. Math. Sci. 5(1), 51-84 (2011) · Zbl 1243.81102
[7] Benatti, F.: Dynamics, Information and Complexity in Quantum Systems. Springer, Berlin (2009) · Zbl 1173.81001
[8] Biane, P., Bouten, L., Cipriani, F., Konno, N., Privault, N., Xu, Q.: Quantum Potential Theory. Lecture Notes in Mathematics, vol. 1954. Springer, Berlin (2008) · Zbl 1296.82028
[9] Bourgain, J., Grünbaum, F.A., Velázquez, L., Wilkening, J.: Quantum recurrence of a subspace and operator-valued Schur functions. Commun. Math. Phys. 329, 1031-1067 (2014) · Zbl 1296.37014 · doi:10.1007/s00220-014-1929-9
[10] Brémaud, P.: Markov Chains: Gibbs Fields, Monte Carlo Simulation and Queues. Texts in Applied Mathematics 31. Springer, Berlin (1999) · Zbl 0949.60009
[11] Burgarth, D., Chiribella, G., Giovannetti, V., Perinotti, P., Yuasa, K.: Ergodic and mixing quantum channels in finite dimensions. New J. Phys. 15, 073045 (2013) · Zbl 1451.81043 · doi:10.1088/1367-2630/15/7/073045
[12] Burgarth, D., Giovannetti, V.: The generalized Lyapunov theorem and its application to quantum channels. New J. Phys. 9, 150 (2007) · doi:10.1088/1367-2630/9/5/150
[13] Carbone, R., Pautrat, Y.: Homogeneous open quantum random walks on a lattice. Preprint arXiv:1408.1113v2 · Zbl 1323.82020
[14] Carbone, R., Pautrat, Y.: Open quantum random walks: reducibility, period, ergodic properties. Preprint arXiv:1405.2214v3 · Zbl 1333.81205
[15] de Oliveira, C.R.: Intermediate Spectral Theory and Quantum Dynamics. Birkhäuser Verlag, Basel (2009) · Zbl 1165.47001 · doi:10.1007/978-3-7643-8795-2
[16] Fagnola, F., Rebolledo, R.: Transience and recurrence of quantum Markov semigroups. Probab. Theory Relat. Fields 126, 289306 (2003) · Zbl 1024.60031 · doi:10.1007/s00440-003-0268-0
[17] Grimmett, G.R., Stirzaker, D.R.: Probability and Random Processes, 3rd edn. Oxford University Press, Oxford (2001) · Zbl 1015.60002
[18] Grünbaum, F.A., Velázquez, L., Werner, A.H., Werner, R.F.: Recurrence for discrete time unitary evolutions. Commun. Math. Phys. 320, 543569 (2013) · Zbl 1276.81087 · doi:10.1007/s00220-012-1645-2
[19] Gudder, S.: Quantum Markov chains. J. Math. Phys. 49, 072105 (2008) · Zbl 1152.81457 · doi:10.1063/1.2953952
[20] Gudder, S.: Transition effect matrices and quantum Markov chains. Found. Phys. 39, 573-592 (2009) · Zbl 1179.81019 · doi:10.1007/s10701-008-9269-2
[21] Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985) · Zbl 0576.15001 · doi:10.1017/CBO9780511810817
[22] Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991) · Zbl 0729.15001 · doi:10.1017/CBO9780511840371
[23] Konno, N., Yoo, H.J.: Limit theorems for open quantum random walks. J. Stat. Phys. 150, 299-319 (2013) · Zbl 1259.82093 · doi:10.1007/s10955-012-0668-6
[24] Kümmerer, B., Maassen, H.: A pathwise ergodic theorem for quantum trajectories. J. Phys. A Math. Gen. 37, 11889-11896 (2004) · Zbl 1063.82021 · doi:10.1088/0305-4470/37/49/008
[25] Lardizabal, C.F.: A quantization procedure based on completely positive maps and Markov operators. Quantum Inf. Process. 12, 1033-1051 (2013) · Zbl 1264.81081 · doi:10.1007/s11128-012-0449-9
[26] Lardizabal, C.F., Souza, R.R.: On a class of quantum channels, open random walks and recurrence. J. Stat. Phys. 159, 772-796 (2015) · Zbl 1328.82025 · doi:10.1007/s10955-015-1217-x
[27] Liu, C., Petulante, N.: On limiting distributions of quantum Markov chains. Int. J. Math. Math. Sci., Vol 2011 (2011). ID 740816 · Zbl 1225.81080
[28] Liu, C., Petulante, N.: Asymptotic evolution of quantum walks on the N-cycle subject to decoherence on both the coin and position degrees of freedom. Phys. Rev. E 81, 031113 (2010) · doi:10.1103/PhysRevE.81.031113
[29] Lozinski, A., Życzkowski, K., Słomczyński, W.: Quantum iterated function systems. Phys. Rev. E 68, 04610 (2003) · doi:10.1103/PhysRevE.68.046110
[30] Lytvynov, E., Mei, L.: On the correlation measure of a family of commuting Hermitian operators with applications to particle densities of the quasi-free representations of the CAR and CCR. J. Funct. Anal. 245, 6288 (2007) · Zbl 1231.47004 · doi:10.1016/j.jfa.2006.12.017
[31] Mukhamedov, F.: Weak ergodicity of nonhomogeneous Markov chains on noncommutative \[L^1\] L1-spaces. Banach J. Math. Anal. 7(2), 53-73 (2013) · Zbl 1276.47012 · doi:10.15352/bjma/1363784223
[32] Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000) · Zbl 1049.81015
[33] Norris, J.R.: Markov Chains. Cambridge University Press, Cambridge (1997) · Zbl 0873.60043 · doi:10.1017/CBO9780511810633
[34] Novotný, J., Alber, G., Jex, I.: Asymptotic evolution of random unitary operations. Cent. Eur. J. Phys. 8(6), 1001-1014 (2010) · Zbl 1201.93127
[35] Novotný, J., Alber, G., Jex, I.: Asymptotic properties of quantum Markov chains. J. Phys. A Math. Theor. 45, 485301 (2012) · Zbl 1278.81117 · doi:10.1088/1751-8113/45/48/485301
[36] Paulsen, V.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2003) · Zbl 1029.47003 · doi:10.1017/CBO9780511546631
[37] Pellegrini, C.: Continuous time open quantum random walks and non-Markovian Lindblad master equations. J. Stat. Phys. 154, 838-865 (2014) · Zbl 1296.82028 · doi:10.1007/s10955-013-0910-x
[38] Petz, D.: Quantum Information Theory and Quantum Statistics. Springer, Berlin (2008) · Zbl 1145.81002
[39] Platis, A., Limnios, N., Le Du, M.: Hitting time in a finite non-homogeneous Markov chain with applications. Appl. Stoch. Models Data Anal. 14, 241-253 (1998) · Zbl 0926.60058 · doi:10.1002/(SICI)1099-0747(199809)14:3<241::AID-ASM354>3.0.CO;2-3
[40] Portugal, R.: Quantum Walks and Search Algorithms. Springer, Berlin (2013) · Zbl 1275.81004 · doi:10.1007/978-1-4614-6336-8
[41] Saloff-Coste, L., Zúñiga, J.: Convergence of some time inhomogeneous Markov chains via spectral techniques. Stoch. Proc. Appl. 117, 961-979 (2007) · Zbl 1124.60057 · doi:10.1016/j.spa.2006.11.004
[42] Szehr, O., Wolf, M.M.: Perturbation bounds for quantum Markov processes and their fixed points. J. Math. Phys. 54, 032203 (2013) · Zbl 1281.81050 · doi:10.1063/1.4795112
[43] Temme, K., Kastoryano, M.J., Ruskai, M.B., Wolf, M.M., Verstraete, F.: The \[\chi^2\] χ2-divergence and mixing times of quantum Markov processes. J. Math. Phys. 51, 12201 (2010) · Zbl 1314.81124 · doi:10.1063/1.3511335
[44] Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quantum Inf. Process. 11, 1015-1106 (2012) · Zbl 1283.81040 · doi:10.1007/s11128-012-0432-5
[45] Watrous, J.: Theory of Quantum Information Lecture Notes from Fall 2011. Institute for Quantum Computing, University of Waterloo
[46] Wolf, M.M.: Quantum Channels & Operations Guided Tour Lecture Notes (unpublished). 5 July 2012
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