Open quantum random walks, quantum Markov chains and recurrence. (English) Zbl 1431.81082
Summary: In the present paper, we construct QMCs (quantum Markov chains) associated with open quantum random walks such that the transition operator of the chain is defined by OQRW and the restriction of QMC to the commutative subalgebra coincides with the distribution \(\mathbb{P}_\rho\) of OQRW. This sheds new light on some properties of the measure \(\mathbb{P}_\rho \). As an example, we simply mention that the measure can be considered as a distribution of some functions of certain Markov processes. Furthermore, we study several properties of QMC and associated measures. A new notion of \(\varphi \)-recurrence of QMC is studied, and the relations between the concepts of recurrence introduced in this paper and the existing ones are established.
MSC:
| 81S25 | Quantum stochastic calculus |
| 60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |
| 46L35 | Classifications of \(C^*\)-algebras |
| 46L55 | Noncommutative dynamical systems |
References:
| [1] | Accardi, L., On noncommutative Markov property, Funct. Anal. Appl.8 (1975) 1-8. · Zbl 0395.60076 |
| [2] | Accardi, L., Local Perturbations of Conditional Expectations, J. Math. Anal. Appl.72 (1979) 34-69. · Zbl 0435.60100 |
| [3] | Accardi, L. and Fidaleo, F., Nonhomogeneous quantum Markov states and quantum Markov fields, J. Funct. Anal.200 (2003) 324-347. · Zbl 1031.46074 |
| [4] | Accardi, L. and Frigerio, A., Markovian cocycles, Proc. Royal Irish Acad. A83 (1983) 251-263. · Zbl 0534.46045 |
| [5] | Accardi, L. and Koroliuk, D., Stopping times for quantum Markov chains, J. Theor. Probab.5 (1992) 521-535. · Zbl 0751.60081 |
| [6] | Accardi, L. and Koroliuk, D., Quantum Markov chains: The recurrence problem, in Quantum Probability and Related Topics, QP-PQ, Vol. VI (World Scientific, 1991), pp. 63-73. · Zbl 0929.60094 |
| [7] | Accardi, L. and Watson, G. S., Quantum random walks, in Quantum Probability and Applications IV, Rome, 1987, eds. Accardi, L. and von Waldenfels, W., , Vol. 1396 (Springer, 1987), pp. 73-88. · Zbl 0674.60099 |
| [8] | Attal, S., Petruccione, F., Sabot, C. and Sinayskiy, I., Open quantum random walks, J. Stat. Phys.147 (2012) 832-852. · Zbl 1246.82039 |
| [9] | Attal, S., Guillotin-Plantard, N. and Sabot, C., Central limit theorems for open quantum random walks and quantum measurement records, Ann. Henri Poincaré16 (2015) 15-43. · Zbl 1319.81056 |
| [10] | Austin, T., Eisner, T. and Tao, T., Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems, Pacific J. Math.250 (2011) 1-60. · Zbl 1219.46058 |
| [11] | Bardet, I., Bernard, D. and Pautrat, Y., Passage times, exit times and Dirichlet problems for open quantum walks, J. Stat. Phys.167 (2017) 173-204. · Zbl 1376.82085 |
| [12] | Beboist, T., Jaksic, V., Pautrat, Y. and Pillet, C. A., On entropy production of repeated quantum measurament I. General Theory, Comm. Math. Phys.57(1) (2018) 77-123. · Zbl 1393.81008 |
| [13] | Bourgain, J., Grünbaum, F. A., Velázquez, L. and Wilkening, J., Quantum recurrence of a subspace and operator-valued Schur functions, Comm. Math. Phys.329 (2014) 1031-1067. · Zbl 1296.37014 |
| [14] | Burgarth, D., Chiribella, G., Giovannetti, V., Perinotti, P. and Yuasa, K., Ergodic and mixing quantum channels in finite dimensions, New J. Phys.15 (2013) 073045. · Zbl 1451.81043 |
| [15] | Burgarth, D. and Giovannetti, V., The generalized Lyapunov theorem and its application to quantum channels, New J. Phys.9 (2007) 150.1-150.18. |
| [16] | Carbone, R. and Pautrat, Y., Homogeneous open quantum random walks on a lattice, J. Stat. Phys.160 (2015) 1125-1152. · Zbl 1323.82020 |
| [17] | Carbone, R. and Pautrat, Y., Open quantum random walks: Reducibility, period, ergodic properties, Ann. Henri Poincaré17 (2016) 99-135. · Zbl 1333.81205 |
| [18] | Carvalho, S. L., Guidi, L. F. and Lardizabal, C. F., Site recurrence of open and unitary quantum walks on the line, Quantum Inf. Process.16 (2017) Art. 17, 32 pp. · Zbl 1374.81063 |
| [19] | Fagnola, F. and Rebolledo, R., Transience and recurrence of quantum Markov semigroups, Probab. Theory Relat. Fields126 (2003) 289-306. · Zbl 1024.60031 |
| [20] | Fannes, M., Nachtergaele, B. and Werner, R. F., Finitely correlated states on quantum spin chains, Comm. Math. Phys.144 (1992) 443-490. · Zbl 0755.46039 |
| [21] | Fannes, M., Nachtergaele, B. and Slegers, L., Functions of Markov processes and algebaraic measures, Rev. Math. Phys.4 (1992) 39-64. · Zbl 0751.60082 |
| [22] | Fidaleo, F. and Mukhamedov, F., Diagonalizability of nonhomogeneous quantum Markov states and associated von Neumann algebras, Probab. Math. Stat.24 (2004) 401-418. · Zbl 1088.46038 |
| [23] | Grünbaum, F. A., Velázquez, L., Werner, A. H. and Werner, R. F., Recurrence for discrete time unitary evolutions.Comm. Math. Phys.320 (2013) 543-569. · Zbl 1276.81087 |
| [24] | Konno, N. and Yoo, H. J., Limit theorems for open quantum random walks, J. Stat. Phys.150 (2013) 299-319. · Zbl 1259.82093 |
| [25] | Lardizabal, C. F. and Souza, R. R., On a class of quantum channels, open random walks and recurrence, J. Stat. Phys.159 (2015) 772-796. · Zbl 1328.82025 |
| [26] | Lardizabal, C. F., A quantization procedure based on completely positive maps and Markov operators, Quantum Inf. Process.12 (2013) 1033-1051. · Zbl 1264.81081 |
| [27] | Liu, C. and Petulante, N., On Limiting distributions of quantum Markov chains, Int. J. Math. and Math. Sciences.2011 (2011) 740816. · Zbl 1225.81080 |
| [28] | Liu, C. and 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. E81 (2010) 031113. |
| [29] | Nachtergaele, B., Working with quantum Markov states and their classical analogoues, in Quantum Probability and Applications V, , Vol. 1442 (Springer-Verlag, 1990), pp. 267-285. · Zbl 0717.60113 |
| [30] | Niculescu, C. P., Ströh, A. and Zsidó, L., Noncommutative extensions of classical and multiple recurrence theorems, J. Operator Theory50 (2003) 3-52. · Zbl 1036.46053 |
| [31] | Nielsen, M. A. and Chuang, I. L., Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000). · Zbl 1049.81015 |
| [32] | Norris, J. R., Markov Chains (Cambridge Univ. Press, 1997). · Zbl 0873.60043 |
| [33] | Novotný, J., Alber, G. and Jex, I., Asymptotic evolution of random unitary operations, Cent. Eur. J. Phys.8 (2010) 1001-1014. |
| [34] | Novotný, J., Alber, G. and Jex, I., Asymptotic properties of quantum Markov chains, J. Phys. A: Math. Theor.45 (2012) 485301. · Zbl 1278.81117 |
| [35] | Park, Y. M. and Shin, H. H., Dynamical entropy of generalized quantum Markov chains over infinite dimensional algebras, J. Math. Phys.38 (1997) 6287-6303. · Zbl 0901.46059 |
| [36] | Pellegrini, C., Continuous time open quantum random walks and non-Markovian Lindblad master equations, J. Stat. Phys.154 (2014) 838-865. · Zbl 1296.82028 |
| [37] | Portugal, R., Quantum Walks and Search Algorithms (Springer, 2013). · Zbl 1275.81004 |
| [38] | Venegas-Andraca, S. E., Quantum walks: A comprehensive review, Quantum Inf. Process.11 (2012) 1015-1106. · Zbl 1283.81040 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
