On a class of quantum channels, open random walks and recurrence. (English) Zbl 1328.82025

Summary: We study a particular class of trace-preserving completely positive maps, called PQ-channels, for which classical and quantum evolutions are isolated in a certain sense. By combining open quantum random walks with a notion of recurrence, we are able to describe criteria for recurrence of the walk related to this class of channels. Positive recurrence for open walks is also discussed in this context.


82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
Full Text: DOI arXiv


[1] Alicki, R., Fannes, M.: Quantum Dynamical Systems. Oxford University Press, Oxford (2000) · Zbl 1140.81308
[2] Andersson, E; Cresser, JD; Hall, MJW, Finding the Kraus decomposition from a master equation and vice versa, J. Mod. Opt., 54, 1695, (2007) · Zbl 1139.81010
[3] Attal, S; Petruccione, F; Sabot, C; Sinayskiy, I, Open quantum random walks, J. Stat. Phys., 147, 832-852, (2012) · Zbl 1246.82039
[4] Benatti, F.: Dynamics Information and Complexity in Quantum Systems. Springer, Berlin (2009) · Zbl 1173.81001
[5] Bourgain, J., Grünbaum, F.A., Velázquez, L., Wilkening, J.: Quantum recurrence of a subspace and operator-valued Schur functions. arXiv:1302.7286v1
[6] Burgarth, D; Chiribella, G; Giovannetti, V; Perinotti, P; Yuasa, K, Ergodic and mixing quantum channels in finite dimensions, New J. Phys., 15, 073045, (2013)
[7] Burgarth, D; Giovannetti, V, The generalized Lyapunov theorem and its application to quantum channels, New J. Phys., 9, 150, (2007)
[8] Durrett, R.: Probability: Theory and Examples, 3rd edn. Duxbury Press, Belmont (1996) · Zbl 0709.60002
[9] Grimmett, G.R., Stirzaker, D.R.: Probability and Random Processes, 2nd edn. Oxford University Press, Oxford (1992) · Zbl 0759.60002
[10] Grünbaum, FA; Velázquez, L; Werner, AH; Werner, RF, Recurrence for discrete time unitary evolutions, Commun. Math. Phys., 320, 543-569, (2013) · Zbl 1276.81087
[11] Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, New York (1991) · Zbl 0729.15001
[12] Landau, LJ; Streater, RF, On birkhoff’s theorem for doubly stochastic completely positive maps of matrix algebras, Lin. Alg. Appl., 193, 107-127, (1993) · Zbl 0797.15021
[13] Liu, C., Petulante, N.: On limiting distributions of quantum Markov chains. Int. J. Math. and Math. Sciences. Volume 2011, ID 740816 · Zbl 1225.81080
[14] Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000) · Zbl 1049.81015
[15] Norris, J.R.: Markov Chains. Cambridge University Press, Cambridge (1997) · Zbl 0873.60043
[16] Novotný, J; Alber, G; Jex, I, Asymptotic evolution of random unitary operations, Cent. Eur. J. Phys., 8, 1001-1014, (2010) · Zbl 1201.93127
[17] Paulsen, V.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2003) · Zbl 1029.47003
[18] Petz, D.: Quantum Information Theory and Quantum Statistics. Springer, Berlin (2008) · Zbl 1145.81002
[19] Raginsky, M, Radon-Nikodym derivatives of quantum operations, J. Math. Phys., 44, 5003, (2003) · Zbl 1062.81011
[20] Štefaňák, M; Jex, I; Kiss, T, Recurrence and Pólya number of quantum walks, Phys. Rev. Lett., 100, 020501, (2008)
[21] Takesaki, M.: Theory of Operator Algebras I. Springer, New York (1979) · Zbl 0436.46043
[22] Watrous, J.: Theory of Quantum Information. Lecture Notes from Fall 2011. Institute for Quantum Computing, University of Waterloo (2011) · Zbl 1149.81007
[23] Wolf, M; Cirac, JI, Dividing quantum channels, Commun. Math. Phys., 279, 147-168, (2008) · Zbl 1149.81007
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.