Open Quantum Random Walks: Ergodicity, Hitting Times, Gambler’s Ruin and Potential Theory
- 193 Downloads
In this work we study certain aspects of open quantum random walks (OQRWs), a class of quantum channels described by Attal et al. (J Stat Phys 147: 832–852, 2012). 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 Saloff-Coste and Zúñiga (Stoch Proc Appl 117: 961–979, 2007), 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.
The authors are grateful to an anonymous referee for several useful suggestions that led to marked improvements of the paper. C.F.L. is partially supported by a CAPES/PROAP grant to the Graduate Program in Mathematics—PPGMat/UFRGS. R.R.S. is partially supported by FAPERGS (proc. 002063-2551/13-0). The authors would like to thank C. Liu, T. Machida, S. E. Venegas-Andraca, N. Petulante, F. Petruccione and F. A. Grünbaum for stimulating discussions on this line of research.
- 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)Google Scholar
- 10.Brémaud, P.: Markov Chains: Gibbs Fields, Monte Carlo Simulation and Queues. Texts in Applied Mathematics 31. Springer, Berlin (1999)Google Scholar
- 13.Carbone, R., Pautrat, Y.: Homogeneous open quantum random walks on a lattice. Preprint arXiv:1408.1113v2
- 14.Carbone, R., Pautrat, Y.: Open quantum random walks: reducibility, period, ergodic properties. Preprint arXiv:1405.2214v3
- 27.Liu, C., Petulante, N.: On limiting distributions of quantum Markov chains. Int. J. Math. Math. Sci., Vol 2011 (2011). ID 740816Google Scholar
- 34.Novotný, J., Alber, G., Jex, I.: Asymptotic evolution of random unitary operations. Cent. Eur. J. Phys. 8(6), 1001–1014 (2010)Google Scholar
- 45.Watrous, J.: Theory of Quantum Information Lecture Notes from Fall 2011. Institute for Quantum Computing, University of WaterlooGoogle Scholar
- 46.Wolf, M.M.: Quantum Channels & Operations Guided Tour Lecture Notes (unpublished). 5 July 2012Google Scholar