TITLE

Perturbation bounds for quantum Markov processes and their fixed points

AUTHOR(S)
Szehr, Oleg; Wolf, Michael M.
PUB. DATE
March 2013
SOURCE
Journal of Mathematical Physics;Mar2013, Vol. 54 Issue 3, p032203
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
We investigate the stability of quantum Markov processes with respect to perturbations of their transition maps. In the first part, we introduce a condition number that measures the sensitivity of fixed points of a quantum channel to perturbations. We establish upper and lower bounds on this condition number in terms of subdominant eigenvalues of the transition map. In the second part, we consider quantum Markov processes that converge to a unique stationary state and we analyze the stability of the evolution at finite times. In this way we obtain a linear relation between the mixing time of a quantum Markov process and the sensitivity of its fixed point with respect to perturbations of the transition map.
ACCESSION #
86446991

 

Related Articles

  • BLOCKWISE PERTURBATION THEORY FOR 2X2 BLOCK MARKOV CHAINS. Xue, Jun-gong; Gao, Wei-guo // Journal of Computational Mathematics;May2000, Vol. 18 Issue 3, p305 

    Presents a study on the perturbation theory of Markov chains. Basic lemmas for Markov chains; Results.

  • Filtering with Discrete State Observations. Dufour, F.; Elliott, R.J. // Applied Mathematics & Optimization;Sep/Oct99, Vol. 40 Issue 2, p259 

    The problem of estimating a finite state Markov chain observed via a process on the same state space is discussed. Optimal solutions are given for both the "weak" and "strong" formulations of the problem. The "weak" formulation proceeds using a reference probability and a measure change for...

  • Convergence properties of perturbed Markov chains. Roberts, Gareth O.; Rosenthal, Jeffrey S.; Schwartz, Peter O. // Journal of Applied Probability;Mar1998, Vol. 35 Issue 1, p1 

    Presents a study that considers the question of which convergence properties of Markov chains are preserved under small perturbations. Properties of Markov chains; Robustness of geometric ergodicity under roundoff error; Robustness of the stationary distributions.

  • Random perturbations of iterated maps. Salazar-Anaya, Gelasio; Urias, Jesus // Journal of Mathematical Physics;Jul96, Vol. 37 Issue 7, p3641 

    Studies random perturbations of Markov processes on systems of contractive maps. Proof of the existence of an invariant measure for the randomly perturbed process under a very weak assumption on perturbations; Types of physical random perturbation.

  • Asymptotic Expansions for Stationary and Quasi-Stationary Distributions of Perturbed Semi-Markov Processes. Silvestrov, Dmitrii; Silvestrov, Sergei // AIP Conference Proceedings;2017, Vol. 1798 Issue 1, p1 

    New algorithms for computing asymptotic expansions, without and with explicit upper bounds for remainders, for stationary and quasi-stationary distributions of nonlinearly perturbed semi-Markov processes are presented. The algorithms are based on special techniques of sequential phase space...

  • STABILITY AND EXPONENTIAL CONVERGENCE OF CONTINUOUS-TIME MARKOV CHAINS. Mitrophanov, A. Yu. // Journal of Applied Probability;Dec2003, Vol. 40 Issue 4, p970 

    For finite, homogeneous, continuous-time Markov chains having a unique stationary distribution, we derive perturbation bounds which demonstrate the connection between the sensitivity to perturbations and the rate of exponential convergence to stationarity. Our perturbation bounds substantially...

  • On the eigenvalue effective size of structured populations. H√∂ssjer, Ola // Journal of Mathematical Biology;Sep2015, Vol. 71 Issue 3, p595 

    A general theory is developed for the eigenvalue effective size ( $$N_{eE}$$ ) of structured populations in which a gene with two alleles segregates in discrete time. Generalizing results of Ewens (Theor Popul Biol 21:373-378, ), we characterize $$N_{eE}$$ in terms of the largest non-unit...

  • Asymptotic evolution of quantum walks with random coin. Ahlbrecht, A.; Vogts, H.; Werner, A. H.; Werner, R. F. // Journal of Mathematical Physics;Apr2011, Vol. 52 Issue 4, p042201 

    We study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain. In the unitary (i.e., nonrandom) case, we allow any unitary operator which commutes with translations and couples...

  • Asymptotic linear programming and policy improvement for singularly perturbed Markov decision processes. Altman, Eitan; Avrachenkov, Konstantin E.; Filar, Jerzy A. // Mathematical Methods of Operations Research;1999, Vol. 49 Issue 1, p97 

    Abstract. In this paper we consider a singularly perturbed Markov decision process with finitely many states and actions and the limiting expected average reward criterion. We make no assumptions about the underlying ergodic structure. We present algorithms for the computation of a uniformly...

Share

Read the Article

Courtesy of THE LIBRARY OF VIRGINIA

Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics