Steady states of continuous-time open quantum walks



Continuous-time open quantum walks (CTOQW) are introduced as the formulation of quantum dynamical semigroups of trace-preserving and completely positive linear maps (or quantum Markov semigroups) on graphs. We show that a CTOQW always converges to a steady state regardless of the initial state when a graph is connected. When the graph is both connected and regular, it is shown that the steady state is the maximally mixed state. As shown by the examples in this article, the steady states of CTOQW can be very unusual and complicated even though the underlying graphs are simple. The examples demonstrate that the structure of a graph can affect quantum coherence in CTOQW through a long-time run. Precisely, the quantum coherence persists throughout the evolution of the CTOQW when the underlying topology is certain irregular graphs (such as a path or a star as shown in the examples). In contrast, the quantum coherence will eventually vanish from the open quantum system when the underlying topology is a regular graph (such as a cycle).


Semigroups on graphs Continuous-time open quantum walks Steady states 



We are pleased to acknowledge that we had useful communication with Carlos F. Lardizabal when we were working on this article. We thank undergraduate research assistant Carlos Sanchez for computing the steady state in the second example. CL gratefully acknowledges the support from ARL through ARL Faculty Fellow Research Team Program.


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsBowie State UniversityBowieUSA
  2. 2.U.S. Army Research LaboratoryComputational and Information Sciences DirectorateAdelphiUSA

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