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. The difference of long-time behaviors
between CTOQW and other two continuous-time processes on graphs is exemplified. The examples
demonstrate that the structure of a graph can affect a quantum coherence effect on CTOQW
through a long time run. Precisely, a quantum coherence effect 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, a quantum coherence effect will eventually vanish from the
open quantum system when the underlying topology is a regular graph (such as a cycle).
Radhakrishnan Balu and Chaobin Liu, "Time-continuous open quantum walks (Conference Presentation)," Proc. SPIE 10193, Ultrafast Bandgap Photonics II, 101931A (Presented at SPIE Defense + Security: April 13, 2017; Published: 7 June 2017); https://doi.org/10.1117/12.2262502.5459357381001.
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