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Abstract:
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Open quantum systems refer to systems that are affected by
interaction with the environment . The effects of these unwanted
interactions , called \emph{quantum noise} , are studied using
dynamical maps . We study the geometry of these maps in this work .
We review the canonical representations of dynamical maps such as
reduced dynamics , $ \mathcal{A} $ and $ \mathcal{B} $ forms and
operator sum representation . We develop a framework for
simplifying the action of dynamical maps in terms of their action
on the coherence vector associated with the density matrix . We use
the framework to describe the geometry of depolarization ,
dephasing and dissipation in the domain of complete positivity . We
give a geometric picture of how two - , three - and four -level
systems are affected by these common forms of quantum noises . We
show useful similarities between two - and four -level depolarizing
maps and give a generalization for $n $ -qubits . We also derive
important results that restrict dephasing and dissipation . |