Thursday, December 27, 2012

1212.6078 (Eman Hamza et al.)

Spectral Transition for Random Quantum Walks on Trees    [PDF]

Eman Hamza, Alain Joye
We define and analyze random quantum walks on homogeneous trees of degree $q\geq 3$. Such walks describe the discrete time evolution of a quantum particle with internal degree of freedom in $\C^q$ hopping on the neighboring sites of the tree in presence of static disorder. The one time step random unitary evolution operator on the Hilbert space of the particle depends on a unitary matrix $C\in U(q)$ which monitors the strength of the disorder. We prove for any $q$ that there exist distinct open sets of matrices in $U(q)$ for which the random evolution is either pure point almost surely or absolutely continuous, thereby showing the existence of a spectral transition driven by $C\in U(q)$. For $q=3$ and $q=4$, we establish some properties of the spectral diagram which allows us to describe the spectral transition.
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