Spectral Transition for Random Quantum Walks on Trees

Abstract

We define and analyze random quantum walks on homogeneous trees of degree $q\ge 3$. 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 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 open sets of matrices in $U(q)$ for which the random evolution has either pure point spectrum almost surely or purely absolutely continuous spectrum, thereby showing the existence of a spectral transition driven by $C\in U(q)$. For $q\in \{3,4\}$, we establish properties of the spectral diagram which provide a description of the spectral transition.

25 pages, 7 figures

References in corpus (6)

• Quantum Walk in Position Space with Single Optically Trapped Atoms
• Three-photon bosonic coalescence in an integrated tritter
• Open Quantum Random Walks
• Asymptotic evolution of quantum walks with random coin
• Quantum Walks on Trees with Disorder: Decay, Diffusion, and Localization
• Localization and Fractality in Inhomogeneous Quantum Walks with Self-Duality

Cited by in corpus (8)

• Spectral Stability of Unitary Network Models
• Exponential Decay of Matrix $\Phi$-Entropies on Markov Semigroups with Applications to Dynamical Evolutions of Quantum Ensembles
• Thermalization of Fermionic Quantum Walkers
• Spectral Properties of Non-Unitary Band Matrices
• Spectral and scattering properties of quantum walks on homogenous trees of odd degree
• Lower Bounds on the Localisation Length of Balanced Random Quantum Walks
• On absolutely continuous spectrum for one-channel unitary operators
• Dynamical Localization and Transport properties of Quantum Walks on the hexagonal lattice