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Quantum random walk approximation in Banach algebra

B. Krishna Das, J. Martin Lindsay

Journal of Mathematical Analysis and ApplicationsFeb 19, 20155 citations

Markov structure of quantum Lévy processes

Uwe Franz, René Schott

Springer eBooksJan 1, 19990 citations

Quantum Random Walk Approximation on Locally Compact Quantum Groups

J. Martin Lindsay, Adam Skalski

A natural scheme is established for the approximation of quantum Lévy processes on locally compact quantum groups by quantum random walks. We work in the somewhat broader context of discrete approximations of completely positive quantum stochastic convolution cocycles on C *-bialgebras.

Letters in Mathematical PhysicsFeb 16, 20134 citations

Quantum random walk approximation on locally compact quantum groups

J. Martin Lindsay, Adam Skalski

A natural scheme is established for the approximation of quantum Levy processes on locally compact quantum groups by quantum random walks. We work in the somewhat broader context of discrete approximations of completely positive quantum stochastic convolution cocycles on C*-bialgebras.

arXiv:1110.3990Oct 18, 20110 citations

A Quantum Decomposition of Lévy Processes

Yuh-Jia Lee, Hsin-Hung Shih

Stochastic Analysis: Classical and QuantumOct 1, 20053 citations

The quantum Lévy walk

Manuel O Cáceres, Marco Nizama

We introduce the quantum Lévy walk to study transport and decoherence in a quantum random model. We have derived from second-order perturbation theory the quantum master equation for a Lévy-like particle that moves along a lattice through scale-free hopping while interacting with a thermal bath of oscillators. The general evolution of the quantum Lévy particle has been solved for different preparations of the system. We examine the evolution of the quantum purity, the localized correlation and the probability to be in a lattice site, all of them leading to important conclusions concerning quantum irreversibility and decoherence features. We prove that the quantum thermal mean-square displacement is finite under a constraint that is different when compared to the classical Weierstrass random walk. We prove that when the mean-square displacement is infinite the density of state has a complex null-set inside the Brillouin zone. We show the existence of a critical behavior in the continuous eigenenergy which is related to its non-differentiability and self-affine characteristics. In general, our approach allows us to study analytically quantum fluctuations and decoherence in a long-range hopping model.We introduce the quantum Lévy walk to study transport and decoherence in a quantum random model. We have derived from second-order perturbation theory the quantum master equation for a Lévy-like particle that moves along a lattice through scale-free hopping while interacting with a thermal bath of oscillators. The general evolution of the quantum Lévy particle has been solved for different preparations of the system. We examine the evolution of the quantum purity, the localized correlation and the probability to be in a lattice site, all of them leading to important conclusions concerning quantum irreversibility and decoherence features. We prove that the quantum thermal mean-square displacement is finite under a constraint that is different when compared to the classical Weierstrass random walk. We prove that when the mean-square displacement is infinite the density of state has a complex null-set inside the Brillouin zone. We show the existence of a critical behavior in the continuous eigenenergy which is related to its non-differentiability and self-affine characteristics. In general, our approach allows us to study analytically quantum fluctuations and decoherence in a long-range hopping model.

arXiv:1112.3881Journal of Physics A Mathematical and TheoreticalOct 25, 201017 citations

Approximation of quantum Lévy processes by quantum random walks

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