Gnacik, Michal (2014) Quantum random walks. PhD thesis, UNSPECIFIED.
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Abstract
In this thesis we investigate the convergence of various quantum random walks to quantum stochastic cocycles defined on a Bosonic Fock space. We prove a quantum analogue of the Donsker invariance principle by invoking the so-called semigroup representation of quantum stochastic cocycles. In contrast to similar results by other authors our proof is relatively elementary. We also show convergence of products of ampliated random walks with different system algebras; in particular, we give a sufficient condition to obtain a cocycle via products of cocycles. The CCR algebra, its quasifree representations and the corresponding quasifree stochastic calculus are also described. In particular, we study in detail gauge-invariant and squeezed quasifree states. We describe repeated quantum interactions between a `small' quantum system and an environment consisting of an infinite chain of particles. We study different cases of interaction, in particular those which occur in weak coupling limits and low density limits. Under different choices of scaling of the interaction part we show that random walks, which are generated by the associated unitary evolutions of a repeated interaction system, strongly converge to unitary quantum stochastic cocycles. We provide necessary and sufficient conditions for such convergence. Furthermore, under repeated quantum interactions, we consider the situation of an infinite chain of identical particles where each particle is in an arbitrary faithful normal state. This includes the case of thermal Gibbs states. We show that the corresponding random walks converge strongly to unitary cocycles for which the driving noises depend on the state of the incoming particles. We also use conditional expectations to obtain a simple condition, at the level of generators, which suffices for the convergence of the associated random walks. Limit cocycles, for which noises depend on the state of the incoming particles, are also obtained by investigating what we refer to as `compressed' random walks. Lastly, we show that the cocycles obtained via the procedure of repeated quantum interactions are quasifree, thus the driving noises form a representation of the relevant CCR algebra. Both gauge-invariant and squeezed representations are shown to occur.