The foundations of superposition and its use in quantum walks on complex networks

In classical physics, it is possible in principle to predict with certainty the outcome of any measurement that we might perform on a system. Where we are not able to predict outcome with certainty, we can attribute the uncertainty to incompleteness in our knowledge about the history of the system (its pre-measurement state), or the behaviour of the apparatus we use to measure it. Quantum theory, on the other hand, describes systems where what we can find out about the pre-measurement state is significantly restricted: we are not generally able to predict with certainty the outcome of an arbitrary measurement, even with full knowledge of that system’s history. Superposition states are at the heart of this phenomenon, and do not have an analogue in classical physics. In this thesis we examine superposition states from two points of view. Firstly, following long-running arguments about whether quantum theory can be considered complete, we examine the possibility of an underlying ‘ontological’ model of quantum theory that explains the quantum measurement statistics. We derive the first such model that has the property that a single state in the underlying model is compatible with distinct quantum states, and recovers the measurement statistics for systems of dimension greater than two. Secondly, we examine the dynamics of continuous time quantum walks in complex networks. Discovering community structure in these networks is a useful task to be able to perform, and there are several algorithms using the classical dynamics of random walks to return that structure. We introduce a new ‘centrality measure’ based on the observable dynamics of a quantum walker, and provide an algorithm using our new measure for community detection.Open Acces