Measurement, Decoherence and Master Equations

Abstract

In the first part of this thesis we concern ourselves with the problem of generating pseudo-random circuits. These are a series of quantum gates chosen at random, with the overall effect of implementing unitary operations with statistical properties close to that of unitaries drawn at random with respect to the Haar measure. Such circuits have a growing number of applications in quantum-information processing, but all known algorithms require an external input of classical randomness. We suggest a scheme to implement random circuits in a weighted graph state. The input state is entangled with the weighted graph state and a random circuit is implemented by performing local measurements in one fixed basis only. A central idea in the analysis of this proposal is the average bipartite entanglement generated by the repeated application of such circuits on a large number of randomly chosen input product states. For a truly random circuit, this should agree with that obtained by applying unitaries at random chosen uniformly with respect to the Haar measure, values which can be calculated using Pages Conjecture. Part II is largely concerned with continuous variables (CV) systems. In particular, we are interested in two descriptions. That of the class of Gaussian states, and that of systems which can be adequately described through the use of Markovian master equations. In the case of the latter, there are a number of approaches one may take in order to derive a suitable equation, all of which require some sort of approximation. These approximations can be made based on a mixture of mathematical and physical grounds. However, unfortunately it is not always clear how justified we are in making a particular choice, especially when the test system we wish to describe includes its own internal interactions. In an attempt to clarify this situation, we derive Markovian master equations for single and interacting harmonic systems under different scenarios, including strong internal coupling. By comparing the dynamics resulting from the corresponding master equations with numerical simulations of the global systems evolution, we assess the robustness of the assumptions usually made in the process of deriving the reduced Markovian dynamics. This serves to clarify the general properties of other open quantum system scenarios subject to treatment within a Markovian approximation. Finally, we extend the notions of the smooth min- and smooth max-entropies to the continuous variable setting. Specifically, we have provided expressions to evaluate these measures on arbitrary Gaussian states. These expressions rely only on the symplectic eigenvalues of the corresponding covariance matrix. As an application, we have considered their use as a suitable measure for detecting thermalisation