This thesis addresses the problem of developing a quantum counter-part of the well established classical theory of control. We dwell on the fundamental fact that quantum states are generally not perfectly distinguishable, and quantum measurements typically introduce noise in the system being measured. Because of these, it is generally not clear whether the central concept of the classical control theory --- that of observing the system and then applying feedback --- is always useful in the quantum setting. We center our investigations around the problem of transforming the state of a quantum system into a given target state, when the system can be prepared in different ways, and the target state depends on the choice of preparation. We call this the "quantum tracking problem" and show how it can be formulated as an optimization problem that can be approached both numerically and analytically. This problem provides a simple route to the characterization of the quantum trade-off between information gain and disturbance, and is seen to have several applications in quantum information. In order to characterize the optimality of our tracking procedures, some figure-of-merit has to be specified. Naturally, distance measures for quantum states are the ideal candidates for this purpose. We investigated several possibilities, and found that there is usually a compromise between physically motivated and mathematically tractable measures. We also introduce an alternative to the Uhlmann-Jozsa fidelity for mixed quantum states, which besides reproducing a number of properties of the standard fidelity, is especially attractive because it is simpler to compute. We employ some ideas of convex analysis to construct optimal control schemes analytically. In particular, we obtain analytic forms of optimal controllers for stabilizing and tracking any pair of states of a single-qubit. In the case of stabilization, we find that feedback control is always useful, but because of the trade-off between information gain and disturbance, somewhat different from the type of feedback performed in classical systems. In the case of tracking, we find that feedback is not always useful, meaning that depending on the choice of states one wants to achieve, it may be better not to introduce any noise by the application of quantum measurements. We also demonstrate that our optimal controllers are immediately applicable in several quantum information applications such as state-dependent cloning, purification, stabilization, and discrimination. In all of these cases, we were able to recover and extend previously known optimal strategies and performances. Finally we show how optimal single-step control schemes can be concatenated to provide multi-step strategies that usually over-perform optimal control protocols based on a single interaction between the controller and the system.
| Identifer | oai:union.ndltd.org:ADTP/287411 |
| Creators | Paulo Marques Furtado de Mendonca |
| Source Sets | Australiasian Digital Theses Program |
| Detected Language | English |
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