Measures of Dynamical Complexity

Abstract

The aim of the thesis is to define, develop, and consider applications of different measures of dynamical complexity, i.e.\ the measures that would quantify complexity of system dynamics. These measures are based on the two fundamental notions of Kolmogorov (or algorithmic) complexity and von Neumann entropy. Our main applications are in the theory of chaos and of open quantum systems. In such applications the interaction of the system with its environment is crucial. Consider a joint quantum state of a system and its environment. A measurement on the environment induces a decomposition of the system state. Using algorithmic information theory, we define the preparation information of a pure or mixed quantum state in a given decomposition. We demonstrate that the minimal value, $I_{\rm min}$, of the average preparation information of the system state characterizes the complexity of system-environment correlations which develop as a result of the system dynamics. Comparing the change of $I_{\rm min}$ with the change of the von~Neumann entropy $\Delta H$ of the system induced by an optimal measurement we introduce a measure of complexity of the system dynamics ($\chi\equiv I_{\rm min} /\Delta H$). We discuss this measure of dynamical complexity in the context of the hypersensitivity approach to quantum chaos. The partial development of a quantum version of symbolic dynamics for the quantum baker's map is one of the achievements presented in this thesis. Although our methods are not yet as powerful and general as the classical symbolic dynamics, we were able to recover the classical symbolic dynamics for the baker's map starting from a purely quantum version of the map and taking the classical limit. We use these results in the framework of the decoherent (consistent) histories approach to introduce a measure of dynamical complexity which is conceptually equivalent to the Kolmogorov-Sinai entropy which quantifies the degree of chaos in classical systems. Often the mathematical formalism of algorithmic measures of complexity is very difficult to apply in a concrete physical setting. In such cases entropy-like measures of dynamical complexity can become the only practical choice. We consider a general setting of homodyne measurements in cavity QED. As our first objective we use the formalism of stochastic master equations to calculate the system entropy reduction due to the measurements. This quantity provides fundamental limits on the experimental resolution of the conditional system dynamics. We go beyond the limitations of the formalism of stochastic master equations end develop analytical tools for calculating the system dynamics conditional on the discrete photocount record.