RHUL-MA-2001-1
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.