In this thesis we explore several areas which lie between the traditional condensed matter
physics and the relatively more modern quantum information processing paradigms. We
move from a single-body problem which displays chaotic behaviour, to two many-body systems
which show a quantum phase transition and sufficient entanglement to perform complex
quantum information processing, respectively.
Firstly, we consider a set of operations which can easily be performed on a single ion in an
ion trap quantum computer, and associate this with an effective Hamiltonian. We determine
the classical and quantum bifurcation structure, and consider the physical signatures of
a bifurcation which may be measured in an experiment. Secondly, we consider a set of
operations which may easily be performed on many ions in an ion trap quantum computer.
We determine that the evolution of the system may be described by an effective Hamiltonian,
and determine the nature of an effective quantum phase transition which may be experienced
by the system. We determine to what extent entanglement is present in this system, and
how even relatively small finite systems may give us insights into quantum phase transitions.
Finally, in the many-body setting, we consider the use of teleportation as a characteristic
of entanglement, and propose a teleportation scheme based on Bell basis measurement. We
determine that non-local, topological order parameters can be used to find the entanglement
content necessary to perform a teleportation, and that we can understand the resources
required for teleportation in terms of a quantum information paradigm that transcends
traditional local order parameters.
In all three systems, we consider models which have “tunable” local and non-local interactions.
Since the “quintessential feature of quantum mechanics” is the superposition of
non-local states, it is natural to refer, time and again, to entanglement to variously describe
the qualitative phenomenon of a system’s non-locality, its phase transition or its ability to
perform a quantum information processing task. In each case, using a quantitative measure
of entanglement, we are able to present concrete evidence for the essential role which entanglement
plays in determining the physics we may observe from these quantum phenomenon.