Entanglem ent and quantum information theory

Quantum information theory is an exciting, inter-disciplinary field, combining elements of condensed matter theory, quantum mechanics and information theory. In this thesis, I shall make a modest contribution to this field by examining entanglement in many-body systems with more than two levels. In the first section, I consider the dynamics of a system of qutrits—three-level quantum systems— which are coupled through an SU(3)-invariant permutation Hamiltonian. Each term in this Hamil­ tonian is a nearest-neighbour permutation operator, and thus this Hamiltonian may be considered a generalisation of the standard SU (2)-invariant Heisenberg Hamiltonian, in which every term (up to the addition of the identity operator) is a nearest-neighbour permutation operator for two-level system. The system considered has the topology of a cross, and thus may be considered (to a limited extent) analogous to a beam-splitter. The aim of the study is to establish a Bell singlet state between two distant parties. Building on this work, I shall go on to consider the ground state of a system made up of many-level systems coupled by the same Hamiltonian; I shall show that this state is a generalisation of the two-level singlet to many levels and many systems. It thus has a high degree of symmetry. I will consider its application in entanglement distribution through measurements (localisable entanglement), and discuss how it may be physically implemented in systems of ultracold atoms, through the Hubbard model. I shall also show that in the famous valence bond solid (the ground state of the Affleck-KennedyLieb-Tasaki spin chain), all the entanglement present in the state may be extracted from a single copy of the chain; this is in contrast to gapless, critical chains, in which only half the total entanglement is extractable from a single copy. This thesis does not exceed 100,000 words.