Pure state entanglement and stabilizer representations

This thesis is concerned with entanglement in quantum systems and representation of the states of these systems using finite Abelian groups called stabilizers. In particular, n-qubit pure states are studied. Because entanglement plays a very important role in the theory of quantum information processing it is a topic of much interest in this field.; Entangled states are classified using physically possible operations under a typical configuration; spatially separated parties who each hold part of the quantum system over which a possibly entangled state is distributed. The most general of these operations are stochastic local operations and classical communications (SLOCC). Entanglement is quantified using functions of the density matrix of a state which behave monotonically under local operations and classical communication (LOCC).; Although entangled states usually depend on an exponential number of parameters, a special subclass of n-qubit pure states known as stabilizer states can efficiently be represented by finite Abelian groups known as stabilizers. This formalism plays a crucial role in quantum error correction [16] and stabilizer states are used in a revolutionary scheme for quantum computation known as measurement based computation [32, 34]. A homomorphism between stabilizer states and simple graphs has been found which opened up a new way of studying their properties [36, 40] in an intuitive and efficient manner. Using this representation, equivalency classes under local unitary operations can be defined by only n2 real parameters. The research done for this project was driven by the wish to generalize the conventional stabilizer formalism for all n-qubit pure states and the idea of using it to distinguish SLOCC-inequivalent states. What is presented in chapters 1, 2 and 3 of this thesis is an overview of the topics that are relevant to the research in this project. The original work and findings are contained in chapter 4.