Orbits and Invariants in Quantum Information Theory

Abstract

In the field of quantum information theory, one studies---among other things---the information processing tasks that can be achieved by taking advantage of a quantum phenomenon known as entanglement. There is a mathematical formalism that captures the notion of entanglement by elements of a vector space called state vectors. The local unitary and SLOCC (stochastic local operations with classical communication) groups act on this space, producing natural equivalence classes of state vectors. In this work, we consider group actions, their invariants, and how these can be used to classify and distinguish state vectors.

In Chapter 1, we give an introduction to quantum information theory. In Chapter 2, we show how results from Lie theory can be used to help find stationary points of invariant polynomials; these points correspond to highly entangled states. In Chapter 3, we discuss the problem of classifying orbits in these state spaces.