Communicating with Transverse Modes of Light

Shannon's theory of communication created a set of tools for studying complex systems in an abstract and powerful way, providing the core foundations for the field of information theory. This thesis uses these ideas to provide a framework for studying the transverse degree of freedom of an optical field, appropriate for both classical and quantum states of light. This degree of freedom is in principle an unbounded space, providing a complex resource for encoding a large amount of information. This work focuses on studying the physical limits to the information of this space, both in terms of fundamental theoretical limitations as well as practical limitations due to experimental implementation and error. This thesis will pay particular interest to the design and implementation of a quantum key distribution system encoded using a particular set of transverse modes for encoding known as orbital angular momentum states, which represent normal modes of a typical free-space optical system. This specific technological implementation provides a motivation that acts to unify many of the themes in this work including quantum state preparation, state detection or discrimination, and state evolution or propagation. Additionally, such a setup gives a specific physical meaning to the abstract tools we will be utilizing as the information that we will be quantifying can be thought of as a measure of the possible complexity or information content of a single photon. Chapter 1 provides a brief introduction to information theory and the basic concepts and tools that are used throughout this work, as well as a basic introduction to quantum key distribution. Chapter 2 theoretically explores the fundamental limits of the information capacity of a channel due to diffraction, as well as computes the communication modes of a channel using a normal mode approach to propagation. Chapter 3 concerns the experimental implementation of a free-space quantum key distribution system including quantum state preparation and detection, as well as demonstration of a working system. Finally, in chapter 4 we consider the effects of a noisy channel on our analysis, especially decoherence due to the presence of atmospheric turbulence.