Upper and lower bounds on quantum codes

This thesis provides bounds on the performance of quantum error correcting codes when used for quantum communication and quantum key distribution. The first two chapters provide a bare-bones introduction to classical and quantum error correcting codes, respectively. The next four chapters present achievable rates for quantum codes in various scenarios. The final chapter is dedicated to an upper bound on the quantum channel capacity. Chapter 3 studies coding for adversarial noise using quantum list codes, showing there exist quantum codes with high rates and short lists. These can be used, together with a very short secret key, to communicate with high fidelity at noise levels for which perfect fidelity is, impossible. Chapter 4 explores the performance of a family of degenerate codes when used to communicate over Pauli channels, showing they can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code and that exceed those of previously known degenerate codes. By studying the scaling of the optimal block length as a function of the channel's parameters, we develop a heuristic for designing even better codes. Chapter 5 describes an equivalence between a family of noisy preprocessing protocols for quantum key distribution and entanglement distillation protocols whose target state belongs to a class of private states called twisted states. In Chapter 6, the codes of Chapter 4 are combined with the protocols of Chapter 5 to provide higher key rates for one-way quantum key distribution than were previously thought possible. Finally, Chapter 7 presents a new upper bound on the quantum channel that is both additive and convex, and which can be interpreted as the of the channel for communication given access to side channels from a class of zero cloning channels. This clone assisted capacity is equal to the unassisted for channels that are degradable, which we use to find new upper bounds on the of a depolarizing channel.