Stabilized distance measures and quantum error correction
Quantum information theory is a quickly-growing area of research that presents no shortage of mathematical challenges. In this thesis, two basic analytic and algebraic problems of interest in quantum information are considered. The first problem considered is that of computing a crucial distance measure for linear maps on finite-dimensional Hilbert space, given by the diamond and completely bounded norms of differences of quantum operations. Based on the theory of completely bounded maps, an algorithm to compute the diamond and completely bounded norms of arbitrary linear maps is formulated and presented. The algorithm is applied to derive a new proof and formula for the distance between arbitrary unitary maps. Finally, an implementation of the algorithm via MATLAB is presented, and its efficiency is discussed. Attention is next turned to quantum error correction, where a new algebraic characterization of error-correcting codes is derived. These results are used to explicitly compute a correction operation, and a new characterization of correctable subsystems in terms of representation theory is obtained.