Classical and Quantum Metric Spaces

Classical error correction has a natural geometric formulation: a classical code is a collection of well-separated points in a metric space, and one of the fundamental bounds is simply a sphere-packing bound. In this talk, I will describe an algebraic reformulation of finite metric spaces, following Kuperberg and Weaver, in terms of filtered relations, and explain how replacing the underlying commutative algebra by a noncommutative one leads to the notion of a quantum metric space. Within this framework, quantum error correction is the natural noncommutative geometric analog of classical error correction. Time permitting, I will explain how association schemes can lead to stronger bounds in the classical setting and explore current efforts to develop an analogous theory for quantum metric spaces.