Quantum information theory as asymptotic geometry

Quantum states are represented as vectors in an inner product space. Because the dimension of that state space grows exponentially with the number of its constituents, quantum information theory is in large part the asymptotic theory of finite dimensional inner product spaces. I'll highlight some examples of how abstract mathematical results on low distortion embeddings of normed spaces manifest themselves in quantum information theory as improvements on the famous "teleportation" procedure, reductions in the amount of shared secret information required to encrypt a quantum message, and counterexamples to the additivity conjecture, among many other applications.