Department colloquium: Quantum Information as Asymptotic Geometry

Speaker: Patrick Hayden (Stanford University)

Title: Quantum Information as Asymptotic Geometry

Abstract:
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, a field with its own long history.

I’ll highlight some examples of how abstract mathematical results from that area, such Dvoretzky’s theorem, manifest themselves in quantum information theory as improvements in quantum teleportation and as the raw material for counterexamples to the field’s famous additivity conjecture.

More recently, this perspective has led to methods for encrypting arbitrarily long messages using constant-sized secret keys. In other circumstances, regularity in information being transmitted leads to natural connections with the asymptotic representation theory of the unitary and symmetric groups, the setting for another fruitful ongoing dialogue.