Greg Kuperberg (UC Davis and Institut Fourier)

"Quantum metric spaces"

One of the uses of metric geometry is in the theory of error correction; an error-detecting or an error-correcting code in a metric space is exactly a sphere packing or a minimum-distance set.  Meanwhile, there exists a notion of quantum error correction, which is a natural and needed generalization of traditional error correction for quantum computers.

In this talk, I will define the axioms of a quantum metric space, a generalization of a metric space that can model both quantum and classical error correction.  The geometry of quantum metric spaces is not very developed.  At the same time, there are many interesting examples and some results to indicate that it is an important kind of geometry.  For instance, there is a notion of a Cauchy completion of a quantum metric space.

(Joint work with Nik Weaver.)