Codes in W*-metric Spaces: Theory and Examples.

摘要

We introduce a W*-metric space, which is a particular approach to non-commutative metric spaces where a quantum metric is defined on a von Neumann algebra. We generalize the notion of a quantum code and quantum error correction to the setting of finite dimensional W*-metric spaces, which includes codes and error correction for classical finite metric spaces. We also introduce a class of W*-metric spaces that come from representations of semi-simple Lie algebras g called g -metric spaces, and present an outline for code constructions. In turn, we produce specific code constructions for su (2, C )-metric spaces that depend upon proving Tverberg's theorem for points on a moment curve constructed from arithmetic sequences. We introduce a quantum distance distribution, and we prove an analogue of the MacWilliam's identities for su (2)-metric spaces.