2012/05/21 by Bumgardner, Christopher
#FOS: Physical sciences #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.1205.4517
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 \mathfrakg called \textit\mathfrakg-metric spaces, and present an outline for code constructions. In turn, we produce specific code constructions for \mathfraksu(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 \mathfraksu(2)-metric spaces.