Codes in W\ast-metric Spaces: Theory and Examples

Abstract

We introduce a $W^*$-metric space, which is a particular approach to non-commutative metric spaces where a \textit{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 $\mathfrak{g}$ called \textit{$\mathfrak{g}$-metric} spaces, and present an outline for code constructions. In turn, we produce specific code constructions for $\mathfrak{su}(2,\mathbb{C})$-metric spaces that depend upon proving Tverberg's theorem for points on a moment curve constructed from arithmetic sequences. We introduce a \textit{quantum distance distribution}, and we prove an analogue of the MacWilliam's identities for $\mathfrak{su}(2)$-metric spaces.