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

We introduce a $W^{\ast }$-metric space, which is a particular approach to non-commutative metric spaces where a $\mathit{\text{quantum}}\mathit{\text{metric}}$ 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 $W^{\ast }$-metric spaces, which includes codes and error correction for classical finite metric spaces. We also introduce a class of $W^{\ast }$-metric spaces that come from representations of semi-simple Lie algebras $\mathfrak{g}$ called $\mathfrak{g}\mathit{\text{-metric}}$ spaces, and present an outline for code constructions. In turn, we produce specific code constructions for $\mathfrak{s}\mathfrak{u}(2,\mathbb{C})$-metric spaces that depend upon proving Tverberg’s theorem for points on a moment curve constructed from arithmetic sequences. We introduce a $\mathit{\text{quantum}}\mathit{\text{distance}}\mathit{\text{distribution}}$, and we prove an analogue of the MacWilliam’s identities for $\mathfrak{s}\mathfrak{u}(2)$-metric spaces.