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

TL;DR

This paper introduces $W^*$-metric spaces as a framework for quantum and classical codes, generalizes quantum error correction to these spaces, and constructs specific codes using Lie algebra representations and geometric theorems.

Contribution

It develops the theory of $W^*$-metric spaces, extends quantum coding concepts to this setting, and provides explicit code constructions based on Lie algebra representations and geometric theorems.

Findings

Defined $W^*$-metric spaces for quantum and classical codes

Extended quantum error correction to $W^*$-metric spaces

Constructed codes using $rak{su}(2)$-metric spaces and geometric theorems

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.