The injective norm of CSS quantum error-correcting codes

TL;DR

This paper calculates the injective norm, a measure of multipartite entanglement, for CSS quantum error-correcting codes, extending previous results and revealing connections to matroid theory and Edmonds' intersection theorem.

Contribution

It generalizes the computation of the injective norm to all CSS codes, linking quantum entanglement measures with matroid theory.

Findings

Injective norm computed for a broad class of CSS codes.

Established a connection between quantum entanglement and matroid theory.

Extended previous results from topological phases to all CSS codes.

Abstract

In this paper, we compute the injective norm - a.k.a. geometric entanglement - of standard basis states of CSS quantum error-correcting codes. The injective norm of a quantum state is a measure of genuine multipartite entanglement. Computing this measure is generically NP-hard. However, it has been computed exactly in condensed-matter theory - notably in the context of topological phases - for the Kitaev code and its extensions, in works by Or\'us and collaborators. We extend these results to all CSS codes and thereby obtain the injective norm for a nontrivial, infinite family of quantum states. In doing so, we uncover an interesting connection to matroid theory and Edmonds' intersection theorem.