The injective norm of CSS quantum error-correcting codes
Stephane Dartois, Gilles Zémor
TL;DR
This work derives an exact formula for the injective norm (geometric entanglement) of basis states of CSS quantum codes by linking the problem to classical code structure. The central result expresses $\|\ket{C}\|_{\mathrm{inj}}=2^{-\frac{1}{2}(k-j(C))}$, with $j(C)$ defined via punctured/shortened codes, and shows that the lower bound matches the upper bound whenever $j(C)=\delta(C)$, a condition proven using matroid theory and Edmonds' intersection theorem. By framing the optimization as a matroid problem, the authors also indicate a polynomial-time algorithm to compute $j(C)$, connecting quantum information quantities to combinatorial optimization. The Kitaev toric code emerges as a notable example where $j(C)=0$, yielding maximal entanglement among CSS-basis states, and the approach generalizes prior condensed-matter results to all CSS codes. Overall, the paper advances understanding of multipartite entanglement in structured quantum states and highlights a deep link between quantum information and matroid theory.
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 Orús 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.