Tailoring Bell inequalities to the qudit toric code and self testing

TL;DR

This paper develops Bell inequalities tailored to the $ ext{Z}_d$ toric code, enabling device-independent certification of topological quantum states and demonstrating self-testing of qutrit subspaces, advancing quantum certification methods.

Contribution

It introduces a general framework for constructing Bell inequalities for the $ ext{Z}_d$ toric code, including self-testing of qutrit subspaces, and proposes schemes to improve robustness against experimental imperfections.

Findings

Bell inequalities maximally violated by ground states of the $ ext{Z}_d$ toric code

First self-testing of a qutrit subspace using Bell inequalities

Schemes to enhance classical-quantum bound ratios for robustness

Abstract

Bell nonlocality provides a robust scalable route to the efficient certification of quantum states. Here, we introduce a general framework for constructing Bell inequalities tailored to the $\mathbb{Z}_d$ toric code for odd prime local dimensions. Selecting a suitable subset of stabilizer operators and mapping them to generalized measurement observables, we compute multipartite Bell expressions whose quantum maxima admit a sum-of-squares decomposition. We show that these inequalities are maximally violated by all states in the ground-state manifold of the $\mathbb{Z}_d$ toric code, and determine their classical (local) bounds through a combination of combinatorial tiling arguments and explicit optimization. As a concrete application, we analyze the case of $d=3$ and demonstrate that the maximal violation self-tests the full qutrit toric-code subspace, up to local isometries and complex conjugation. This constitutes, to our knowledge, the first-ever example of self-testing a qutrit subspace. Extending these constructions, we further present schemes to enhance the ratio of classical--quantum bounds and thus improve robustness to experimental imperfections. Our results establish a pathway toward device-independent certification of highly entangled topological quantum matter and provide new tools for validating qudit states in error-correcting codes and quantum simulation platforms.