Tailoring Bell inequalities to the qudit toric code and self testing

• URL: http://arxiv.org/abs/2512.00146v1
• Date: Fri, 28 Nov 2025 19:00:00 GMT
• Title: Tailoring Bell inequalities to the qudit toric code and self testing
• Authors: Eloïc Vallée, Owidiusz Makuta, Patrick Emonts, Rhine Samajdar, Jordi Tura,
• Abstract summary: We introduce a general framework for constructing Bell inequalities tailored to the $mathbbZ_d$ toric code for odd prime local dimensions.<n>We show that these inequalities are maximally violated by all states in the ground-state manifold of the $mathbbZ_d$ toric code.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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