Tailoring Bell inequalities to the qudit toric code and self testing
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| Format: | Preprint |
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2025
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| _version_ | 1866918223884058624 |
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| author | Vallée, Eloïc Makuta, Owidiusz Emonts, Patrick Samajdar, Rhine Tura, Jordi |
| author_facet | Vallée, Eloïc Makuta, Owidiusz Emonts, Patrick Samajdar, Rhine Tura, Jordi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00146 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tailoring Bell inequalities to the qudit toric code and self testing Vallée, Eloïc Makuta, Owidiusz Emonts, Patrick Samajdar, Rhine Tura, Jordi Quantum Physics 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. |
| title | Tailoring Bell inequalities to the qudit toric code and self testing |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2512.00146 |