Tailoring Bell inequalities to the qudit toric code and self testing

TL;DR

The work addresses device-independent certification of topological qudit states by constructing Bell inequalities tailored to the $\mathbb{Z}_d$ toric code (odd prime $d$). The authors map stabilizer generators to generalized observables and derive a sum-of-squares quantum bound, showing that maximal violations occur for states in the toric-code ground subspace and enabling self-testing of the full $\mathbb{Z}_3$ toric-code subspace up to local isometries and complex conjugation. By introducing multiple special sites, they demonstrate a tunable improvement in the classical--quantum ratio, enhancing robustness to experimental imperfections. The results offer a principled route to device-independent certification of topological quantum matter and provide tools for validating qudit states in error-correcting codes and quantum simulation platforms.

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.

Figures (7)

• Figure 1: Schematic illustration of the $\mathbb{Z}_d$ toric code on a directed toroidal lattice. Orange arrows indicate the directionality of the links. The two main types of stabilizer operators, $V$ and $P$, are marked in blue and yellow, respectively. An additional operator, $E(x)$, is shown in red and indexed by the "special site" (dark red); this operator plays an important role in the construction of the Bell inequality. The indexing convention is depicted in the bottom right corner with dashed lines.
• Figure 2: (a) To compute the local bound, the global optimization routine is subdivided into local optimizations. Terms in \ref{['eq:bell_inequality']} that reside outside the colored area are optimized individually, while those inside are jointly optimized. The special site is marked by the dark red ring at the center. The $V$ and $P$ operators are represented by colored squares, while the $E$ operator is indicated by bright red rings. (b) Example configuration with multiple special sites, shown in red. All stabilizing operators appearing in the Bell expression in \ref{['eq:bell_inequality_general']} include at most one special site.
• Figure 3: Ratio between the quantum and local bounds, $\Lambda$, as a function of the number of special sites, $R$. The local dimension is $d = 3$, and the total number of sites is fixed to $N = 200$. The ratio increases monotonically with $R$, evidencing that a larger number of special sites enhances the robustness of the inequality against experimental imperfections.
• Figure 4: The five elementary polyominoes used in decomposition.
• Figure 5: Visualization of the variables appearing in the Bell expression given in \ref{['eq:bell_inequality_general']} in the specific case of $d=3$. The red dots are variables associated to special sites. The black dots are variables appearing in the expression of the special tile.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• Theorem 4
• proof
• Theorem 5
• Lemma 2
• Theorem 6