Robust self-testing with CHSH mod 3

Abstract: The CHSH mod 3 Bell inequality is a natural testbed for higher-dimensional quantum nonlocality, yet its maximal quantum violation and self-testing properties have remained unresolved. We determine its exact maximal quantum value and show that, up to unitary equivalence and the natural symmetries of the inequality, it admits a unique optimal irreducible strategy; equivalently, there are four symmetry-related optimal irreducible strategies. Each of these strategies uses a maximally entangled two-qutrit state. We further prove that any strategy whose value is within $\varepsilon$ of the optimum is $O(\sqrt{\varepsilon})$-close, up to local isometries, to a direct sum of optimal irreducible strategies.

• The paper determines the maximal quantum violation for the CHSH mod 3 inequality and proves a unique optimal two-qutrit strategy up to unitary equivalence.
• It employs exact SOS certificates via semidefinite programming and group-theoretic methods to extract optimal strategy details without full polynomial construction.
• The work establishes robust self-testing by showing that near-optimal quantum strategies are O(√ε)-close to the unique optimal approach using maximally entangled qutrit states.

Exact and Robust Self-Testing for the CHSH mod 3 Bell Inequality

Introduction

Self-testing constitutes a foundational approach in device-independent quantum information processing, enabling strict certification of quantum states and measurements solely from the statistics of observed correlations. Central to achieving self-testing are Bell inequalities, observables whose maximal quantum violation identifies both the state and measurement operators, up to known equivalences. The canonical Clauser-Horne-Shimony-Holt (CHSH) inequality is a well-studied paradigm for two-dimensional (qubit) systems, with established self-testing character. However, generalizations to higher-dimensional systems, particularly CHSH inequalities with outcomes modulo $d$ (CHSH mod $d$), remain less understood. In the present work, the CHSH mod $3$ inequality is addressed, and its exact quantum value, optimal strategies, and robust self-testing properties are rigorously established.

Main Results

The key contributions of the work are threefold:

1. Exact Quantum Value and Unique Optimal Strategies: The precise maximal quantum violation for the CHSH mod $3$ inequality is determined analytically. It is shown that, up to unitary equivalence and the natural symmetries of the Bell functional, there exists a unique optimal irreducible quantum strategy. All optimal strategies correspond to projective measurements on a maximally entangled two-qutrit state. Formally, the maximal quantum value is:

$$\beta_q = \frac{1}{3} + \frac{2\cos(\pi/18)}{3\sqrt{3}}$$

Four symmetry-related optimal irreducible strategies are identified, each realized by appropriately permuted or relabelled measurements acting on the standard maximally entangled qutrit state.

1. Exact SOS Certificates via Semidefinite and Group Theoretic Methods: An exact sum-of-squares (SOS) certificate for the optimal value is constructed using advanced high-precision semidefinite programming techniques, leveraging symmetry reduction and the rounding methodology of Cohn, de Laat, and Leijenhorst (Cohn et al., 2024). Notably, the extraction of the matrix factors required for the self-testing proof is shown to be feasible directly from the rectangular matrix factor $T$ in the decomposition $Z = T\hat{Z}T^\top$, bypassing the need for explicit construction of the complete SOS polynomials, which is nontrivial for complex high-dimensional cases.
2. Robust Self-Testing: A robust quantitative self-testing statement is proved: any quantum strategy achieving a value within $\varepsilon$ of the quantum optimum is, up to local isometry, $O(\sqrt{\varepsilon})$-close (in norm) to a direct sum of the four optimal strategies. Each irreducible block uses a maximally entangled two-qutrit state. This robust self-testing result is enabled by combining the exact SOS certificate with the group-theoretic 'inverse theorem' of Gowers–Hatami, ensuring that approximate group representations induce proximity to true representations under the relevant induced norm.

Methodology

The analysis deploys a combination of noncommutative polynomial optimization and group representation theory, structured as follows:

• Noncommutative Polynomial Optimization and the NPA Hierarchy:

The CHSH mod $3$ functional is encoded as a polynomial optimization problem over noncommuting operator variables, subject to projection relations and commutation constraints. Upper bounds on the quantum value are obtained via the Navascués–Pironio–Acín (NPA) hierarchy, a convergent sequence of semidefinite programs whose dual solutions correspond to moment matrices.

• Symmetry Reduction and Group Structure:

Exploiting the large symmetry group of the CHSH mod $d$0 Bell functional, the size of the semidefinite program is vastly reduced by block-diagonalization according to irreducible representations of the symmetry group. This reduction is critical for tractability of high-precision computations required by the rounding algorithm.

• Exact Rounding and Strategy Extraction:

The high-precision numerical solution to the SDP is converted into an exact solution defined over a number field of low degree. From the matrix factor $d$1 in the SOS decomposition, a system of polynomial equations vanishing on optimal strategies is determined. All optimal strategies are then classified as representations of a specified finite group, and all irreducible representations are explicitly extracted using algebraic software and computational group libraries.

• Connection with Flat Extensions:

The work establishes a technical equivalence between optimizer extraction via the SOS approach and via flat extensions of the dual moment matrix, validating the completeness of the solution set.

• Robustness via Approximate Representation Theory:

The group-theoretic Gowers–Hatami theorem is applied to the approximate relations arising from nearly-optimal strategies, guaranteeing robust stability of the self-testing statements.

Implications

The results provide a definitive resolution for quantum realizability and self-testing properties of the CHSH mod $d$2 Bell inequality. The demonstration that exact SOS certificates and analytic characterizations of optimal strategies can be attained for higher-dimensional, non-binary-output Bell inequalities is an important milestone. Practically, the robust self-testing statements established here enable device-independent certification of maximally entangled two-qutrit states with a minimal measurement scenario (modulo the minimal choice for $d$3), a crucial ingredient for scalable quantum information protocols such as certified randomness expansion and verified quantum computation.

The theoretical implications include:

• The group-theoretic structure of optimal strategies for CHSH mod $d$4 is fully elucidated, providing a template for analysis of more general CHSH mod $d$5 inequalities.
• The connection between SOS-based certificates and dual flat extensions gives methodological insight of broad relevance for quantum optimization and operator algebra, clarifying when and how numerical optimal certificates admit analytic rationalization.
• The robust self-testing characterization closes a previously open case which, unlike the $d$6 CHSH scenario, is not reducible to qubit subspaces.

Future Directions

Several avenues are opened by this work:

• Extension to CHSH mod $d$7 for $d$8: The present exact methods may, with appropriate computational resources, be extended to higher $d$9. However, the complexity of SOS certificates and group representations increases rapidly. Identifying structure theorems or patterns governing optimal certificates for general $3$0 remains a major open challenge.
• Applications to Other Device-Independent Primitives: The computational pipeline established here provides a methodological framework for certifying optimality and robustness in other Bell-type inequalities, particularly those for which only numerical upper bounds are presently known.
• Refinement of Robustness Constants: Determination of the exact constants involved in the robustness (rather than merely $3$1 scaling) would allow for tighter finite-sample and noise-tolerance estimates in experimental implementations.

Conclusion

This work rigorously determines the maximal quantum value and self-testing properties of the CHSH mod $3$2 Bell inequality, affirming the optimality and robustness of quantum correlations in this minimal ternary system. The synergy of algebraic, semi-definite, and group-theoretic techniques presented here provides a solid foundation for analytical and computational approaches to device-independent quantum certification in high-dimensional systems.

---

Reference:

"Robust self-testing with CHSH mod 3" (2604.03700)