Quantum correlations, exhibiting Bell nonlocality, are a valuable resource for device-independent (DI) applications like quantum cryptography and randomness generation. DI certification methods, particularly self-testing, allow for verifying the properties of an unknown quantum system based solely on observed nonlocal correlations without assumptions about its internal structure. While self-testing protocols for two-qubit entangled states have been developed, extending these to higher-dimensional qudit systems using a single *d*-outcome Bell inequality and minimal measurements remains a challenge. This paper addresses this challenge by proposing a self-testing protocol for maximally entangled qudit states using the minimum number of measurements (two per party) and a *d*-outcome Bell inequality, making it experimentally appealing and suitable for applications such as randomness expansion.

Existing self-testing schemes primarily focus on two-qubit systems, often utilizing the violation of multiple two-outcome Bell inequalities. Previous attempts to extend these to higher dimensions either rely on combining qubit self-testing results or use a non-optimal number of measurements. The paper highlights the lack of self-testing statements based on a single *d*-outcome Bell inequality for qudit systems and emphasizes the need for protocols with the minimum number of measurements (two) for practical implementation.

The paper introduces a self-testing protocol based on the Salavrakos-Augusiak-Tura-Wittek-Acín-Pironio (SATWAP) Bell inequality, a generalization of the CHSH inequality for *d*-outcome measurements. The authors employ the correlator picture, using the Fourier transform of conditional probabilities to represent correlations. The core of the methodology involves showing that maximal violation of the SATWAP inequality by a quantum state and measurements implies that the state and measurements are, up to local unitary transformations and the addition of an auxiliary system, equivalent to the maximally entangled state and specific d-outcome observables (CGLMP measurements). The proof relies on a sum-of-squares decomposition of the Bell operator and analysis of the eigenvalue multiplicities of the involved observables. The paper details the mathematical formulation of the SATWAP inequality and the specific forms of the d-outcome observables Z and T, which are generalizations of Pauli Z matrices and are shown to be unitarily equivalent to CGLMP measurements. The self-testing statement asserts that maximal violation of the SATWAP inequality leads to a state equivalent to the maximally entangled state and measurements that are unitarily equivalent to the desired forms.

The paper's main contribution is the proof of a self-testing theorem. This theorem establishes that maximal violation of the SATWAP inequality by a quantum state and measurements implies that the state is equivalent to the maximally entangled state of two qudits, and the measurements are unitarily equivalent to specific d-outcome observables (up to local unitary transformations and the addition of an auxiliary system). This result holds for any local dimension *d*. The minimal number of measurements required (two per party) is a crucial aspect of this result. The authors derive the explicit form of these equivalent observables, showing their unitary equivalence to CGLMP measurements. A significant application of this self-testing is DI randomness certification. The authors show that maximal violation of the SATWAP inequality certifies log₂*d* bits of perfect randomness in the measurement outcomes, implying unbounded randomness expansion as *d* can be arbitrarily large. This contrasts with previous approaches that required more measurements. The analysis uses the local guessing probability to quantify the certified randomness. The traceless nature of the observables used is key to determining the guessing probability and certifying the maximal amount of randomness.

The self-testing protocol presented significantly advances the field by providing a self-testing statement for arbitrary local dimensions that uses only two measurements per party and a single *d*-outcome Bell inequality. This contrasts favorably with previous methods, which relied on results for qubits or employed a higher number of measurements. The minimal measurement requirement makes this protocol experimentally more feasible and efficient. The connection to randomness expansion further highlights its practical significance. The results are important for DI quantum information processing tasks where randomness is a crucial resource.

This work provides a novel self-testing protocol for maximally entangled states of any local dimension using the minimal number of measurements. This protocol utilizes the maximal violation of a single *d*-outcome Bell inequality (SATWAP) and certifies the state and measurements up to local unitary transformations. The result leads to a DI randomness certification scheme, demonstrating unbounded randomness expansion. Future research could explore the robustness of this self-testing and investigate its applicability to partially entangled states or other DI tasks like delegated quantum computation.

The paper notes that while the robustness of the self-testing statement has been numerically studied for *d* = 3, a general analytical method for robustness bounds for arbitrary dimensions is lacking. This is identified as an area for future work. Furthermore, the exploration of self-testing for partially entangled states using modified versions of the SATWAP inequality is also suggested as a potential future research direction.