Scalable Self-Testing of Mutually Anticommuting Observables and Maximally Entangled Two-Qudits

The next frontier in device-independent quantum information lies in the certification of scalable and parallel quantum resources, which underpin advanced quantum technologies. We put forth a simultaneous self-testing framework for maximally entangled two-qudit state of local dimension $m_*=2^{\lfloor n/2 \rfloor}$ (equivalently $\lfloor n/2 \rfloor$ copies of maximally entangled two-qubit pairs), together with $n$ numbers of anti-commuting observables on one side. To this end, we employ an $n$-settings Bell inequality comprising two space-like separated observers, Alice and Bob, having $2^{n-1}$ and $n$ number of measurement settings, respectively. We derive the local ontic bound of this inequality and, crucially, employ the Sum-of-Squares decomposition to determine the optimal quantum bound without presupposing the dimension of the state or observables. We then establish that any physical realisation achieving the maximal quantum violation must, up to local isometries and complex conjugation, correspond to a reference strategy consisting of a maximally entangled state of local dimension of at least $2^{\lfloor n/2 \rfloor}$ and local observables forming an irreducible representation of the Clifford algebra. This construction thereby demonstrates that the minimal dimension compatible with $n$ mutually anticommuting observables is naturally self-tested by the maximal violation of the proposed Bell functional. Finally, we analyse the robustness of the protocol by establishing quantitative bounds relating deviations in the observed Bell value to the fidelity between the realised and the ideal strategies. Our results thus provide a scalable, dimension-independent route for the certification of high-dimensional entanglement and Clifford measurements in a fully device-independent framework.