Self-testing of mutually anticommuting observables and maximally entangled two-qudits
The device-independent (DI) certification of high-dimensional entanglement and complex measurement structures is central to scalable quantum information processing. While existing approaches to high-dimensional self-testing have largely relied on Bell tests with multi-outcome measurements, achieving such certification using only binary-outcome Bell tests remains an open challenge. Here, we put forth a simultaneous self-testing framework for maximally entangled two-qudit state of local dimension m∗=2⌊n/2⌋ (equivalently, ⌊n/2⌋ copies of maximally entangled two-qubit pairs), together with n mutually anticommuting observables on one side. To this end, we employ a family of n-settings Bell inequalities comprising two spatially separated observers, Alice and Bob, with 2n−1 and n binary-outcome measurement settings, respectively. We first derive the optimal local and quantum bounds of these inequalities without presupposing the dimension of the underlying state or observables. We then prove that any physical realisation achieving the maximal quantum violation must, up to local isometries and complex conjugation, correspond to a maximally entangled state of local dimension of at least 2⌊n/2⌋, together with local observables forming an irreducible representation of the Clifford algebra. Consequently, the maximal violation self-tests the minimal-dimension quantum realisation compatible with n mutually anticommuting observables. Finally, we establish robustness by proving that if the observed Bell violation deviates from the optimal quantum value by δ, then the realised state and measurements are Oδ-close to the ideal strategy. Our results thus provide a unified DI route for the certification of high-dimensional entanglement and Clifford measurements using only binary-outcome Bell tests.