Robustly self-testing all maximally entangled states in every finite dimension

Abstract We establish a device-independent, noise-tolerant certification of maximally entangled states in every finite dimension $d$. The core ingredient is a $d$-input, $d$-outcome Bell experiment that recovers the genuinely qudit Heisenberg–Weyl structure. Yielding the Tsirelson bound in closed form, the associated Bell operator has an exact sum-of-positive-operators decomposition for $d\geq 3$ prime, from which we reconstruct the Heisenberg–Weyl commutation relations on the support of the state. Representation-theoretically, once the twisted-commutation relations are established, the finite Stone–von Neumann theorem identifies the realization with the canonical Heisenberg–Weyl model; in addition, we provide an explicit SWAP-type local isometry that yields the robustness constants. This gives a genuinely qudit certification: rather than reducing to qubit subtests, the higher-dimensional Heisenberg-Weyl algebra itself is recovered. We show that any $\epsilon$-near-optimal strategy is, up to local isometries, within trace distance $\delta=\mathcal{O}(\sqrt{\epsilon})$ of the ideal maximally entangled state (with dimension-explicit constants), and the implemented measurements are correspondingly close to the target observables. Via a tensor-factor argument, the prime-dimensional result extends the self-testing protocol to every composite dimension $d$, thereby yielding a fully rigorous, robust, and genuinely qudit self-test of maximally entangled states in all finite dimensions.