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 generalizes the Clauser-Horne-Shimony-Holt test from qubits to qudits, where each setting is a non-diagonal Heisenberg-Weyl observable. For every odd prime $d \geq 3$, the associated Bell operator has an exact sum-of-positive-operators decomposition, yielding the Cirelson bound in closed form, from which we reconstruct the Heisenberg-Weyl commutation relations on the support of the state. We then extend the Mayers-Yao local isometry from qubits to prime-dimensional systems and show that any $\epsilon$-near-optimal strategy below that bound is, up to local isometries, within trace distance $\delta = \mathcal{O}(\sqrt{\epsilon})$ of the ideal maximally entangled state; the implemented measurements are correspondingly close to the target observables. Via a tensor-factor argument, the prime-dimension result extends the self-testing protocol to every composite dimension $d$. The protocol uses standard Heisenberg-Weyl operations and non-Clifford phase gates that are diagonal in the computational basis, making it directly applicable to high-dimensional photonic and atomic platforms.