Probing non-Gaussian nuclear spin baths with entangled NV-center qutrits

Nitrogen-vacancy (NV) center decoherence is often interpreted with effective Gaussian noise models, which encode only second-order environmental correlations and can obscure intrinsically non-Gaussian nuclear-bath dynamics. Here we show that entangled NV qutrit pairs provide an operational probe of this non-Gaussianity via the time-dependent d = 3 Collins-Gisin-Linden-Massar-Popescu (CGLMP) Bell parameter $I_{3}(t)$ , built from joint outcome probabilities using two fixed local measurement settings per party. We derive a cumulant expansion for $I_{3}(t)$ and isolate contributions from third- and fourth-order Overhauser-field cumulants that are absent in a Gaussian model whose second cumulant is fixed by standard qubit-coherence inference under identical conditions. Exact simulations of few-spin central-spin clusters show that these higher-order bath statistics generate a structured residual at the few-percent level relative to the qubit-constrained Gaussian prediction, even when qubit coherence is accurately described by a Gaussian envelope. Our results provide an operational non-Gaussianity witness beyond qubit-constrained Gaussian inference.