Hypothesis Tests for Observing Quantum Entanglement in HWW at the LHC

We present a novel experimental strategy for testing quantum entanglement in Higgs boson decays to $W$ boson pairs at the Large Hadron Collider. Unlike theoretical approaches that rely on expectation values of Bell operators, which are highly sensitive to outliers and detector effects, we introduce a continuous formulation of the CGLMP inequality that enables standard hypothesis testing between entangled and separable states. To overcome the fundamental challenge of reconstructing invisible neutrino momenta in the $H \rightarrow WW^* \rightarrow \ell\nu\ell\nu$ channel, we employ conditional denoising diffusion probabilistic models (cDDPM), which provide unbiased, multidimensional unfolding applicable to the full measured dataset, including backgrounds. We evaluate the diffusion-based reconstruction against analytical methods through profile likelihood hypothesis tests implemented in RooFit, with systematic uncertainties from background normalisation and unfolding shape fully propagated. Our results demonstrate that the diffusion-based approach enables robust hypothesis testing of quantum entanglement in a realistic collider environment, with 3$\sigma$ evidence of quantum entanglement projected at approximately 555~fb$^{-1}$ and exceeding 5$\sigma$ at 1600~fb$^{-1}$ to be well within the expected limits of the HL-LHC luminosity targets.