Testing entanglement and Bell inequalities in $H \to ZZ$

We discuss quantum entanglement and violation of Bell inequalities in the $H\rightarrow ZZ$ decay, in particular when the two $Z-$bosons decay into light leptons. Although such process implies an important suppression of the statistics, this is traded by clean signals from a "quasi maximally-entangled" system, which makes it very promising to check these crucial phenomena at high energy. In this paper we devise a novel framework to extract from $H \to ZZ$ data all significant information related to this goal, in particular spin correlation observables. In this context we derive sufficient and necessary conditions for entanglement in terms of only two parameters. Likewise, we obtain a sufficient and improved condition for the violation of Bell-type inequalities. The numerical analysis shows that with a luminosity of $L = 300 \text{fb}^{-1}$ entanglement can be probed at $> 3σ$ level. For $L = 3 \text{ab}^{-1}$ (HL-LHC) entanglement can be probed beyond the $5σ$ level, while the sensitivity to a violation of the Bell inequalities is at the $4.5σ$ level