  In this article, we study quantum entanglement properties of the three-body
$H\to\gamma l\bar{l}$ decays (for $l=e,\mu,\tau$) within the context of the
Standard Model augmented with CP-violating interactions in the lepton Yukawa
sector. Our aim is to elucidate the distribution of entanglement between the
final photon, lepton and antilepton across the phase-space. These rare Higgs
boson decays occur at 1-loop level, presenting a unique opportunity to
scrutinize quantum correlations of fundamental interactions in tripartite
systems by computing concurrence measures and investigating Bell non-locality.
Moreover, we explore post-decay and autodistillation phenomena. Multipartite
entanglement measures have much richer structure than those in the bipartite
case, thus deserve more attention in collider phenomenology. In this line, we
analyze here novel observables for these three-body Higgs boson decays, which
can be extended to other multiparticle systems within the high-energy regime.
We found that entanglement manifests among final particles, occasionally
achieving a maximally entangled state in specific kinematical configurations.
Also, these decay channels are promising for Bell non-locality tests but
CP-effects are suppressed by lepton masses in this kind of observables.
