Quantum Entanglement Theory and Its Generic Searches in High Energy Physics

We propose a new formalism for quantum entanglement (QE), and study its generic searches at the colliders. For a general quantum system with $N$ particles, we show that the quantum space (the total spin polarization parameter space) is complex projective space, and the classical space (the spin polarization parameter space for classical theory) is the cartesian product of the complex projective spaces. Thus, the quantum entanglement space is the difference of these two spaces. For the $ff$, $AA$, $Af$, $fff$, and $ffA$ systems, we propose their discriminants $\Delta_i$. The corresponding classical spaces are the discriminant locus $\Delta=0$ for $ff$ system, and intersections of the discriminant loci $\Delta_i=0$ for $AA$, $Af$, $fff$, and $ffA$ systems in the quantum space. In particular, for two fermion $ff$ system, we prove that our discriminant criterion is equivalent to the original Peres-Horodecki criterion and the CHSH criterion. And thus our quantum entanglement space is indeed Bell non-local. With the collider searches, we can reconstruct the discriminants from various measurements, and probe the quantum entanglement spaces via a fundamental approach at exact level. In addition, for the specific approach, we present a comprehensive framework to detect quantum entanglement in high-energy multi-particle systems, spanning fermion pairs ($t\bar{t}$, $\tau^{+}\tau^{-}$), bosonic pairs ($W^{-}W^{+}$), and hybrid or three-body systems ($W^{-}t$, $ttt$, $t\bar{t}W^{-}$), by diverse observables through angular correlations in decay products. These results establish model-independent methodologies for probing QE across collider experiments, bridging quantum information principles with high-energy phenomenology, while offering novel pathways to explore exotic particles and quantum properties in multi-particle systems.