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
Δi. The corresponding classical
spaces are the discriminant locus
Δ=0 for
ff system, and intersections
of the discriminant loci
Δi=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 (
ttˉ,
τ+τ−), bosonic pairs (
W−W+), and hybrid or three-body
systems (
W−t,
ttt,
ttˉW−), 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.