Abstract: We propose a new formalism for quantum entanglement, 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 calculate their discrimants \Delta_i. The corresponding classical spaces are the discrimant 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. We show that our criterion \Delta\not=0 is equivalent to the original Peres-Horodecki criterion for ff system. At the colliders, we can reconstruct the discriminants from various measurements, and probe the quantum entanglement spaces at exact level. This provides a fundamental approach to test the quantum entanglement. In addition, for the specific approach, we propose a generic method to calculate the quantum range and classical range for the expectation value of any physics observable, and can probe the quantum entanglement spaces which the previous way cannot test for some cases. Furthermore, we define the quantum non-locality tests as the tests for quantum entanglement space via the space-like separated measurements, which can be done at colliders as well. Event recording: