  In this paper, we study cohomology rings and cohomological pairings over
Abelian symplectic quotients of special Hamiltonian tori manifolds. The
Hamiltonian group actions appear in quantum information theory where the tori
are maximal tori of some compact semi-simple Lie groups, so called the Local
Unitary groups, which act effectively and with non-trivial characters on
specific complex projective varieties. The complex projective spaces are in
fact spaces of pure multi-partite quantum states. By studying the geometry of
associated moment polytopes, we explicitly obtain cohomology rings, in terms of
elementary symmetric functions, and a recursive wall-crossing algorithm to
compute cohomological pairings over the corresponding Abelian symplectic
quotients. We propose algorithms for general $r$-qubit case and elaborate
discussions with explicit examples for the cases with $r=2,3$.
