Poincar\'e polynomials for Abelian symplectic quotients of pure $r$-qubits via wall-crossings

In this paper, we compute a recursive wall-crossing formula for the Poincar'e polynomials and Euler characteristics of Abelian symplectic quotients of a complex projective manifold under a special effective action of a torus with non-trivial characters. An analogy can be made with the space of pure states of a composite quantum system containing $r$ quantum bits under action of the maximal torus of Local Unitary operations.