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

TL;DR

This paper develops a recursive wall-crossing formula to compute Poincaré polynomials and Euler characteristics of Abelian symplectic quotients, with applications to quantum systems of multiple qubits.

Contribution

It introduces a novel recursive method for calculating topological invariants of symplectic quotients, linking geometric techniques to quantum state spaces.

Findings

Derived explicit formulas for Poincaré polynomials of Abelian quotients.

Established connections between symplectic geometry and quantum information theory.

Provided computational tools for analyzing quantum state spaces.

Abstract

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.