One Application of Duistermaat-Heckman Measure in Quantum Information Theory

Abstract While the exact separability probability of 8/33 for two-qubit states under the Hilbert-Schmidt measure has been reported by Huong and Khoi, detailed derivations remain inaccessible for general audiences. This paper provides a comprehensive, self-contained derivation of this result, elucidating the underlying geometric and probabilistic structures. We achieve this by developing a framework centered on the computation of Hilbert-Schmidt volumes for key components: the quantum state space, relevant flag manifolds, and regular (co)adjoint orbits. Crucially, we establish and leverage the connection between these Hilbert-Schmidt volumes and the symplectic volumes of the corresponding regular co-adjoint orbits, formalized through the Duistermaat-Heckman measure. By meticulously synthesizing these volume computations—specifically, the ratios defining the relevant probability measures—we reconstruct and rigorously verify the 8/33 separability probability. Our approach offers a transparent pathway to this fundamental constant, detailing the interplay between symplectic geometry, representation theory, and quantum probability.