One application of Duistermaat-Heckman measure in quantum information theory
Lin Zhang, Xiaohan Jiang, Bing Xie
TL;DR
The paper addresses the problem of determining the exact separability probability for two-qubit states under the Hilbert-Schmidt measure. It develops a cohesive framework that connects Hilbert-Schmidt volumes of the state space and its submanifolds to the symplectic volumes of regular coadjoint orbits via the Duistermaat-Heckman measure, using the Harish-Chandra formula and jump formulas to obtain piecewise polynomial densities. By explicitly computing HS volumes of flag manifolds, adjoint orbits, and the full state space, and relating them to symplectic volumes, the authors derive the separable-set volume and confirm the exact probability $P^{(2\times2)}_{sep}=\tfrac{8}{33}$, aligning with prior, later-proved results. The work provides a transparent, self-contained derivation, strengthening the geometric understanding of entanglement prevalence and offering a methodology extendable to higher dimensions and alternative metrics, with potential impact on quantum information geometry and related computational techniques.
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 [\href{https://doi.org/10.1088/1751-8121/ad8493}{J.Phys.A:Math.Theor.{\bf57}, 445304(2024)}], 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.