One application of Duistermaat-Heckman measure in quantum information theory

TL;DR

This paper provides a detailed geometric derivation of the 8/33 separability probability for two-qubit states under the Hilbert-Schmidt measure, connecting symplectic geometry and quantum information theory.

Contribution

It introduces a framework using Duistermaat-Heckman measures to compute Hilbert-Schmidt volumes and rigorously derive the separability probability, enhancing understanding of quantum state geometry.

Findings

Derived the 8/33 separability probability using geometric methods

Established the link between Hilbert-Schmidt volumes and symplectic volumes

Provided a self-contained, rigorous derivation accessible to a broad audience

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.