Quantum Information & Computation
Volume 25, Issue 6
RESEARCH ARTICLE Open access

One Application of Duistermaat-Heckman Measure in Quantum Information Theory

Lin Zhang, Xiaohan Jiang, Bing Xie

Author affiliations
  • Lin Zhang1
  • Xiaohan Jiang1
  • Bing Xie1
  1. School of Science, Hangzhou Dianzi University, Hangzhou, China

pp. 598-632 · 10.2478/qic-2025-0033

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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.
Open accessCC BY-NC-ND 4.0

Keywords

  • separability probability
  • Duistermaat-Heckman measure
  • Hilbert-Schmidt volume
  • symplectic volume

Suggested citation

Lin Zhang, Xiaohan Jiang, Bing Xie (2025). One Application of Duistermaat-Heckman Measure in Quantum Information Theory. Quantum Information & Computation, 25(6), 598-632. https://doi.org/10.2478/qic-2025-0033

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