Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2507.02369 · Quantum Information and Computation

One Application of Duistermaat-Heckman Measure in Quantum Information Theory

Lin Zhang, Xiaohan Jiang, Bing Xie

Published Dec 1, 2025• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.2478/qic-2025-0033

Related papers

Separability probability of two-qubit states

Ha Thanh Huong, Vu The Khoi

Abstract Many works have been devoted to determining the volume of the set of separable states and the separability probability both analytically and numerically. However, computing the exact separability probability remains a highly nontrivial open problem. In these notes, we examine this problem in the simplest context of two-qubit states. A novel aspect of our method is the use of the Duistermaat-Heckman measure, which is the push-forward of Liouville measure on the coadjoint orbit along the moment map. On each coadjoint orbit, the Hilbert-Schmidt measure differs from the Duistermaat-Heckman measure by a constant. The latter measure can be explicitly computed by using tools from symplectic geometry. We confirm the conjecture that the separability probability of two-qubit states with Hilbert-Schmidt measure is 8 33 .

Journal of Physics A Mathematical and Theoretical•Oct 8, 2024•3 citations

Duistermaat-Heckman measure and the mixture of quantum states

Zhang, L., Jiang, Y., Wu, J.

In this paper, we present a general framework to solve a fundamental problem in Random Matrix Theory (RMT), i.e., the problem of describing the joint distribution of eigenvalues of the sum $\bsA+\bsB$ of two independent random Hermitian matrices $\bsA$ and $\bsB$. Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of Duistermaat-Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of $n$ random density matrices chosen from a random state ensemble (specified in the text) increases with the number $n$. Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components $n$ in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels.

•Jan 1, 2019•3 citations

Duistermaat–Heckman measure and the mixture of quantum states

Lin Zhang, Yixin Jiang, Junde Wu

Abstract In this paper, we present a general framework to solve a fundamental problem in random matrix theory (RMT), i.e. the problem of describing the joint distribution of eigenvalues of the sum of two independent random Hermitian matrices and . Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of the Duistermaat–Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of n random density matrices chosen from a random state ensemble (specified in the text) increases with the number n . Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components n in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels.

arXiv:1810.02630•Journal of Physics A Mathematical and Theoretical•Oct 30, 2019•6 citations

Separability probability formulas and their proofs for generalized two-qubit X-matrices endowed with Hilbert–Schmidt and induced measures

Charles F. Dunkl, Paul B. Slater

Two-qubit X-matrices have been the subject of considerable recent attention, as they lend themselves more readily to analytical investigations than two-qubit density matrices of arbitrary nature. Here, we maximally exploit this relative ease of analysis to formally derive an exhaustive collection of results pertaining to the separability probabilities of generalized two-qubit X-matrices endowed with Hilbert–Schmidt and, more broadly, induced measures. Further, the analytical results obtained exhibit interesting parallels to corresponding earlier (but, contrastingly, not yet fully rigorous) results for general two-qubit states — deduced on the basis of determinantal moment formulas. Geometric interpretations can be given to arbitrary positive values of the random-matrix Dyson-index-like parameter [Formula: see text] employed.

arXiv:1501.02289•Random Matrices Theory and Application•Aug 21, 2015•8 citations

Hilbert-Schmidt Separability Probabilities from Bures Ensembles and vice versa: Applications to Quantum Steering Ellipsoids and Monotone Metrics

Paul B. Slater

We reexamine a recent analysis in which, using the volume of the associated quantum steering ellipsoid (QES) as a measure, we sought to estimate the probability that a two-qubit state is separable. In the estimation process, we, in effect, sought to attach to states random with respect to Hilbert-Schmidt (HS) measure, the corresponding QES volumes. However, a study of the relations between HS and Bures ensembles and their well-supported separability probabilities of $\frac{8}{33}$ and $\frac{25}{341}$, respectively, now lead us to explore as a possible alternative measure, the QES volume divided by the $Π_{j

arXiv:2101.07716•Jan 19, 2021•0 citations

Quantum Steering Ellipsoid Volume as a Measure on the Two-Qubit States and Associated Separability and Absolute Separability Ratios

Paul B. Slater

Employing the volume of quantum steering ellipsoids (QSEs) as a measure on the fifteen-dimensional convex set of two-qubit states, we estimate the ratio of the integral of the measure over the separable states to its integral over all (separable and entangled) states to be 0.0288. This can be contrasted with the considerably larger separability ratios (probabilities) of $\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424$ and $\frac{25}{341}=\frac{5^2}{11 \cdot 31} \approx 0.0733138$ that various forms of evidence point to with the use of the prominent Hilbert-Schmidt and Bures measures, respectively. The questions of whether the ratio in the QSE setting can be more precisely obtained or even exactly computed, as well as whether a metric can be constructed, the volume element of which yields the measure, remain to be addressed. We also investigate related issues pertaining to absolute separability. Further, we examine the behavior of the separability probability--constant in the Hilbert-Schmidt case and decreasing in the Bures--as a function of the Bloch vector norm in the QSE instance. It appears to increase approaching the pure state boundary.

arXiv:2012.09152•Dec 16, 2020•0 citations
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.
Abstract Many works have been devoted to determining the volume of the set of separable states and the separability probability both analytically and numerically. However, computing the exact separability probability remains a highly nontrivial open problem. In these notes, we examine this problem in the simplest context of two-qubit states. A novel aspect of our method is the use of the Duistermaat-Heckman measure, which is the push-forward of Liouville measure on the coadjoint orbit along the moment map. On each coadjoint orbit, the Hilbert-Schmidt measure differs from the Duistermaat-Heckman measure by a constant. The latter measure can be explicitly computed by using tools from symplectic geometry. We confirm the conjecture that the separability probability of two-qubit states with Hilbert-Schmidt measure is 8 33 .
In this paper, we present a general framework to solve a fundamental problem in Random Matrix Theory (RMT), i.e., the problem of describing the joint distribution of eigenvalues of the sum $\bsA+\bsB$ of two independent random Hermitian matrices $\bsA$ and $\bsB$. Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of Duistermaat-Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of nnn random density matrices chosen from a random state ensemble (specified in the text) increases with the number nnn. Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components nnn in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels.
Abstract In this paper, we present a general framework to solve a fundamental problem in random matrix theory (RMT), i.e. the problem of describing the joint distribution of eigenvalues of the sum of two independent random Hermitian matrices and . Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of the Duistermaat–Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of n random density matrices chosen from a random state ensemble (specified in the text) increases with the number n . Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components n in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels.
Two-qubit X-matrices have been the subject of considerable recent attention, as they lend themselves more readily to analytical investigations than two-qubit density matrices of arbitrary nature. Here, we maximally exploit this relative ease of analysis to formally derive an exhaustive collection of results pertaining to the separability probabilities of generalized two-qubit X-matrices endowed with Hilbert–Schmidt and, more broadly, induced measures. Further, the analytical results obtained exhibit interesting parallels to corresponding earlier (but, contrastingly, not yet fully rigorous) results for general two-qubit states — deduced on the basis of determinantal moment formulas. Geometric interpretations can be given to arbitrary positive values of the random-matrix Dyson-index-like parameter [Formula: see text] employed.
We reexamine a recent analysis in which, using the volume of the associated quantum steering ellipsoid (QES) as a measure, we sought to estimate the probability that a two-qubit state is separable. In the estimation process, we, in effect, sought to attach to states random with respect to Hilbert-Schmidt (HS) measure, the corresponding QES volumes. However, a study of the relations between HS and Bures ensembles and their well-supported separability probabilities of 833\frac{8}{33}338​ and 25341\frac{25}{341}34125​, respectively, now lead us to explore as a possible alternative measure, the QES volume divided by the $Π_{j
Employing the volume of quantum steering ellipsoids (QSEs) as a measure on the fifteen-dimensional convex set of two-qubit states, we estimate the ratio of the integral of the measure over the separable states to its integral over all (separable and entangled) states to be 0.0288. This can be contrasted with the considerably larger separability ratios (probabilities) of 833=233⋅11≈0.242424\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424338​=3⋅1123​≈0.242424 and 25341=5211⋅31≈0.0733138\frac{25}{341}=\frac{5^2}{11 \cdot 31} \approx 0.073313834125​=11⋅3152​≈0.0733138 that various forms of evidence point to with the use of the prominent Hilbert-Schmidt and Bures measures, respectively. The questions of whether the ratio in the QSE setting can be more precisely obtained or even exactly computed, as well as whether a metric can be constructed, the volume element of which yields the measure, remain to be addressed. We also investigate related issues pertaining to absolute separability. Further, we examine the behavior of the separability probability--constant in the Hilbert-Schmidt case and decreasing in the Bures--as a function of the Bloch vector norm in the QSE instance. It appears to increase approaching the pure state boundary.