© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Geometric Analysis of Density Matrices on the Manifold of Symmetric Positive Definite Matrices

Y. Wang

Published Jan 7, 2026• 0 citations

Abstract

Density matrices are fundamental mathematical objects for describing quantum states, and their set can be viewed as a subset of the manifold of symmetric positive definite (SPD) matrices. Based on the Riemannian geometric structure on the SPD manifold, this paper analyzes geometric paths and distance functions on the space of density matrices. We adopt the affine-invariant Riemannian metric and review the associated geodesics and Riemannian distance, comparing them with linear interpolation and the Frobenius distance defined in the ambient Euclidean space. Numerical experiments are conducted to visualize geodesic paths between density matrices and to quantitatively compare several commonly used matrix distances, including the Frobenius, Riemannian, and Bures distances. The results demonstrate that distances induced by different geometric structures exhibit fundamentally different behaviors in characterizing discrepancies between density matrices, reflecting the intrinsic non-Euclidean geometry of the space. These analyses and numerical results provide a coherent geometric perspective for understanding the structure and similarity of density matrices.

DOI: 10.5281/zenodo.18174636

Related papers

Geodesic distances on density matrices

Anna Jenčová

We find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices.

arXiv:math-ph/0312044•Journal of Mathematical Physics•Apr 16, 2004•23 citations

GEODESIC DISTANCES ON DENSITY MATRICES

Anna Jenčová

Abstract.We find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices. 1.

•Jan 1, 2003•15 citations

Quantum Riemannian Geometry: Curved Geometry of Quantum State Manifolds and its Physical Implications

Rowan Brad Quni-Gudzinas

This manuscript explores the Riemannian geometry of quantum state manifolds and its physical implications. We establish that quantum state manifolds, as submanifolds of Hilbert space, possess intrinsic Riemannian geometry characterized by a metric tensor derived from the Hilbert space inner product. This geometric framework reveals fundamental connections: the distance between nearby states scales with quantum fluctuations, curvature indicates state robustness, and geometric phase transitions occur in many-body systems. We review historical developments from early quantum geometry to modern quantum information geometry, and present methodologies for computing geometric quantities and their physical interpretations. Results include distance-fluctuation relations for coherent states, curvature calculations for SU(2) manifolds, and applications to quantum optimization and metrology via the quantum Fisher information metric. The discussion interprets geometric quantities in terms of uncertainty principles, state robustness, and quantum criticality, while outlining future research directions in quantum chaos-geometry correlations, non-associative geometry, and geometric quantum algorithm design. We conclude that Riemannian geometry provides a powerful framework for understanding and exploiting the structure of quantum state space, with applications across quantum information processing, metrology, and foundational physics.

•Jan 31, 2026•0 citations

Metrics induced by certain Hilbert-Schmidt fidelities on positive semi-definite matrices

Ho Minh Toan, Vu The Khoi

Motivated by measuring the degree of similarity of a pair of quantum states (density matrices), we consider the metric property of the modified Bures angles and modified Bures distances of symmetric functions which are extensions of some fidelity measures on the spaces [Formula: see text] of nonzero positive semi-definite matrices. We use the positive semi-definiteness of the Gram-type matrices to characterize the metric property of the modified Bures angles. As a consequence, we can show that the modified Bures angles induced by the geometric mean, harmonic mean, minimum and maximum of two positive numbers are metrics on [Formula: see text]. In addition, we can also show that the metric property of the modified Bures angles is stronger than that of the modified Bures distances.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Apr 23, 2023•0 citations

Bures–Wasserstein geometry for positive-definite Hermitian matrices and their trace-one subset

Jesse van Oostrum

Abstract In his classical argument, Rao derives the Riemannian distance corresponding to the Fisher metric using a mapping between the space of positive measures and Euclidean space. He obtains the Hellinger distance on the full space of measures and the Fisher distance on the subset of probability measures. In order to highlight the interplay between Fisher theory and quantum information theory, we extend this construction to the space of positive-definite Hermitian matrices using Riemannian submersions and quotient manifolds. The analog of the Hellinger distance turns out to be the Bures–Wasserstein (BW) distance, a distance measure appearing in optimal transport, quantum information, and optimisation theory. First we present an existing derivation of the Riemannian metric and geodesics associated with this distance. Subsequently, we present a novel derivation of the Riemannian distance and geodesics for this metric on the subset of trace-one matrices, analogous to the Fisher distance for probability measures.

arXiv:2001.08056•Information Geometry•Sep 22, 2022•9 citations

Bures-Wasserstein Geometry

Jesse van Oostrum

The Bures-Wasserstein distance is a Riemannian distance on the space of positive definite Hermitian matrices and is given by: $d(\Sigma,T) = \left[\text{tr}(\Sigma) + \text{tr}(T) - 2 \text{tr} \left(\Sigma^{1/2}T\Sigma^{1/2}\right)^{1/2}\right]^{1/2}$. This distance function appears in the fields of optimal transport, quantum information, and optimisation theory. In this paper, the geometrical properties of this distance are studied using Riemannian submersions and quotient manifolds. The Riemannian metric and geodesics are derived on both the whole space and the subspace of trace-one matrices. In the first part of the paper a general framework is provided, including different representations of the tangent bundle for the SLD Fisher metric. The last part of the paper unifies up till now independent arguments and results from quantum information theory and optimal transport. The Bures-Wasserstein geometry is related to the Fubini-Study metric and the Wigner-Yanase information.

•Jan 21, 2020•1 citation

Density matrices are fundamental mathematical objects for describing quantum states, and their set can be viewed as a subset of the manifold of symmetric positive definite (SPD) matrices. Based on the Riemannian geometric structure on the SPD manifold, this paper analyzes geometric paths and distance functions on the space of density matrices. We adopt the affine-invariant Riemannian metric and review the associated geodesics and Riemannian distance, comparing them with linear interpolation and the Frobenius distance defined in the ambient Euclidean space. Numerical experiments are conducted to visualize geodesic paths between density matrices and to quantitatively compare several commonly used matrix distances, including the Frobenius, Riemannian, and Bures distances. The results demonstrate that distances induced by different geometric structures exhibit fundamentally different behaviors in characterizing discrepancies between density matrices, reflecting the intrinsic non-Euclidean geometry of the space. These analyses and numerical results provide a coherent geometric perspective for understanding the structure and similarity of density matrices.

This manuscript explores the Riemannian geometry of quantum state manifolds and its physical implications. We establish that quantum state manifolds, as submanifolds of Hilbert space, possess intrinsic Riemannian geometry characterized by a metric tensor derived from the Hilbert space inner product. This geometric framework reveals fundamental connections: the distance between nearby states scales with quantum fluctuations, curvature indicates state robustness, and geometric phase transitions occur in many-body systems. We review historical developments from early quantum geometry to modern quantum information geometry, and present methodologies for computing geometric quantities and their physical interpretations. Results include distance-fluctuation relations for coherent states, curvature calculations for SU(2) manifolds, and applications to quantum optimization and metrology via the quantum Fisher information metric. The discussion interprets geometric quantities in terms of uncertainty principles, state robustness, and quantum criticality, while outlining future research directions in quantum chaos-geometry correlations, non-associative geometry, and geometric quantum algorithm design. We conclude that Riemannian geometry provides a powerful framework for understanding and exploiting the structure of quantum state space, with applications across quantum information processing, metrology, and foundational physics.

Motivated by measuring the degree of similarity of a pair of quantum states (density matrices), we consider the metric property of the modified Bures angles and modified Bures distances of symmetric functions which are extensions of some fidelity measures on the spaces [Formula: see text] of nonzero positive semi-definite matrices. We use the positive semi-definiteness of the Gram-type matrices to characterize the metric property of the modified Bures angles. As a consequence, we can show that the modified Bures angles induced by the geometric mean, harmonic mean, minimum and maximum of two positive numbers are metrics on [Formula: see text]. In addition, we can also show that the metric property of the modified Bures angles is stronger than that of the modified Bures distances.

Abstract In his classical argument, Rao derives the Riemannian distance corresponding to the Fisher metric using a mapping between the space of positive measures and Euclidean space. He obtains the Hellinger distance on the full space of measures and the Fisher distance on the subset of probability measures. In order to highlight the interplay between Fisher theory and quantum information theory, we extend this construction to the space of positive-definite Hermitian matrices using Riemannian submersions and quotient manifolds. The analog of the Hellinger distance turns out to be the Bures–Wasserstein (BW) distance, a distance measure appearing in optimal transport, quantum information, and optimisation theory. First we present an existing derivation of the Riemannian metric and geodesics associated with this distance. Subsequently, we present a novel derivation of the Riemannian distance and geodesics for this metric on the subset of trace-one matrices, analogous to the Fisher distance for probability measures.

The Bures-Wasserstein distance is a Riemannian distance on the space of positive definite Hermitian matrices and is given by:

d(\Sigma,T) = \left[\text{tr}(\Sigma) + \text{tr}(T) - 2 \text{tr} \left(\Sigma^{1/2}T\Sigma^{1/2}\right)^{1/2}\right]^{1/2}

. This distance function appears in the fields of optimal transport, quantum information, and optimisation theory. In this paper, the geometrical properties of this distance are studied using Riemannian submersions and quotient manifolds. The Riemannian metric and geodesics are derived on both the whole space and the subspace of trace-one matrices. In the first part of the paper a general framework is provided, including different representations of the tangent bundle for the SLD Fisher metric. The last part of the paper unifies up till now independent arguments and results from quantum information theory and optimal transport. The Bures-Wasserstein geometry is related to the Fubini-Study metric and the Wigner-Yanase information.