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arXiv:2001.08056 · Information Geometry

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

Jesse van Oostrum

Published Sep 22, 2022• 9 citations
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Abstract

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.

DOI
10.1007/s41884-022-00069-7

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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(Σ,T)=[tr(Σ)+tr(T)−2tr(Σ1/2TΣ1/2)1/2]1/2d(\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}d(Σ,T)=[tr(Σ)+tr(T)−2tr(Σ1/2TΣ1/2)1/2]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.
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.
The Hellinger distance between quantum states is a significant measure in quantum information theory, known for its Riemannian and monotonic properties. It is easier to compute than the Bures distance, another measure that shares these properties, particularly in information-theoretic applications. Furthermore, in these applications, random quantum states are crucial for securing communication, studying entanglement, benchmarking quantum systems, aiding in quantum state tomography, and understanding how quantum systems behave in noisy environments. Recent works have computed exact results for distance measures such as Bures and Hilbert-Schmidt for random density matrices; however, no such results exist for the Hellinger distance. In this work, we derive the mean and variance of the Hellinger distance between pairs of density matrices, where one or both matrices are random. Our derivation utilizes Weingarten functions to perform the necessary unitary group integrals and is supported by existing results for eigenvalue moments of random density matrices. Along the way, we also obtain exact results for the mean affinity and mean square affinity. The first two cumulants of the Hellinger distance allow us to propose an approximation for the corresponding probability density function based on the gamma distribution. Our analytical results are corroborated through Monte Carlo simulations, showing excellent agreement.
Given a pure state vector | x ⟩ and a density matrix , the function defines a probability density on the space of pure states parameterised by density matrices. The associated Fisher–Rao information measure is used to define a unitary invariant Riemannian metric on the space of density matrices. An alternative derivation of the metric, based on square-root density matrices and trace norms, is provided. This is applied to the problem of quantum-state estimation. In the simplest case of unitary parameter estimation, new higher-order corrections to the uncertainty relations, applicable to general mixed states, are derived.
The indispensability of the complex structure in quantum mechanics has beenexperimentally confirmed, but the geometric description of the latent world (modaldimension) still lacks a rigorous metric theory. This paper elevates the latent worldalgebraic framework to the level of information geometry, proving that the naturalmetric carried by the latent world subalgebra Alat is precisely the Bures-Wassersteinmetric, and that the measurement projection π : A → Aobs corresponds to a gradient flow from the Wasserstein-2 space to the Fisher-Rao space. The core discoveryis that the critical condition for the information potential difference ∆µ ≥ ln2 isequivalent to the curvature singularity of the Bures distance, and the projection dynamics is exactly the blow-up solution of the Wasserstein gradient flow at the divergence of the Ricci curvature. This paper establishes the exact relation Klat = −14FQ between the latent world curvature Klat and the quantum Fisher information FQ,proposes testable predictions of information-geometric phase transitions, and extends the framework to noncommutative geometry (spectral triples) and interfaceswith quantum gravity. This work provides a unified perspective of differential geometry and optimal transport for understanding the quantum-classical transition.
We introduce a family of fidelities, termed generalized fidelity, which are based on the Riemannian geometry of the Bures-Wasserstein manifold. We show that this family of fidelities generalizes standard quantum fidelities such as Uhlmann-, Holevo-, and Matsumoto-fidelity and demonstrate that it satisfies analogous celebrated properties. The generalized fidelity naturally arises from a generalized Bures distance, the natural distance obtained by linearizing the Bures-Wasserstein manifold. We prove various invariance and covariance properties of generalized fidelity as the point of linearization moves along geodesic-related paths. We also provide a Block-matrix characterization and prove an Uhlmann-like theorem, as well as provide further extensions to the multivariate setting and to quantum Rényi divergences, generalizing Petz-, Sandwiched-, Reverse sandwiched-, and Geometric-Rényi divergences of order α.