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Bures-Wasserstein Geometry

Jesse van Oostrum

Published Jan 21, 2020• 1 citation
OpenAlex

Abstract

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.

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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.
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.
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.
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.
We define a local Riemannian metric tensor in the manifold of Gaussian channels and the distance that it induces. We adopt an information-geometric approach and define a metric derived from the Bures-Fisher metric for quantum states. The resulting metric inherits several desirable properties from the Bures-Fisher metric and is operationally motivated by distinguishability considerations: It serves as an upper bound to the attainable quantum Fisher information for the channel parameters using Gaussian states, under generic constraints on the physically available resources. Our approach naturally includes the use of entangled Gaussian probe states. We prove that the metric enjoys some desirable properties like stability and covariance. As a by-product, we also obtain some general results in Gaussian channel estimation that are the continuous-variable analogs of previously known results in finite dimensions. We prove that optimal probe states are always pure and bounded in the number of ancillary modes, even in the presence of constraints on the reduced state input in the channel. This has experimental and computational implications. It limits the complexity of optimal experimental setups for channel estimation and reduces the computational requirements for the evaluation of the metric: Indeed, we construct a converging algorithm for its computation. We provide explicit formulas for computing the multiparametric quantum Fisher information for dissipative channels probed with arbitrary Gaussian states and provide the optimal observables for the estimation of the channel parameters (e.g., bath couplings, squeezing, and temperature).
Abstract We study a quantum analogue of the 2-Wasserstein distance as a measure of proximity on the set Ω N of density matrices of dimension N. We show that such (semi-)distances do not induce Riemannian metrics on the tangent bundle of Ω N and are typically not unitarily invariant. Nevertheless, we prove that for N = 2 dimensional Hilbert space the quantum 2-Wasserstein distance (unique up to rescaling) is monotonous with respect to any single-qubit quantum operation and the solution of the quantum transport problem is essentially unique. Furthermore, for any N ⩾ 3 and the quantum cost matrix proportional to a projector we demonstrate the monotonicity under arbitrary mixed unitary channels. Finally, we provide numerical evidence which allows us to conjecture that the unitary invariant quantum 2-Wasserstein semi-distance is monotonous with respect to all CPTP maps for dimension N = 3 and 4.
In this paper, we describe a possible generalization of the Wasserstein-2 metric, originally defined on the space of scalar probability densities, to the space of Hermitian matrices with trace one and to the space of matrix-valued probability densities. Our approach follows a control-theoretic optimization formulation of the Wasserstein-2 metric, having its roots in fluid dynamics, and utilizes certain results from the quantum mechanics of open systems, in particular the Lindblad equation. It allows determining the gradient flow for the quantum entropy relative to this matricial Wasserstein metric.