Bures-Wasserstein Geometry

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