Note on Explicit Formulae for the Bures Metric

TL;DR

This paper derives explicit, diagonalization-free expressions for the Bures metric on the manifold of finite-dimensional nondegenerate density matrices. By solving the Sylvester equation $\rho X+X\rho=Y$ and reformulating the solution as a polynomial in $\rho$, it yields multiple matrix-based representations of the metric that rely only on traces, determinants and invariants rather than eigenvalue decompositions. It presents a general coefficient matrix formulation, a Cayley-Hamilton based reduction, and a local decomposition aligned with the structure $\mathcal D \approx \mathbb{R}^n\times U(n)/T^n$, plus concrete $n=2$ and $n=3$ examples. The results enable efficient computation of the Bures metric in higher dimensions and have relevance for quantum information geometry and state distinguishability.

Abstract

The aim of this paper is to derive explicit formulae for the Riemannian Bures metric on the manifold of (finite dimensional) nondegenerate density matrices. The computation of the Bures metric using the presented equations does not require any diagonalization procedure and uses matrix products, determinants and traces, only.