Bures-Wasserstein Visual Intuitions

The Bures-Wasserstein (BW) metric defines the geometry under which the invariant hierarchy is measured. This page walks through the metric's components on 2×2 SPD matrices: eigenvalue resolution, the conjugation sandwich A^½BA^½, the BW geodesic, and fidelity as a principal angle. Formulas follow Bhatia, Jain, and Lim (2019), "On the Bures-Wasserstein distance between positive definite matrices."

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Stage 01 / 04

Eigenvalue Resolution

A 2×2 symmetric positive definite matrix Σ = QΛQ^T can be continuously rotated into its own eigenframe. The slider below interpolates from the original matrix (with off-diagonal entries) to the diagonal form. The ellipse on the left shows the covariance shape; the matrix on the right shows the entries. Notice that the trace tr(Σ) = λ₁ + λ₂ remains constant throughout, since eigenvalues are basis-invariant.

Σ(t) = R(tθ)^T Λ R(tθ)
t = 0: original frame. t = 1: eigenframe (diagonal).

Diagonalization t = 0.00

OriginalEigenframe

Stage 02 / 04

The Conjugation Sandwich

The BW distance between two SPD matrices A and B involves the product A^½ B A^½. This "sandwich" conjugates B into A's eigenframe. The animation below shows B (right panel) morphing continuously into A^½BA^½. A is fixed on the left. The ghost outline of the original B persists for comparison. A's eigenaxes are drawn as dashed lines on the right panel so you can see the target frame.

A^½ B A^½
Conjugates B into A's geometry. The eigenvalues of this product determine the BW distance.

Conjugation t = 0.00

B originalA^½BA^½

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The BW Geodesic

The shortest path between two SPD matrices under the BW metric is not a straight line in matrix space. The geodesic γ(t) interpolates between A (t=0) and B (t=1) through the curved SPD manifold. The formula (equation 39 in Bhatia-Jain-Lim) involves the matrix square root and inverse square root of A.

γ(t) = A^½ [(1−t)I + t(A^−½BA^−½)^½]² A^½
The covariance ellipse morphs along this path, not along the Euclidean interpolation.

Geodesic parameter t = 0.00

A0.5B

Drag the slider or press auto-play to watch the ellipse traverse the BW geodesic. A faint trail shows intermediate positions.

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Fidelity as Principal Angle

The BW distance decomposes into trace terms and a fidelity term: d²(A, B) = tr(A) + tr(B) − 2F(A, B), where F(A, B) = tr(A^½BA^½)^½. For 2×2 matrices, fidelity captures how aligned the two ellipses are. Rotate B's eigenframe and adjust its aspect ratio to see how fidelity and distance respond.

d²_BW(A, B) = tr(A) + tr(B) − 2 tr(A^½BA^½)^½
Fidelity F = tr(A^½BA^½)^½ measures overlap. Maximum when A = B.

B rotation angle θ = 0°

0°45°90°

B aspect ratio λ₂/λ₁ = 0.30

ElongatedCircular

The same conjugation pattern shows up in quantum measurement: ⟨O⟩ = tr(U^†OU ρ). The A-Gate is a specific U built from PN-neuron-derived gates. The QNFM framework uses this connection to read tangent-space projections of the patient-specific covariance shift through a fixed quantum measurement basis.

The BW metric is one of two natural metrics on the SPD manifold (the other being affine-invariant). Both are used in Riemannian BCI literature. The A-Gate's measurement geometry corresponds to the BW metric because the quantum fidelity between density matrices is the Bures fidelity.

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