Bures Geodesic Arcs

• Bures geodesic arcs are distance-minimizing curves defined on SPD matrices, quantum states, and covariance matrices with clear criteria for uniqueness based on rank and support conditions.
• They are derived via optimal square-root alignments or purifications, providing explicit formulations that distinguish between unique and multiple minimizers in various regimes.
• These arcs have broad implications in optimization, quantum information, and thermodynamics, enabling improved convergence rates and optimal estimation protocols.

Bures geodesic arcs are geodesic segments determined by the Bures or Bures–Wasserstein geometry on spaces of positive operators, covariance matrices, and quantum states. In the manifold of symmetric positive definite matrices they are the distance-minimizing curves of the Bures–Wasserstein metric; in the manifold of mixed quantum states they are shortest arcs for the Bures angle $d_B(\rho,\sigma)=\arccos\sqrt{F(\rho,\sigma)}$; and in rank-deficient or boundary regimes they can fail to be unique, with full families of minimizers controlled by support or rank data (Han et al., 2021, Spehner, 2023, Thanwerdas et al., 2022, Carrasco et al., 4 Jun 2026). A plausible unifying description is that these arcs arise from interpolation in a lifted space—square-root factors, purifications, or quotient representatives—followed by projection to the original state space.

1. Positive definite matrices and the Bures–Wasserstein arc

On the SPD manifold $M=\mathbb{S}_{++}^n$, the Bures–Wasserstein distance between $P,Q\in\mathbb{S}_{++}^n$ is

$$d_{\rm bw}(P,Q) = \left( \operatorname{tr}(P)+\operatorname{tr}(Q) -2\,\operatorname{tr}\!\left( (P^{1/2} Q P^{1/2})^{1/2} \right) \right)^{1/2}.$$

The same distance has the Procrustes form

$$d_{\rm bw}(P,Q) = \min_{O\in O(n)} \|P^{1/2}-Q^{1/2}O\|_{\rm F},$$

with the minimum attained at the orthogonal polar factor $O$ of $P^{1/2}Q^{1/2}$. The corresponding Bures geodesic arc is

$$\gamma(t) = \big((1-t)P^{1/2}+t\,Q^{1/2}O\big) \big((1-t)P^{1/2}+t\,Q^{1/2}O\big)^\top, \qquad t\in[0,1].$$

This is the explicit “straight-line in factor space” representation: one aligns the square roots optimally and then projects the Euclidean segment back to $\mathbb{S}_{++}^n$. The same path coincides with the covariance interpolation of the Wasserstein geodesic between centered Gaussians,

$$\omega(t) = \big((1-t)I+tT\big)\,P\,\big((1-t)I+tT\big), \qquad T = P^{-1/2}\big(P^{1/2} Q P^{1/2}\big)^{1/2} P^{-1/2}.$$

The local differential structure is equally explicit. In the neighborhood where the geodesic is unique,

$M=\mathbb{S}_{++}^n$0

and

$M=\mathbb{S}_{++}^n$1

with inverse exponential only on the open neighborhood

$M=\mathbb{S}_{++}^n$2

The paper explicitly notes that the BW geometry is not globally uniquely geodesic. It also states that the BW manifold has non-negative sectional curvature, in contrast with the Affine-Invariant manifold, and uses this to obtain the sharp comparison constant $M=\mathbb{S}_{++}^n$3 in the distance inequality used for optimization analysis (Han et al., 2021).

2. Rank stratification and minimizing arcs between covariance matrices

For covariance matrices, the relevant ambient space is

$M=\mathbb{S}_{++}^n$4

equipped with the Bures–Wasserstein distance as a quotient metric from $M=\mathbb{S}_{++}^n$5 under the right action of $M=\mathbb{S}_{++}^n$6. The orbit stratification is exactly the stratification by rank, so the geometry is not a single smooth manifold but a union of rank strata $M=\mathbb{S}_{++}^n$7. This stratified structure governs both existence and multiplicity of Bures geodesic arcs.

Between arbitrary covariance matrices $M=\mathbb{S}_{++}^n$8, every minimizing geodesic segment has the quadratic form

$M=\mathbb{S}_{++}^n$9

where $P,Q\in\mathbb{S}_{++}^n$0, $P,Q\in\mathbb{S}_{++}^n$1, and $P,Q\in\mathbb{S}_{++}^n$2 is chosen from an admissible orthogonal family. If

$P,Q\in\mathbb{S}_{++}^n$3

then the full set of minimizing geodesics is parametrized by the closed unit ball

$P,Q\in\mathbb{S}_{++}^n$4

The canonical choice $P,Q\in\mathbb{S}_{++}^n$5 yields the canonical Bures–Wasserstein geodesic

$P,Q\in\mathbb{S}_{++}^n$6

The uniqueness criterion is exact: $P,Q\in\mathbb{S}_{++}^n$7 If this condition fails, there are infinitely many minimizing geodesics. The paper also emphasizes that minimizing arcs may cross strata. For

$P,Q\in\mathbb{S}_{++}^n$8

it gives the geodesic

$P,Q\in\mathbb{S}_{++}^n$9

which passes through positive definite matrices for $$d_{\rm bw}(P,Q) = \left( \operatorname{tr}(P)+\operatorname{tr}(Q) -2\,\operatorname{tr}\!\left( (P^{1/2} Q P^{1/2})^{1/2} \right) \right)^{1/2}.$$0. This directly shows that singular endpoints do not force the interior of a minimizing Bures arc to remain singular (Thanwerdas et al., 2022).

3. Quantum-state geodesics, purifications, and angular parametrization

For mixed quantum states on the interior

$$d_{\rm bw}(P,Q) = \left( \operatorname{tr}(P)+\operatorname{tr}(Q) -2\,\operatorname{tr}\!\left( (P^{1/2} Q P^{1/2})^{1/2} \right) \right)^{1/2}.$$1

the Bures arccos distance is

$$d_{\rm bw}(P,Q) = \left( \operatorname{tr}(P)+\operatorname{tr}(Q) -2\,\operatorname{tr}\!\left( (P^{1/2} Q P^{1/2})^{1/2} \right) \right)^{1/2}.$$2

A geodesic arc is a finite segment of a constant-speed curve that is locally length-minimizing, and the shortest arc has length equal to the geodesic distance. The geometric construction uses purifications: $$d_{\rm bw}(P,Q) = \left( \operatorname{tr}(P)+\operatorname{tr}(Q) -2\,\operatorname{tr}\!\left( (P^{1/2} Q P^{1/2})^{1/2} \right) \right)^{1/2}.$$3 The purification sphere carries the Euclidean metric, the projection $$d_{\rm bw}(P,Q) = \left( \operatorname{tr}(P)+\operatorname{tr}(Q) -2\,\operatorname{tr}\!\left( (P^{1/2} Q P^{1/2})^{1/2} \right) \right)^{1/2}.$$4 is a Riemannian submersion, and a Bures geodesic is the projection of a great-circle arc

$$d_{\rm bw}(P,Q) = \left( \operatorname{tr}(P)+\operatorname{tr}(Q) -2\,\operatorname{tr}\!\left( (P^{1/2} Q P^{1/2})^{1/2} \right) \right)^{1/2}.$$5

whose tangent is horizontal at $$d_{\rm bw}(P,Q) = \left( \operatorname{tr}(P)+\operatorname{tr}(Q) -2\,\operatorname{tr}\!\left( (P^{1/2} Q P^{1/2})^{1/2} \right) \right)^{1/2}.$$6.

For invertible endpoints $$d_{\rm bw}(P,Q) = \left( \operatorname{tr}(P)+\operatorname{tr}(Q) -2\,\operatorname{tr}\!\left( (P^{1/2} Q P^{1/2})^{1/2} \right) \right)^{1/2}.$$7 and $$d_{\rm bw}(P,Q) = \left( \operatorname{tr}(P)+\operatorname{tr}(Q) -2\,\operatorname{tr}\!\left( (P^{1/2} Q P^{1/2})^{1/2} \right) \right)^{1/2}.$$8, the full family of Bures geodesics is labeled by unitary self-adjoint operators $$d_{\rm bw}(P,Q) = \left( \operatorname{tr}(P)+\operatorname{tr}(Q) -2\,\operatorname{tr}\!\left( (P^{1/2} Q P^{1/2})^{1/2} \right) \right)^{1/2}.$$9 satisfying

$$d_{\rm bw}(P,Q) = \min_{O\in O(n)} \|P^{1/2}-Q^{1/2}O\|_{\rm F},$$0

The shortest arc corresponds to $$d_{\rm bw}(P,Q) = \min_{O\in O(n)} \|P^{1/2}-Q^{1/2}O\|_{\rm F},$$1. The paper states that for generic invertible $$d_{\rm bw}(P,Q) = \min_{O\in O(n)} \|P^{1/2}-Q^{1/2}O\|_{\rm F},$$2, if $$d_{\rm bw}(P,Q) = \min_{O\in O(n)} \|P^{1/2}-Q^{1/2}O\|_{\rm F},$$3 is nondegenerate, there are $$d_{\rm bw}(P,Q) = \min_{O\in O(n)} \|P^{1/2}-Q^{1/2}O\|_{\rm F},$$4 distinct geodesic arcs, with the shortest one the unique arc for which all eigenvalue signs are $$d_{\rm bw}(P,Q) = \min_{O\in O(n)} \|P^{1/2}-Q^{1/2}O\|_{\rm F},$$5. In the same framework, the reduced mixed-state geodesic can be realized as the projection of a two-dimensional unitary rotation on system plus ancilla, so the arc is the reduced dynamics of a physically implemented non-Markovian evolution (Spehner, 2023).

A closely related angle-based formulation persists in the $$d_{\rm bw}(P,Q) = \min_{O\in O(n)} \|P^{1/2}-Q^{1/2}O\|_{\rm F},$$6-algebraic setting. There, geodesic arcs are explicit curves with normal parametrization

$$d_{\rm bw}(P,Q) = \min_{O\in O(n)} \|P^{1/2}-Q^{1/2}O\|_{\rm F},$$7

and every such arc has length

$$d_{\rm bw}(P,Q) = \min_{O\in O(n)} \|P^{1/2}-Q^{1/2}O\|_{\rm F},$$8

The paper further states that the state space, with the Bures metric, is a length space and that shortest paths are geodesic arcs up to reparameterization (Alberti, 2020).

4. Boundary states, non-faithful endpoints, and rank-changing geometry

When at least one endpoint is faithful, the shortest Bures geodesic arc from $$d_{\rm bw}(P,Q) = \min_{O\in O(n)} \|P^{1/2}-Q^{1/2}O\|_{\rm F},$$9 to $O$0 is

$O$1

with $O$2 from the polar decomposition

$O$3

For non-faithful states, the same trigonometric structure survives after regularization, but the limiting operator generally decomposes as

$O$4

or equivalently through a contraction term $O$5 acting on the kernel-overlap sector. The paper presents the condition

$O$6

as the criterion under which the regularization-dependent term disappears and the shortest geodesic is unique. When this fails, there are infinitely many shortest geodesic arcs, all with the same length, in direct analogy with great-circle arcs between antipodal points. Any such shortest geodesic has support inside

$O$7

and its rank satisfies

$O$8

For pure states, the formula reduces to the Fubini–Study geodesic (Carrasco et al., 4 Jun 2026).

The local geometry near rank-changing points is dimension-dependent. For qubits,

$O$9

and the Bures line element is

$P^{1/2}Q^{1/2}$0

Although $P^{1/2}Q^{1/2}$1 diverges at the pure-state boundary $P^{1/2}Q^{1/2}$2, the coordinate change

$P^{1/2}Q^{1/2}$3

regularizes the metric and yields

$P^{1/2}Q^{1/2}$4

so the divergence is a coordinate artifact. Radial Bures geodesics satisfy

$P^{1/2}Q^{1/2}$5

For $P^{1/2}Q^{1/2}$6, the paper states that the transverse geometry near a pure state is conical. Writing

$P^{1/2}Q^{1/2}$7

the Bures metric reduces to

$P^{1/2}Q^{1/2}$8

With $P^{1/2}Q^{1/2}$9, this becomes

$$\gamma(t) = \big((1-t)P^{1/2}+t\,Q^{1/2}O\big) \big((1-t)P^{1/2}+t\,Q^{1/2}O\big)^\top, \qquad t\in[0,1].$$0

The geodesic equations then force angular motion to freeze asymptotically near the cone tip, so geodesics reaching the rank-changing point become radial. This underlies a common misconception: singular-looking behavior at the boundary is not uniform across dimensions; it is a coordinate artifact for $$\gamma(t) = \big((1-t)P^{1/2}+t\,Q^{1/2}O\big) \big((1-t)P^{1/2}+t\,Q^{1/2}O\big)^\top, \qquad t\in[0,1].$$1, but a genuine curvature singularity for $$\gamma(t) = \big((1-t)P^{1/2}+t\,Q^{1/2}O\big) \big((1-t)P^{1/2}+t\,Q^{1/2}O\big)^\top, \qquad t\in[0,1].$$2 (Huang et al., 27 May 2026).

5. Adapted, generalized, and constrained Bures-type arcs

In the adapted Bures–Wasserstein geometry of Gaussian processes, discrete-time processes are represented by block-lower triangular factors $$\gamma(t) = \big((1-t)P^{1/2}+t\,Q^{1/2}O\big) \big((1-t)P^{1/2}+t\,Q^{1/2}O\big)^\top, \qquad t\in[0,1].$$3, and the metric space is the quotient

$$\gamma(t) = \big((1-t)P^{1/2}+t\,Q^{1/2}O\big) \big((1-t)P^{1/2}+t\,Q^{1/2}O\big)^\top, \qquad t\in[0,1].$$4

with

$$\gamma(t) = \big((1-t)P^{1/2}+t\,Q^{1/2}O\big) \big((1-t)P^{1/2}+t\,Q^{1/2}O\big)^\top, \qquad t\in[0,1].$$5

Geodesics are exactly affine interpolations in factor space, modulo optimal adapted orthogonal alignment: $$\gamma(t) = \big((1-t)P^{1/2}+t\,Q^{1/2}O\big) \big((1-t)P^{1/2}+t\,Q^{1/2}O\big)^\top, \qquad t\in[0,1].$$6 The paper proves that every constant-speed geodesic has this form. It also gives a uniqueness criterion: $$\gamma(t) = \big((1-t)P^{1/2}+t\,Q^{1/2}O\big) \big((1-t)P^{1/2}+t\,Q^{1/2}O\big)^\top, \qquad t\in[0,1].$$7 The full space is Alexandrov with non-negative curvature, and the regular subspace $$\gamma(t) = \big((1-t)P^{1/2}+t\,Q^{1/2}O\big) \big((1-t)P^{1/2}+t\,Q^{1/2}O\big)^\top, \qquad t\in[0,1].$$8 is geodesically convex (Acciaio et al., 31 Jan 2026).

A different extension uses the Bures–Wasserstein manifold as the base for generalized fidelities. There the relevant “Bures geodesic arcs” are base-point paths $$\gamma(t) = \big((1-t)P^{1/2}+t\,Q^{1/2}O\big) \big((1-t)P^{1/2}+t\,Q^{1/2}O\big)^\top, \qquad t\in[0,1].$$9 along which the generalized fidelity becomes rigid. The BW geodesic between $\mathbb{S}_{++}^n$0 and $\mathbb{S}_{++}^n$1 is

$\mathbb{S}_{++}^n$2

If the base point itself moves along the BW geodesic between $\mathbb{S}_{++}^n$3 and $\mathbb{S}_{++}^n$4,

$\mathbb{S}_{++}^n$5

then

$\mathbb{S}_{++}^n$6

If the inverse base moves along the BW geodesic between $\mathbb{S}_{++}^n$7 and $\mathbb{S}_{++}^n$8, the same Uhlmann lock-in holds. By contrast, certain AI and Euclidean geodesic paths lock the generalized fidelity to Matsumoto fidelity. In this formulation, geodesic arcs are not the paths between the fixed states $\mathbb{S}_{++}^n$9 and $$\omega(t) = \big((1-t)I+tT\big)\,P\,\big((1-t)I+tT\big), \qquad T = P^{-1/2}\big(P^{1/2} Q P^{1/2}\big)^{1/2} P^{-1/2}.$$0 themselves, but the distinguished trajectories of the linearization base point (Afham et al., 2024).

Constrained submanifolds can behave much more rigidly than the ambient SPD space. For determinant-normalized Kronecker positive definite matrices, the ambient Bures geodesic

$$\omega(t) = \big((1-t)I+tT\big)\,P\,\big((1-t)I+tT\big), \qquad T = P^{-1/2}\big(P^{1/2} Q P^{1/2}\big)^{1/2} P^{-1/2}.$$1

remains in the Kronecker model if and only if one of the two one-factor conditions holds: $$\omega(t) = \big((1-t)I+tT\big)\,P\,\big((1-t)I+tT\big), \qquad T = P^{-1/2}\big(P^{1/2} Q P^{1/2}\big)^{1/2} P^{-1/2}.$$2 The paper proves the stronger equivalence that local membership near one endpoint already implies membership of the whole segment. Outside these one-factor leaves, the ambient Bures geodesic exits the model immediately (Yang et al., 4 May 2026).

6. Optimization, metrology, thermodynamics, and broader significance

In Riemannian optimization over $$\omega(t) = \big((1-t)I+tT\big)\,P\,\big((1-t)I+tT\big), \qquad T = P^{-1/2}\big(P^{1/2} Q P^{1/2}\big)^{1/2} P^{-1/2}.$$3, Bures geodesic arcs are the curves along which geodesic convexity is tested and along which convergence estimates are derived. The paper proves that

$$\omega(t) = \big((1-t)I+tT\big)\,P\,\big((1-t)I+tT\big), \qquad T = P^{-1/2}\big(P^{1/2} Q P^{1/2}\big)^{1/2} P^{-1/2}.$$4

are geodesically convex on the BW manifold under the stated assumptions, and more generally that

$$\omega(t) = \big((1-t)I+tT\big)\,P\,\big((1-t)I+tT\big), \qquad T = P^{-1/2}\big(P^{1/2} Q P^{1/2}\big)^{1/2} P^{-1/2}.$$5

is geodesically convex when $$\omega(t) = \big((1-t)I+tT\big)\,P\,\big((1-t)I+tT\big), \qquad T = P^{-1/2}\big(P^{1/2} Q P^{1/2}\big)^{1/2} P^{-1/2}.$$6 is increasing and convex. Because BW geometry has non-negative curvature, the trigonometric constant in the distance comparison inequality is $$\omega(t) = \big((1-t)I+tT\big)\,P\,\big((1-t)I+tT\big), \qquad T = P^{-1/2}\big(P^{1/2} Q P^{1/2}\big)^{1/2} P^{-1/2}.$$7, whereas the AI geometry has $$\omega(t) = \big((1-t)I+tT\big)\,P\,\big((1-t)I+tT\big), \qquad T = P^{-1/2}\big(P^{1/2} Q P^{1/2}\big)^{1/2} P^{-1/2}.$$8. The same paper reports faster and more robust convergence for BW on weighted least squares, Lyapunov equations, trace regression, and metric learning, while also noting that AI can outperform BW for log-det maximization and Gaussian mixture model estimation (Han et al., 2021).

In quantum information and metrology, Bures geodesics are operationally distinguished curves. One paper shows that geodesic evolutions are the reduced dynamics of a system coupled to an ancilla, and that when the unknown parameter is encoded along such a geodesic, the estimation error from measurements on the system alone can equal the smallest error achievable from joint measurements on system and ancilla; in that regime the optimal measurement on the system is $$\omega(t) = \big((1-t)I+tT\big)\,P\,\big((1-t)I+tT\big), \qquad T = P^{-1/2}\big(P^{1/2} Q P^{1/2}\big)^{1/2} P^{-1/2}.$$9-independent and determined by the intersections of the geodesic with the boundary of quantum states (Spehner, 2023). A second paper shows that shortest Bures geodesic arcs for non-faithful states are exactly the curves that saturate the generalized Mandelstam–Tamm quantum speed limit, so every shortest arc constructed there corresponds to an optimal speed-limit-saturating evolution (Carrasco et al., 4 Jun 2026).

In equilibrium quantum thermodynamics, the Bures angle functions as an arc-length quantity relative to the maximally mixed state

$M=\mathbb{S}_{++}^n$00

For commuting thermal states, the paper gives the exact relation

$M=\mathbb{S}_{++}^n$01

and interprets the Bures angle as measuring “the length of the curve within $M=\mathbb{S}_{++}^n$02.” In that thermal setting, Bures angle and distance reduce respectively to the Bhattacharyya and Hellinger distances, and the paper derives explicit relations linking Bures position relative to $M=\mathbb{S}_{++}^n$03 to free energy, work moments, and Carnot-efficiency bounds (Hardal et al., 2016).

Across these settings, the recurrent structural feature is explicitness: Bures geodesic arcs are often available in closed form, yet their uniqueness, rank behavior, and confinement to submodels depend sharply on support conditions, stratification by rank, and the ambient geometry of the space in question.