Coherence-Enhanced Orthogonalization Dynamics

• Coherence-enhanced orthogonalization dynamics is a unifying concept showing that structured phase information reshapes the effective orthogonality of quantum states and numerical bases.
• It demonstrates that initial coherent superpositions can accelerate state orthogonalization in nonequilibrium quenches, as quantified by metrics like the Loschmidt amplitude and scaling laws.
• Applications span quantum state engineering, environment-induced dipole coupling, symmetric resource theories, and optimized numerical algorithms in machine learning.

Searching arXiv for the specified papers to ground the article in the cited literature. Coherence-enhanced orthogonalization dynamics denotes a family of mechanisms in which coherence, structured anisotropy, or symmetry-preserving orthogonalization changes how states, transitions, updates, or bases become effectively orthogonal. In current arXiv usage, the phrase appears most explicitly in nonequilibrium quench dynamics, where initial coherence accelerates orthogonalization after a localized perturbation, but related ideas also recur in environment-induced coupling of nominally orthogonal dipoles, heralded and filter-based state orthogonalization, Löwdin-based bridges between coherence and superposition, and matrix-valued optimization or numerical linear algebra (Donelli et al., 28 Jul 2025, Bennett, 2020, Torun, 2022, Khaled et al., 19 Oct 2025). This suggests a unifying theme: orthogonality is not fixed solely by bare Euclidean overlap, but can be reshaped by the reservoir, by conditional filtering, by basis geometry, or by the metric used to orthogonalize.

1. Conceptual structure

The most literal use of the term concerns sudden-quench quantum dynamics. In "Orthogonalization speed-up from quantum coherence after a sudden quench" (Donelli et al., 28 Jul 2025), the phenomenon is that a coherent superposition of many energy eigenstates orthogonalizes faster after a local defect quench than the corresponding incoherent diagonal state. In "Impact of quantum coherence on the dynamics and thermodynamics of quenched free fermions coupled to a localized defect" (Donelli et al., 2 Aug 2025), the same idea is extended to free fermions, coherent states, and cat states, with Loschmidt dynamics and work quasiprobabilities as the main observables.

A second usage concerns nominally orthogonal radiative channels. "Inverse design of environment-induced coherence" shows that orthogonal dipole moments do not by themselves forbid interference: in an anisotropic electromagnetic environment, the relevant overlap is reservoir-mediated and depends on the dyadic Green tensor rather than on the bare scalar product of the dipoles (Bennett, 2020).

A third usage appears in quantum state engineering. "Heralded orthogonalisation of coherent states and their conversion to discrete-variable superpositions" realizes a conditional optical map from the nonorthogonal pair $\{|\alpha\rangle,|-\alpha\rangle\}$ to an orthogonal output pair without destroying the usable output state (Kruse et al., 2017). "Orthogonalization of partly unknown quantum states" shows that exact orthogonalization becomes possible once one knows the expectation value of a single operator and applies a suitable quantum filter (Jezek et al., 2014).

A fourth usage is resource-theoretic. "Coherence, superposition, and Löwdin symmetric orthogonalization" and "Symmetric orthogonalization and probabilistic weights in resource quantification" use Löwdin symmetric orthogonalization to connect coherence in orthonormal bases with superposition in nonorthogonal bases, while "Quantifying the Coherence Between Coherent States" introduces an orthogonalization procedure that makes Baumgratz-type coherence quantification applicable to coherent-state decompositions (Torun, 2022, Torun, 18 Aug 2025, Tan et al., 2017).

In optimizer and numerical linear algebra work, the language of coherence is partly interpretive rather than formal. MuonBP alternates local shardwise orthogonalization with periodic full orthogonalization to restore whole-matrix geometry at scale, while recent analyses of simplified Muon show that spectral orthogonalization decouples matrix dynamics into scalar spectral recursions (Khaled et al., 19 Oct 2025, Ma et al., 20 Jan 2026). A plausible implication is that some algorithmic orthogonalization schemes can be read as coherence-shaping mechanisms over spectral modes or distributed shards.

2. Nonequilibrium quench orthogonalization

The canonical physical setup is a particle in a one-dimensional harmonic trap with Hamiltonian

$$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$

subjected at $t=0$ to a localized repulsive defect

$\hat V = k\,\delta(\hat x), \qquad k>0,$

so that the post-quench Hamiltonian is $\hat H'=\hat H+k\,\delta(\hat x)$. Because the defect is centered at $x=0$, only even-parity oscillator states are affected (Donelli et al., 28 Jul 2025).

The main diagnostic is the Loschmidt amplitude

$$\nu(t)=\mathrm{Tr}\!\left[\hat U'(t)\,\hat\rho(0)\,\hat U(-t)\right],$$

with $\hat U(t)=e^{-i\hat H t}$ and $\hat U'(t)=e^{-i\hat H' t}$. For pure input states, $|\nu(t)|$ is the overlap between the unperturbed and quenched evolutions, and the Bures angle is

$$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$0

Orthogonality corresponds to $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$1, equivalently $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$2 (Donelli et al., 28 Jul 2025).

The core result is a scaling law

$$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$3

where $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$4 is the number of coherently occupied even eigenstates. For coherent initial superpositions, $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$5; for the diagonal state with the same populations, $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$6 at short times. Thus increasing $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$7 accelerates orthogonalization only in the coherent case. For strong coupling $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$8, the reported asymptotic value is $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$9 (Donelli et al., 28 Jul 2025).

The work-quasiprobability description makes the mechanism more explicit. The Loschmidt amplitude is expanded as

$t=0$0

and the corresponding Kirkwood-Dirac quasidistribution is

$t=0$1

For coherent initial states, $t=0$2 need not be positive or real; its real part is the Margenau-Hill quasiprobability. The non-positivity functional

$t=0$3

grows with $t=0$4, and for the single-particle example shown the paper reports

$t=0$5

The average work satisfies

$t=0$6

and for the equal superposition state its large-$t=0$7 asymptotic form is

$t=0$8

Because the quantum-speed-limit bound is

$t=0$9

the growth of $\hat V = k\,\delta(\hat x), \qquad k>0,$0 lowers the minimum orthogonalization time; the paper states that the orthogonalization time can decrease “as $\hat V = k\,\delta(\hat x), \qquad k>0,$1 at maximum” (Donelli et al., 28 Jul 2025).

The few-body free-fermion extension sharpens this picture. In the strong-defect regime, standard cat states do not necessarily maximize orthogonalization for most times, whereas cat or coherent states with an alternating phase factor $\hat V = k\,\delta(\hat x), \qquad k>0,$2 produce the strongest suppression of the Loschmidt echo. The paper identifies these alternating-phase coherent superpositions as the most effective cases for fast and sustained orthogonalization under the $\hat V = k\,\delta(\hat x), \qquad k>0,$3-defect quench (Donelli et al., 2 Aug 2025). This suggests that the decisive quantity is not coherence in the abstract, but coherence with the right phase structure relative to the post-quench overlap matrix.

3. Environment-induced coherence of orthogonal transitions

In the photonic setting, the relevant system is a three-level emitter in a $\hat V = k\,\delta(\hat x), \qquad k>0,$4 configuration, with excited state $\hat V = k\,\delta(\hat x), \qquad k>0,$5 decaying to two nearly degenerate lower states $\hat V = k\,\delta(\hat x), \qquad k>0,$6 and $\hat V = k\,\delta(\hat x), \qquad k>0,$7. The transition dipoles are $\hat V = k\,\delta(\hat x), \qquad k>0,$8 and $\hat V = k\,\delta(\hat x), \qquad k>0,$9, and the environment is encoded by the dyadic Green tensor $\hat H'=\hat H+k\,\delta(\hat x)$0 (Bennett, 2020).

The spontaneous rates are

$\hat H'=\hat H+k\,\delta(\hat x)$1

$\hat H'=\hat H+k\,\delta(\hat x)$2

while the cross-damping term is

$\hat H'=\hat H+k\,\delta(\hat x)$3

This $\hat H'=\hat H+k\,\delta(\hat x)$4 is the signature of environment-induced coherence. If the environment is isotropic, $\hat H'=\hat H+k\,\delta(\hat x)$5, and $\hat H'=\hat H+k\,\delta(\hat x)$6, which vanishes for orthogonal dipoles. In anisotropic environments, $\hat H'=\hat H+k\,\delta(\hat x)$7 is tensorial, and the bilinear form $\hat H'=\hat H+k\,\delta(\hat x)$8 can be nonzero even when $\hat H'=\hat H+k\,\delta(\hat x)$9 (Bennett, 2020).

The steady-state coherence is written compactly as

$x=0$0

with

$x=0$1

This makes clear that maximizing coherence is not the same as maximizing a Purcell factor: the relevant objective is the ratio of cross-coupling to total decay (Bennett, 2020).

The benchmark geometry is a perfectly reflecting plane. For an atom above a mirror, the equal-point Green tensor is anisotropic, with different tangential and normal responses. For orthogonal circularly rotating dipoles in the $x=0$2-plane,

$x=0$3

the induced coherence oscillates with distance from the mirror, vanishes at $x=0$4, and has its first antinode at $x=0$5 (Bennett, 2020).

The inverse-design contribution uses adjoint methods based on source-observer symmetry to maximize $x=0$6 directly. The implementation uses FDTD in Meep, dielectric cubes of side length $x=0$7, permittivity $x=0$8, an optimization region three wavelengths wide and one block deep in $x=0$9, and 12 pixels per wavelength. Two geometry classes are considered, with and without a perfect-reflector backplate. The best-performing case, perpendicular dipole rotation with a backplate, produces approximately double the coherence of the infinitely extended perfect reflector benchmark, and the paper’s abstract summarizes the overall outcome as approximately double the coherence obtained from simple planar geometries (Bennett, 2020).

4. Conditional and heralded orthogonalization of quantum states

One line of work studies how nonorthogonal states can be converted into orthogonal ones without sacrificing the output state. For coherent states, the starting pair is

$$\nu(t)=\mathrm{Tr}\!\left[\hat U'(t)\,\hat\rho(0)\,\hat U(-t)\right],$$0

whose overlap is

$$\nu(t)=\mathrm{Tr}\!\left[\hat U'(t)\,\hat\rho(0)\,\hat U(-t)\right],$$1

The protocol in "Heralded orthogonalisation of coherent states and their conversion to discrete-variable superpositions" uses quantum catalysis or photon replacement: the input coherent state interferes with an ancilla single photon on a beam splitter of transmissivity $$\nu(t)=\mathrm{Tr}\!\left[\hat U'(t)\,\hat\rho(0)\,\hat U(-t)\right],$$2, and success is heralded by detecting one photon in an auxiliary output mode (Kruse et al., 2017).

Conditioned on success, the outputs satisfy an orthogonality constraint

$$\nu(t)=\mathrm{Tr}\!\left[\hat U'(t)\,\hat\rho(0)\,\hat U(-t)\right],$$3

For every $$\nu(t)=\mathrm{Tr}\!\left[\hat U'(t)\,\hat\rho(0)\,\hat U(-t)\right],$$4 there is one physical solution $$\nu(t)=\mathrm{Tr}\!\left[\hat U'(t)\,\hat\rho(0)\,\hat U(-t)\right],$$5. At $$\nu(t)=\mathrm{Tr}\!\left[\hat U'(t)\,\hat\rho(0)\,\hat U(-t)\right],$$6, the paper reports $$\nu(t)=\mathrm{Tr}\!\left[\hat U'(t)\,\hat\rho(0)\,\hat U(-t)\right],$$7, and the successful outputs have fidelity $$\nu(t)=\mathrm{Tr}\!\left[\hat U'(t)\,\hat\rho(0)\,\hat U(-t)\right],$$8 with

$$\nu(t)=\mathrm{Tr}\!\left[\hat U'(t)\,\hat\rho(0)\,\hat U(-t)\right],$$9

respectively. Even and odd cat-like superpositions are mapped, again conditionally, to highly recognizable discrete-variable states: the even combination has $\hat U(t)=e^{-i\hat H t}$0 fidelity with a $\hat U(t)=e^{-i\hat H t}$1 squeezed vacuum, and the odd combination has $\hat U(t)=e^{-i\hat H t}$2 fidelity with $\hat U(t)=e^{-i\hat H t}$3 (Kruse et al., 2017). The crucial point is that success is heralded without measuring away the logical output, so the operation is a state-conversion primitive rather than a terminal discriminator.

A different conditional route appears in "Orthogonalization of partly unknown quantum states". Here the state is not completely unknown: one knows the expectation value

$\hat U(t)=e^{-i\hat H t}$4

of a single operator $\hat U(t)=e^{-i\hat H t}$5. Then

$\hat U(t)=e^{-i\hat H t}$6

is orthogonal to $\hat U(t)=e^{-i\hat H t}$7, because

$\hat U(t)=e^{-i\hat H t}$8

The implementation is probabilistic, with success probability

$\hat U(t)=e^{-i\hat H t}$9

where $\hat U'(t)=e^{-i\hat H' t}$0 is the largest singular value of $\hat U'(t)=e^{-i\hat H' t}$1 (Jezek et al., 2014).

For qubits, the paper takes $\hat U'(t)=e^{-i\hat H' t}$2 and a state of known latitude,

$\hat U'(t)=e^{-i\hat H' t}$3

The implemented filter is

$\hat U'(t)=e^{-i\hat H' t}$4

with success probability

$\hat U'(t)=e^{-i\hat H' t}$5

At the equator, $\hat U'(t)=e^{-i\hat H' t}$6, so orthogonalization is deterministic; near an eigenstate of $\hat U'(t)=e^{-i\hat H' t}$7, the success probability becomes small (Jezek et al., 2014).

Experimentally, the single-qubit input purity exceeded $\hat U'(t)=e^{-i\hat H' t}$8, the orthogonalized output purity was at least $\hat U'(t)=e^{-i\hat H' t}$9, and the input-output overlap was always below $|\nu(t)|$0. The same principle extends to entangled states: $|\nu(t)|$1 so perfect orthogonalization of partly unknown two-qubit entangled states can be performed by filtering only one subsystem. The reported two-qubit overlaps remained low, and the entanglement of formation was largely preserved (Jezek et al., 2014).

5. Symmetric orthogonalization, superposition, and coherence measures

In resource-theoretic work, the central distinction is between coherence relative to an orthonormal basis and superposition relative to a nonorthogonal one. "Coherence, superposition, and Löwdin symmetric orthogonalization" formulates this explicitly. If $|\nu(t)|$2 is a linearly independent nonorthogonal basis with Gram matrix

$|\nu(t)|$3

Löwdin symmetric orthogonalization defines an orthonormal basis through

$|\nu(t)|$4

Its defining property is minimal distortion: $|\nu(t)|$5 Because the transformation is invertible, the same physical state can be represented as a superposition state in $|\nu(t)|$6 and as a coherent state in the Löwdin basis $|\nu(t)|$7. The paper’s main claim is that maximally coherent states turn into states with maximal superposition under this symmetric orthogonalization (Torun, 2022).

For qubits with overlap $|\nu(t)|$8, the maximally coherent states $|\nu(t)|$9 map to the golden superposition states

$$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$00

with the relevant branch depending on the sign range of $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$01. In symmetric $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$02-dimensional overlap models, the equal-amplitude maximally coherent state maps to the equal-amplitude maximal superposition state with overlap-dependent normalization (Torun, 2022).

"Symmetric orthogonalization and probabilistic weights in resource quantification" develops this further. It contrasts Löwdin symmetric orthogonalization with Gram-Schmidt orthogonalization, emphasizing that Gram-Schmidt is order dependent while Löwdin treats all vectors equally and minimizes deviation from the original basis. If a pure state is written

$$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$03

then the orthogonalized amplitudes are

$$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$04

and the Löwdin weights are

$$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$05

For mixed states,

$$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$06

These weights define entropy and participation-ratio diagnostics,

$$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$07

and provide a basis-consistent probabilistic reading of nonorthogonal expansions (Torun, 18 Aug 2025).

A related extension applies coherence theory directly to coherent states. "Quantifying the Coherence Between Coherent States" introduces a state-dependent orthogonalization map based on a Gram-Schmidt unitary that labels selected coherent states with orthogonal ancilla branches. This yields an $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$08-dimensional orthogonalized subspace on which ordinary finite-dimensional coherence monotones can be evaluated, and defines $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$09-coherence as the corresponding asymptotic limit. The decisive theorem is that

$$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$10

where classicality is defined by the Glauber-Sudarshan $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$11 representation. The paper thus identifies orthogonalization as the bridge by which nonclassicality of light becomes expressible as coherence in an enlarged orthogonal basis (Tan et al., 2017).

6. Algorithmic and numerical analogues

In machine learning and numerical linear algebra, orthogonalization dynamics reappear in matrix-structured form. Muon uses operator-norm geometry, with update

$$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$12

"MuonBP: Faster Muon via Block-Periodic Orthogonalization" replaces full orthogonalization at every step by mostly local shardwise orthogonalization plus a global step every $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$13 iterations. With $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$14, the paper reports about a $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$15 reduction in optimizer-step communication volume, and on an 8B model with eight-way tensor parallelism and ZeRO optimizer state sharding it reports an 8% throughput increase compared to Muon with no degradation in performance (Khaled et al., 19 Oct 2025). The same paper explicitly states that it does not define or analyze “coherence” as a formal quantity; a plausible interpretation is that periodic full orthogonalization restores cross-shard alignment that would otherwise be lost under purely local blockwise orthogonalization.

"Preconditioning Benefits of Spectral Orthogonalization in Muon" makes the spectral mechanism explicit. Simplified Muon uses

$$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$16

for $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$17. The paper proves linear convergence with iteration complexity independent of the relevant condition number in two case studies, and shows that the dynamics decouple into independent scalar recursions in the spectral domain (Ma et al., 20 Jan 2026). The authors explicitly note that they do not provide a formal coherence theory; the technically justified statement is that spectral orthogonalization preserves singular directions while flattening singular values, thereby reducing harmful cross-mode coupling.

Structured orthogonalization also remains central in numerical linear algebra. "On Two-Stage Householder Orthogonalization" considers the problem of orthogonalizing a block $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$18 against a matrix $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$19 with orthonormal columns. Instead of reorthogonalizing the full concatenated matrix $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$20, the proposed algorithm uses a generalized Householder transformation and only needs to orthogonalize a square submatrix of $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$21. The paper proves unconditional stability and shows that, unlike block Gram-Schmidt, the method does not inherit instability from the condition number of $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$22 (He et al., 16 Feb 2026).

"Randomized orthogonalization and Krylov subspace methods: principles and algorithms" changes the orthogonalization metric itself. Rather than constructing $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$23 with $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$24, it constructs a basis whose sketch is orthonormal: $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$25 If $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$26 is an $$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$27-embedding, then

$$\hat H = -\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}\omega^2 \hat x^2,$$28

The paper argues that randomized orthogonalization can reduce cost and communication while preserving, and in some cases improving, numerical stability, and uses this idea to derive randomized Arnoldi relations and sketched Krylov methods (Damas et al., 17 Dec 2025). Here again, any language of coherence is interpretive rather than formal. A plausible implication is that randomized sketching suppresses the dominance of a few problematic directions by enforcing orthogonality in a compressed geometry that approximately preserves the active subspace.

Taken together, these works suggest that coherence-enhanced orthogonalization dynamics is best understood as a recurring technical principle rather than a single standardized theory. In quench physics, initial coherence changes orthogonalization rates and work statistics; in nanophotonics, anisotropic reservoirs alter the dynamical orthogonality of nominally orthogonal transitions; in quantum information, conditional filters and heralded operations turn nonorthogonal inputs into usable orthogonal outputs; in resource theory, symmetric orthogonalization exposes a precise bridge between coherence and superposition; and in matrix algorithms, orthogonalization frequency, spectral equalization, or sketch-based metrics reshape conditioning and stability. The common limitation is equally clear: these effects are highly structure dependent. They typically rely on specific phase patterns, specific prior information, specific geometry, specific basis choices, or specific embedding quality, and they are therefore powerful precisely where orthogonality is a designed dynamical property rather than a fixed background notion.