Random Orbital Rotations

• Random orbital rotations are defined as various transformations acting on mode or orbital degrees of freedom, with differing mathematical and physical implementations.
• In optics and inverse problems, they are modeled as random draws or latent group actions to enhance precision in rotation sensing and orbit recovery.
• In electronic structure, orbital rotations refer to deterministic basis optimizations or disorder-induced mixing, highlighting non-invariance and symmetry breaking.

Searching arXiv for papers on random rotations, orbital angular momentum, orbit recovery, and orbital rotations to ground the article in current literature. Random orbital rotations denote several distinct constructions in contemporary research, unified by the appearance of rotations acting on orbital, mode, or basis degrees of freedom but differing sharply in mathematical meaning and physical implementation. In optical metrology, the term can refer to an unknown axis-angle rotation $$R(\boldsymbol{\Omega})=e^{i\omega \mathbf{J}\cdot \mathbf{u}}$$ acting on a finite-dimensional orbital-angular-momentum or spatial-mode encoding of a spin-$J$ system, with $\boldsymbol{\Omega}=(\omega,\Theta,\Phi)$ treated as an arbitrary or random draw of rotation parameters (Eriksson et al., 2023). In harmonic analysis and inverse problems, it refers to latent random elements of $SO(3)$ acting on spherical or volumetric signals, where only the orbit under the rotation group is identifiable (Bendory et al., 24 Feb 2026, Bendory et al., 30 Jun 2025). In geometric sampling, random rotations are used to generate equidistributed nets on spheres or randomized low-energy point sets on $SO(3)$ itself (Chakraborty et al., 2018, Beltrán et al., 16 Jun 2025). In electronic-structure theory and correlated oxides, by contrast, orbital rotations usually mean deterministic orbital-basis optimization or disorder-induced local orbital mixing rather than stochastic sampling (Chakraborty et al., 2024, Moreno et al., 2023, Horsch et al., 2020).

1. Terminological scope and conceptual distinctions

The phrase combines at least three non-equivalent notions of “orbital” and “rotation.” One is the literal orbital or azimuthal degree of freedom of light, where spatial modes carrying orbital angular momentum are rotated in angle or transformed within a finite mode basis (Eriksson et al., 2023, Magana-Loaiza et al., 2013). A second is group-theoretic rotation in $SO(3)$ or $SU(2)$, where an unknown rotation acts on spherical harmonics, Wigner $D$-matrices, or other irreducible representations (Bendory et al., 24 Feb 2026, Bendory et al., 30 Jun 2025). A third is orbital-basis rotation in many-electron theory, where single-particle orbitals are changed by a basis transformation and observables become basis-sensitive when the ansatz is not invariant under those transformations (Chakraborty et al., 2024, Moreno et al., 2023).

A further distinction concerns whether the rotation is physically realized, statistically latent, or variationally chosen. In the multiplane-light-conversion experiment, the rotation is an implemented finite-dimensional unitary on a selected spatial-mode basis rather than a literal rigid-body rotation of the beam in real space (Eriksson et al., 2023). In orbit recovery, the rotations are unobserved nuisance variables drawn from an unknown distribution on $SO(3)$ (Bendory et al., 24 Feb 2026). In pCCD and related methods, the orbital rotations are deterministically optimized and are explicitly described as purposeful and variationally motivated rather than random (Chakraborty et al., 2024).

This distinction resolves a common ambiguity. “Random orbital rotations” in optics and inverse problems usually means random draws of rotation parameters or random latent orientations. In quantum chemistry, the same phrase would be misleading unless one explicitly means arbitrary or uncontrolled basis changes, because the central methodological object is ordinarily orbital optimization rather than stochastic sampling (Chakraborty et al., 2024, Moreno et al., 2023).

2. Random rotations in orbital-angular-momentum and spatial-mode optics

A particularly explicit formulation appears in rotation sensing with multiplane light conversion. There the probe lives in a $(2J+1)$-dimensional Hilbert space

$J$0

embedded experimentally into transverse spatial modes of light, specifically Laguerre–Gauss or OAM modes (Eriksson et al., 2023). The unknown transformation is a general three-parameter rotation, represented by the corresponding $J$1 unitary,

$J$2

with axis-angle parameters $J$3 and

$J$4

Within that framework, a random orbital rotation is naturally interpreted as a random draw of $J$5 acting on the selected finite OAM-mode subspace (Eriksson et al., 2023).

The metrological treatment is genuinely multiparameter. For pure states under unitary evolution, the quantum Fisher information matrix is written as

$J$6

where $J$7 is the symmetrized covariance matrix of the angular-momentum generators and $J$8 separates parameter dependence from state dependence (Eriksson et al., 2023). Choosing $J$9, the paper gives the intrinsic bound

$\boldsymbol{\Omega}=(\omega,\Theta,\Phi)$0

with the optimum achieved for isotropic states satisfying

$\boldsymbol{\Omega}=(\omega,\Theta,\Phi)$1

The optimal probes are the Kings of Quantumness, or anticoherent states, which are first-order unpolarized and second-order isotropic (Eriksson et al., 2023).

Experimentally, the crucial point is that MPLC realizes arbitrary unitary transformations on a finite set of spatial modes. The “rotation” is therefore an abstract mode-space unitary matched to the spin-$\boldsymbol{\Omega}=(\omega,\Theta,\Phi)$2 rotation operator, not simply a real-space turning of the beam profile (Eriksson et al., 2023). The practical measurement is based on five spin-coherent-state projections, interpreted as Husimi-$\boldsymbol{\Omega}=(\omega,\Theta,\Phi)$3-function samples, followed by maximum-likelihood inference. For full three-parameter estimation of 37 rotations, the reported average deviations between true and estimated rotations were

$\boldsymbol{\Omega}=(\omega,\Theta,\Phi)$4

which were stated to be much smaller than the average deviation $\boldsymbol{\Omega}=(\omega,\Theta,\Phi)$5 expected for a completely random guess (Eriksson et al., 2023). The paper does not claim exact saturation of the quantum Cramér–Rao bound; rather, the achieved precision is reported as within about a factor of $\boldsymbol{\Omega}=(\omega,\Theta,\Phi)$6 of the ultimate limit (Eriksson et al., 2023).

A related but distinct optical use concerns small azimuthal rotations in weak-value metrology. There the estimated parameter is a small relative orbital rotation angle $\boldsymbol{\Omega}=(\omega,\Theta,\Phi)$7, generated by the OAM operator $\boldsymbol{\Omega}=(\omega,\Theta,\Phi)$8 through a spin-orbit coupling

$\boldsymbol{\Omega}=(\omega,\Theta,\Phi)$9

and read out either as an angular shift

$SO(3)$0

or as a centroid shift in the conjugate OAM basis

$SO(3)$1

That paper treats primarily small deterministic rotations, but explicitly notes that the formalism is naturally interpreted as a local estimator for unknown, shot-to-shot varying, or fluctuating rotations in the weak regime (Magana-Loaiza et al., 2013). This suggests a local linear-response meaning of random orbital rotations in the azimuthal coordinate rather than in finite-dimensional mode space.

3. Random rotations as latent nuisance variables in $SO(3)$2 orbit recovery

In inverse problems on spherical and volumetric signals, random orbital rotations are formalized as unknown group actions. One model observes

$SO(3)$3

where $SO(3)$4 are random elements drawn from a distribution $SO(3)$5 on $SO(3)$6, $SO(3)$7 is an unknown band-limited three-dimensional signal, and $SO(3)$8 is Gaussian noise (Bendory et al., 24 Feb 2026). Because the rotations are unobserved, the recoverable object is the orbit

$SO(3)$9

not the absolute orientation of $SO(3)$0 (Bendory et al., 24 Feb 2026).

The central representation-theoretic structure uses spherical harmonics for $SO(3)$1, Wigner $SO(3)$2-functions for $SO(3)$3, and Clebsch–Gordan decompositions for tensor products (Bendory et al., 24 Feb 2026). The signal is expanded as

$SO(3)$4

while the unknown rotation distribution is parameterized by Fourier coefficients

$SO(3)$5

The first and second moments are then

$SO(3)$6

The notable result is that when the rotation distribution is non-uniform and its Fourier coefficients are invertible up to bandlimit $SO(3)$7, the first and second moments determine the pair $SO(3)$8, up to a global rotation, for generic signals with $SO(3)$9 radial shells (Bendory et al., 24 Feb 2026). The paper states that this improves the high-noise sample-complexity scaling from the uniform-rotation $SO(3)$0 regime to a $SO(3)$1 regime, because identifiability drops from third to second moment (Bendory et al., 24 Feb 2026).

A closely related invariant-theoretic perspective studies generic orbit recovery under rotational group actions using degree-three invariants, or the bispectrum (Bendory et al., 30 Jun 2025). For functions on the sphere and finite-dimensional shell discretizations,

$SO(3)$2

the paper proves that generic orbits are determined by invariants of degree at most three once the number of radial shells is large enough (Bendory et al., 30 Jun 2025). In the $SO(3)$3 case the result is especially sharp: if $SO(3)$4, then degree-$SO(3)$5 invariants separate generic $SO(3)$6-orbits in $SO(3)$7, independently of bandlimit $SO(3)$8 (Bendory et al., 30 Jun 2025). The bispectrum is written explicitly in terms of Clebsch–Gordan coefficients,

$SO(3)$9

and supports a frequency-marching reconstruction by successive linear solves (Bendory et al., 30 Jun 2025).

These two orbit-recovery frameworks use random rotations in different statistical regimes. One emphasizes non-uniform latent orientation distributions and low-order moment identifiability (Bendory et al., 24 Feb 2026); the other emphasizes invariant separation under generic rotations and the sufficiency of cubic invariants (Bendory et al., 30 Jun 2025). A plausible implication is that “random orbital rotations” in high-dimensional inverse problems is best understood through the harmonic analysis of the group action rather than through coordinate-angle heuristics.

4. Randomized constructions of rotation sets and sphere coverings

A different line of work studies random rotations not as latent nuisances but as design primitives for well-distributed point sets. On the sphere $SU(2)$0, one can sample a small number of Haar-random rotations in $SU(2)$1, form all words of a prescribed length in those rotations and their inverses, and act them on a fixed point $SU(2)$2 (Chakraborty et al., 2018). The resulting orbit-like set

$SU(2)$3

is shown, with high probability, to be both an $SU(2)$4-net and equidistributed at scale $SU(2)$5 (Chakraborty et al., 2018).

The construction is governed by two parameters. The paper’s abstract states that one picks

$SU(2)$6

random rotations and takes all possible words of length

$SU(2)$7

in the same alphabet (Chakraborty et al., 2018). The theorem in the body gives an explicit condition

$SU(2)$8

with

$SU(2)$9

and a word length of the form

$D$0

for the Wasserstein equidistribution theorem (Chakraborty et al., 2018). Equidistribution is measured by the $D$1-Wasserstein distance,

$D$2

where $D$3 is the uniform counting measure on the finite orbit set and $D$4 is uniform surface measure on $D$5 (Chakraborty et al., 2018). By Kantorovich–Rubinstein duality, this controls integration error for all $D$6-Lipschitz functions.

The mechanism combines spherical harmonic decomposition, averaging operators associated with the sampled rotations, concentration of the empirical operator around its Haar expectation, and heat-kernel smoothing (Chakraborty et al., 2018). This suggests that random orbital rotations can generate structured low-randomness surrogates for direct random sampling, while still producing dense and statistically uniform configurations.

A related but intrinsically group-level construction concerns point sets on $D$7 itself. The logarithmic energy of rotations $D$8 is defined by

$D$9

and equivalently

$SO(3)$0

using the Frobenius norm (Beltrán et al., 16 Jun 2025). The best randomized construction analyzed in that paper chooses $SO(3)$1 points on $SO(3)$2 from zeros of a random degree-$SO(3)$3 polynomial and places $SO(3)$4 equally spaced rotations on the $SO(3)$5 fiber over each point, giving $SO(3)$6 rotations (Beltrán et al., 16 Jun 2025). The resulting expected energy is

$SO(3)$7

with

$SO(3)$8

for an infinite subsequence of $SO(3)$9 (Beltrán et al., 16 Jun 2025). This is explicitly described as the best-performing randomized construction in the paper.

These results treat random rotations as a means of regularization rather than pure randomness. They are not ordinary Haar-i.i.d. samplers; instead, they generate random but anti-clustered ensembles. A plausible implication is that many applications calling for “random rotations” in fact require controlled coverage or repulsion rather than independence.

5. Statistical and dynamical behavior under repeated random rotations

Repeated random rotations also define nontrivial stochastic and dynamical processes. On the circle, one model chooses at each step either the rotation $(2J+1)$0 or $(2J+1)$1, each with probability $(2J+1)$2, producing the Markov chain

$(2J+1)$3

on $(2J+1)$4 (Czudek, 2022). For an observable $(2J+1)$5, one studies the additive functional

$(2J+1)$6

and asks whether a central limit theorem holds. The transition operator

$(2J+1)$7

diagonalizes on Fourier modes $(2J+1)$8 with eigenvalues $(2J+1)$9 (Czudek, 2022). The paper proves that if $J$00 is Diophantine of type $J$01 and $J$02 with $J$03, then the normalized sums satisfy a CLT (Czudek, 2022). It also proves the opposite extreme: for every Liouville angle there exists a smooth observable $J$04 such that the CLT fails, and there exists a Liouville angle with an analytic observable such that the CLT fails (Czudek, 2022). The point is that arithmetic resonance survives randomization; the process is ergodic and reversible, yet additive statistics can remain non-Gaussian.

In celestial mechanics, irregular rotation can arise without stochasticity at all. For a rigid triaxial ellipsoid in a Kepler orbit, the full 3D Euler equations in the body frame are

$J$05

and generate deterministic chaotic tumbling under time-varying gravitational torque (Makarov et al., 2022). The paper reports that the main $J$06 spin-orbit resonance disappears for specific moderately prolate shapes already at eccentricities as low as $J$07, and that the island of short-term stability around the main $J$08 resonance completely vanishes at approximately $J$09 (Makarov et al., 2022). Near the $J$10 resonance, trajectories become chaotic at smaller eccentricities, but separated enclaves of orderly rotation emerge at eccentricities as high as $J$11 (Makarov et al., 2022). This is not a stochastic model, but it is directly relevant to a misconception sometimes attached to the phrase “random rotations”: apparent randomness can be either genuinely random, as in the $J$12 chain, or purely deterministic chaos, as in triaxial attitude dynamics.

An additional caution appears in the study of Trotterized orbital rotations on $J$13. There the target unitary $J$14 is approximated by

$J$15

and the state-dependent error

$J$16

is shown to scale as $J$17 for regular states $J$18, but convergence can be arbitrarily slow for states that do not lie in the domains of all three orbital angular momentum operators simultaneously (Facchi et al., 21 Jul 2025). This establishes that approximating a desired orbital rotation by composing small rotations can be sharply state-dependent.

6. Orbital rotations in electronic structure: deterministic optimization, disorder, and non-invariance

In electronic-structure theory, orbital rotations usually do not mean random rotations of physical space. They mean changes of the single-particle basis. The distinction is explicit in the pCCD dipole-moment study, which states that the paper does not discuss random orbital rotations explicitly, and that there is no stochastic sampling of orbital rotations or analysis of arbitrary or randomly chosen rotated orbital bases (Chakraborty et al., 2024). What it studies instead is the effect of deterministic orbital optimization within pCCD, contrasted against canonical Hartree–Fock orbitals, on dipole moments, density matrices, and the quality of linearized coupled-cluster corrections (Chakraborty et al., 2024). The motivation is that pCCD is strongly orbital-dependent because it retains only pair excitations; changing the orbital basis changes which electron pairs are considered paired (Chakraborty et al., 2024).

A related variational framework in neural quantum states and hardware-efficient VQE makes the single-particle basis itself variational through a Gaussian fermionic unitary

$J$19

with the rotated Hamiltonian

$J$20

and one-particle rotation matrix

$J$21

acting on creation operators as

$J$22

The paper argues that many ansätze are not invariant under such single-particle basis transformations, so their expressive power depends strongly on basis choice (Moreno et al., 2023). It reports that joint optimization of orbital rotations and variational parameters improves both expressivity and the optimization landscape across chemistry and condensed-matter examples, including the first active-space calculation using neural quantum states with basis transformations applied to all orbitals in the basis set (Moreno et al., 2023). The paper does study 128 random initializations of circuit parameters for HF, but it does not study random orbital rotations directly (Moreno et al., 2023).

In correlated oxides, orbital rotations may instead be induced by disorder fields. In doped vanadates, charged Ca defects and doped holes create local Coulomb fields that mix the $J$23 orbitals

$J$24

and suppress orbital order (Horsch et al., 2020). The defect-induced orbital-polarization term is

$J$25

which is explicitly off-diagonal in orbital space and therefore constitutes literal local orbital mixing (Horsch et al., 2020). The paper’s main conclusion is that the gradual decline of orbital order with doping has its origin not predominantly in the charge carriers, but in the off-diagonal couplings of orbital rotations induced by the charges of the doped ions (Horsch et al., 2020). Because defect positions are random substitutional positions and results are disorder-averaged over many realizations, this is an instance where orbital rotations are effectively random in space even though they are not stochastic variables inserted by hand.

The common lesson across these electronic-structure examples is that orbital rotations are usually purposeful, variationally optimized, or disorder-induced. Calling them random without qualification obscures the central issue, which is non-invariance of approximate ansätze or local symmetry breaking by defects rather than stochastic group action (Chakraborty et al., 2024, Moreno et al., 2023, Horsch et al., 2020).