Rotationally Invariant Constellations

• Rotationally invariant constellations are signal sets whose properties remain unchanged under specified rotations, ensuring robustness against channel impairments.
• Optimization techniques, including geodesic flow and local diversity metrics, are employed to enhance mutual information and lower error probability.
• Practical implementations in wireless, fiber-optic, and quantum systems demonstrate performance gains, improved carrier phase recovery, and alignment-free operations.

Rotationally invariant constellations are structured signal sets for communications and quantum systems whose statistical or geometric properties remain unchanged under a group of rotations in the relevant signal space. These constructions facilitate robustness to channel impairments, enable frame-independent transmission or measurement, and naturally arise in settings where the noise or channel model itself exhibits rotational invariance. This article covers core mathematical definitions, design methodologies, analysis of achievable rates, implications for physical-layer optimization, and quantum-optical analogues.

1. Mathematical Foundations and Symmetry

A finite or continuous constellation $\mathcal{X} \subset \mathbb{R}^n$ (or $\mathbb{C}^n$, as appropriate) is said to be rotationally invariant if either (i) its distribution is statistically invariant under the action of the special orthogonal group $SO(n)$ (or a subgroup), or (ii) its set structure is permuted onto itself under a prescribed group of rotations.

In the discrete signal processing literature, an important subclass is the $2\pi/M$-rotationally symmetric (M-fold invariant) complex constellations. For $C = \{c_k\}_{k=0}^{L-1} \subset \mathbb{C}$,

$$\forall m \in \{0,\dots,M-1\},\quad \{c\,e^{j2\pi m/M}: c \in C\} = C.$$

Examples include QPSK (M=4), 8-PSK (M=8), and certain nonuniform QAM variants such as ITU's V.29 (Slimane et al., 2012).

For more general multidimensional real constellations, e.g., in Rayleigh fading or fiber-optic scenarios, rotational invariance is defined with respect to $SO(n)$ acting on $\mathbb{R}^n$. For Majorana constellations, the invariance is under $SU(2)$ Möbius transformations of the Riemann sphere (Torres-Leal et al., 2024).

2. Construction and Optimization Techniques

Numerical Optimization on Lie Groups

High-dimensional rotation-optimized constellations are constructed by formulating an objective such as the tight upper bound on pairwise error probability (PEP) or the mutual information, then minimizing (or maximizing) it over $SO(n)$:

• For a finite $\mathbb{C}^n$0 and SNR, the objective

$\mathbb{C}^n$1

is minimized subject to $\mathbb{C}^n$2, $\mathbb{C}^n$3, $\mathbb{C}^n$4 (Karpuk et al., 2013).

Gradient flows are implemented on the group manifold, with the tangent direction at $\mathbb{C}^n$5 given by

$\mathbb{C}^n$6

followed by geodesic updates $\mathbb{C}^n$7 for step-size $\mathbb{C}^n$8.

Parametric Families and Local Diversity

Analysis in (Karpuk et al., 2015) constructs a recursively defined one-parameter subgroup $\mathbb{C}^n$9, with $SO(n)$0 a specific skew-symmetric generator, such that $SO(n)$1 is optimized for cutoff rate or local error properties. Surprisingly, maximization of finite-SNR performance—especially at low-to-moderate SNR—is better predicted by "local diversity" metrics and local minimum product distance $SO(n)$2 (within radius $SO(n)$3) than the classical (global) diversity criterion. The optimal rotation angle $SO(n)$4 can be found in closed form at low SNR.

Majorana Constellations in Quantum Optics

In the quantum and structured-light context, the Majorana representation associates each spin-$SO(n)$5 state $SO(n)$6 with a set of "stars" on the Riemann sphere, via roots $SO(n)$7 of a degree-$SO(n)$8 Majorana polynomial. Rotationally invariant (isotropic) states—sometimes termed "kings of quantumness"—are those with stars at vertices of Platonic solids, satisfying

$SO(n)$9

used to guarantee rotational invariance under $2\pi/M$0 (Torres-Leal et al., 2024).

3. Information-Theoretic Properties and Mutual Information

In channels where both noise and signal sets are rotationally invariant, the mutual information analysis reduces to the statistics of the induced norm (radius) (Karout et al., 2016). For multisphere constellations (uniform over concentric shells of radii $2\pi/M$1, occupation probabilities $2\pi/M$2),

$2\pi/M$3

and the mutual information can be expressed in terms of the PDF of $2\pi/M$4.

The pre-log of achievable rate $2\pi/M$5 at high SNR for $2\pi/M$6-dimensional $2\pi/M$7-shell multisphere constellations is $2\pi/M$8, in contrast to $2\pi/M$9 for unconstrained Gaussian signaling. Notably, in 4D dual-polarization optical systems, a single 4D multisphere yields asymptotic pre-log $C = \{c_k\}_{k=0}^{L-1} \subset \mathbb{C}$0, outperforming two independent 2D rings (pre-log $C = \{c_k\}_{k=0}^{L-1} \subset \mathbb{C}$1) (Karout et al., 2016).

4. Practical Applications and Performance Gains

Coherent Communication and Wireless Channels

Rotation-optimized constellations are crucial for fading channels and coded modulation with bit-interleaved coded modulation (BICM). Empirical findings include:

• 2D 16-NUQAM (non-uniform QAM) with optimized rotation exhibits a $C = \{c_k\}_{k=0}^{L-1} \subset \mathbb{C}$20.3 dB coding gain at CER $C = \{c_k\}_{k=0}^{L-1} \subset \mathbb{C}$3, and up to 0.1 bit/s/Hz BICM capacity improvement (Karpuk et al., 2013).
• 4D and 8D rotations—either via geodesic flow or parametric subgroups—outperform prior algebraic rotations, often with additional gains at finite SNR, even if global diversity is not maximized (Karpuk et al., 2015).
• Local diversity and local minimum product distance provide a finer criterion for optimization at finite SNR than global criteria.

Fiber-Optic Systems and Polarization-Multiplexing

Multisphere/rotationally invariant constellations leverage channel symmetry in coherent optical systems, providing robustness to polarization rotations and nonlinear effects. 4D rotational invariance enables channels to be "shell" indexed, reducing receiver adaptation complexity, and improves rate versus parallel 2D designs (Karout et al., 2016).

Quantum and Optical Reference-Frame Alignment

In quantum optics, rotationally invariant Majorana constellations underpin reference-frame alignment, Heisenberg-limited rotation sensing, and alignment-free QKD. Physical implementations involve scalar beams (Laguerre–Gaussian), vector beams, and hybrid "kings of quantumness" states, all mapped to Platonic solid star arrangements for isotropy (Torres-Leal et al., 2024).

5. Symmetry and Carrier Phase Recovery

For phase-modulated communications, rotational invariance under finite subgroups (e.g., $C = \{c_k\}_{k=0}^{L-1} \subset \mathbb{C}$4) informs blind phase recovery algorithms. The $C = \{c_k\}_{k=0}^{L-1} \subset \mathbb{C}$5th-order Phase Metric Method (PMM) exploits symmetry to provide unbiased, near-MCRB carrier phase estimates: $C = \{c_k\}_{k=0}^{L-1} \subset \mathbb{C}$6 with variance $C = \{c_k\}_{k=0}^{L-1} \subset \mathbb{C}$7 at high SNR, and superior mean-square error to traditional methods for $C = \{c_k\}_{k=0}^{L-1} \subset \mathbb{C}$8-PSK and symmetric QAM (Slimane et al., 2012).

6. Algorithmic Summaries and Adoption Guidelines

Construction Domain	Key Principles	Recommended Algorithms
Multidimensional QAM	Optimize cutoff rate / PEP via SO(n)	Geodesic-flow descent, 1-parameter subgroup search (Karpuk et al., 2013, Karpuk et al., 2015)
Multisphere Constellations	Exploit radial symmetry for mutual info	Shell optimization with Blahut–Arimoto; uniform angular codebooks (Karout et al., 2016)
Majorana Constellations	Platonic solid star placement in S²	Orthogonal polynomial root placement, Vieta’s formula (Torres-Leal et al., 2024)

Optimization should target local diversity and mode-packing for finite SNR, especially in applications sensitive to nearest-neighbor error patterns rather than asymptotic error rates. At high SNR, increasing the number of shells/points and matching Gaussian radial distributions further closes the gap to channel capacity (Karout et al., 2016).

7. Extensions and Emerging Research Directions

Ongoing research addresses:

• Generalization of local diversity to broader channel models, as classical full diversity may be suboptimal for modern coding/interleaving schemes (Karpuk et al., 2015).
• Application of rotationally invariant methods to lattice-based and non-uniform constellations beyond QAM.
• Exotic structured-light states, such as nonclassical and "cat" codes, realized as rotationally invariant Majorana constellations for quantum communication (Torres-Leal et al., 2024).
• Computational complexity reduction by exploiting angular invariance and radial decoupling in multidimensional decoding.

By integrating geometric, group-theoretic, and information-theoretic insights, rotationally invariant constellations provide a unifying framework for robust, symmetry-exploiting communication and metrology in both classical and quantum technologies.