Majorana-Stellar Representation

• Majorana-Stellar Representation is a geometric framework that maps quantum states to constellations of stars on the Bloch sphere, defining state geometry and symmetries.
• It employs a polynomial construction where the stereographically projected roots expose key quantum properties like entanglement, Berry phases, and topological invariants.
• The representation extends to mixed states and multipartite systems, offering clear geometric insights into quantum optics, information processing, and phase transitions.

The Majorana-Stellar Representation (MSR) provides a bijective geometric mapping between pure (and in generalized frameworks, mixed) finite-dimensional quantum states and constellations of points—termed "Majorana stars"—on the Bloch (or Riemann) sphere. Introduced by Ettore Majorana in 1932 for spin systems, MSR has evolved into a foundational apparatus across quantum information, geometric phase theory, quantum optics, the study of entanglement, and topological phases. The following sections survey the mathematical construction, transformational properties, applications, and recent generalizations of the MSR, with explicit formulas and conventions used throughout contemporary literature.

1. Mathematical Structure and Constructive Recipes

Let $|j,m\rangle$ denote the standard angular-momentum basis for a spin-$j$ system. An arbitrary pure state in this $(2j+1)$-dimensional Hilbert space is

$$|\psi\rangle = \sum_{m=-j}^{j} a_m |j, m\rangle, \qquad \sum_{m=-j}^j |a_m|^2 = 1.$$

Majorana's construction associates to $|\psi\rangle$ a degree-$2j$ polynomial: $$P(\zeta) = \sum_{m=-j}^{j} (-1)^{j-m} \sqrt{\binom{2j}{j + m}}\, a_m\, \zeta^{j + m}.$$ The $2j$ complex roots $\{\zeta_k\}$ of $P(\zeta)$ are stereographically projected to points on the unit sphere: $$\zeta_k = \tan\left(\frac{\theta_k}{2}\right) e^{i\phi_k}, \qquad \mathbf{n}_k = (\sin\theta_k \cos\phi_k,\, \sin\theta_k \sin\phi_k,\, \cos\theta_k).$$ The unordered set $\{\mathbf{n}_1, \dots, \mathbf{n}_{2j}\}$ fully determines the projective ray $[|\psi\rangle]$ up to normalization and global phase (Kam et al., 2020, Bruno, 2012). For general $d$-dimensional pure states, a similar polynomial of degree $d-1$ gives $d-1$ stars on the Bloch sphere (Ferraz et al., 2022, Liu et al., 2014).

2. Symmetry, Transformations, and Generalizations

SU(2) Covariance and Rigid Rotations

The MSR is SU(2)-covariant: a rotation of the quantum state induces a Möbius transformation of the roots, amounting to a rigid rotation of the Majorana constellation on the sphere. This property renders the representation uniquely suited to analyzing rotational symmetries and quantum phase transitions (Chryssomalakos et al., 2019, Roy et al., 3 Jul 2025).

Mixed States and Generalized Structures

For spin-$j$ mixed states, the density matrix $\rho$ is mapped to a set of polynomials for the multipole components: $$p^{(\sigma)}_\rho(\zeta) = \sum_{\mu = -\sigma}^\sigma (-1)^{\sigma-\mu} \sqrt{\binom{2\sigma}{\sigma - \mu}}\, \rho^{(\sigma)}_\mu\, \zeta^{\sigma + \mu}.$$ Each block of rank $\sigma$ yields $2\sigma$ stars (with Hermiticity enforcing antipodal pairing) and has a geometric “radius” set by the multipole norm. The transformation law under rotations continues to be the Möbius action (Serrano-Ensástiga et al., 2019, Serrano-Ensástiga et al., 2022).

For composite or multipartite symmetric states (e.g., $k$-fold symmetric qudits), the representation extends to so-called "multiconstellations," assigning principal constellations for each irreducible spin sector, potentially augmented by a "spectator" constellation that encodes relative complex weights (Chryssomalakos et al., 2021).

Grassmannians and Antisymmetric States

The MSR also admits a generalization to $k$-planes in Grassmannians, assigning $k(n-k)$ stars to a $k$-dimensional subspace of $\mathbb{C}^n$ via the Wronskian of vector-valued polynomials (Chryssomalakos et al., 2019).

3. Physical Applications: Berry Phases and Topology

Geometric Phase Decomposition

For cyclic evolution of a quantum state, the accumulated Berry phase decomposes into a sum involving the solid angles swept out by the individual Majorana stars' trajectories plus explicit pairwise correlation terms: $$\gamma_B = -\frac{1}{2} \sum_{k=1}^{2j} \Omega_k + \frac{1}{2} \sum_{i < j} \int \beta_{ij}\, \Omega(d\mathbf{u}_{ij}),$$ where $\Omega_k$ is the solid angle for star $k$ and the correlation terms $\beta_{ij}$ depend on the normalization structure (Liu et al., 2014, Kam et al., 2020, Bruno, 2012). In spin-$1/2$ or coherent states, only the solid angle term survives; higher spins require the additional internal-twist (self-rotation) contribution, which is related to multipartite entanglement (Kam et al., 2020).

Topological Invariants in Condensed Matter

In multiband Bloch systems, the translation from Bloch eigenstates to Majorana stars enables computation of topological invariants such as Zak or Berry phases by tracking azimuthal windings or trajectories of the stars in the Brillouin zone. Topologically trivial and nontrivial phases correspond to distinct winding patterns and parities of the star distributions at high-symmetry points (Yang et al., 2015, Xu et al., 2020).

4. Quantum Information: Entanglement, SICs, and MUBs

Entanglement Geometry

For symmetric $n$-qubit states, inter-star distances encode entanglement measures such as concurrence (for $n=2$) and the three-tangle (for $n=3$): $$\mathcal{C} \propto 1 - \mathbf{u}_1 \cdot \mathbf{u}_2, \qquad \tau_3 \propto \prod_{i < j} d_{ij}$$ where $d_{ij}$ are chordal distances between stars (Kam et al., 2019, Liu et al., 2014). For arbitrary (not necessarily symmetric) three-qubit states, a suitable sequence of local $SL(2,\mathbb{C})$ transformations brings the state into symmetric form, preserving the three-tangle and allowing geometric quantification via Majorana stars. This symmetrization procedure fails for generic four- or five-qubit states, for which no uniform Majorana representation of genuine entanglement exists.

SIC-POVMs and MUBs

In $d=3$ (spin-1), the MSR provides explicit geometric constructions of mutually unbiased bases (MUBs) and symmetric informationally complete POVMs (SIC-POVMs). For instance, the star configurations of spin-1 MUBs correspond to points positioned on equatorial double-cones or vertices of regular polyhedra, attesting to the utility of MSR in visualizing quantum measurement structure (Aravind, 2017). For higher $d$, analogous patterns relate to extremal geometries on the sphere.

5. Quantum Optics, Polarization, and Wigner-Negativity

Polarization of Classical and Quantum Fields

Polarization states of electromagnetic (spin-1) and gravitational (spin-2) waves are mapped to constellations on the sphere via their decomposition into spherical tensor components. For a coherent monochromatic EM wave, the roots of the associated polynomial correspond to the directions of vanishing probability in a detector's response, and observables including energy, momentum, and helicity densities can be expressed in terms of these stars (Bruno, 2019).

Majorana Constellations in Structured Optical Fields

In optical beams, the MSR describes the spatial and spin degrees of freedom. For Laguerre-Gaussian modes, the polynomial prescription assigns stars at the poles of the sphere, while vector (polarization ⊗ OAM) fields yield richer constellations with SU(2) manipulations controlling the spatial distribution of stars. Hermite-Gaussian beams, as SU(2)-rotated LG states, generate constellations on the equator, corresponding to linear polarization (Torres-Leal et al., 26 Aug 2024).

Wigner-Negativity and Nonclassicality

States maximizing the SU(2)-covariant Wigner negativity are found by optimizing the arrangement of the Majorana stars. For low spin, familiar symmetric constellations (e.g., tetrahedral, octahedral) appear, but as $j$ increases, the extremal configurations become irregular and non-polyhedral. Notably, all pure spin coherent states admit nonzero Wigner negativity, and extremal negative states are rare in Hilbert space, distinguishable by their unique stellar geometry (Davis et al., 2022).

6. Extensions: Mixed States, General Symmetries, and Dynamical Evolution

Mixed State and Operator Extensions

Any spin-$s$ density matrix $\rho$ admits a decomposition into a set of polynomials labeled by multipole rank, corresponding to constellations on spheres of characteristic radii. Under SU(2) rotations, these transform as rigid body rotations, and under partial trace, the constellation radii are rescaled but the star directions are preserved for the invariant blocks (Serrano-Ensástiga et al., 2019, Serrano-Ensástiga et al., 2022). Anticoherence and other operator properties can be reformulated directly at the level of polynomial coefficients.

Non-SU(2) Generalizations and Nonlinear Evolution

The coherent-state method allows extension of the MSR to Lie algebras beyond SU(2), such as Heisenberg-Weyl and SU(1,1), by selecting appropriate reference coherent states to define the star-equation. For squeezed and cat states, distributions of stars represent squeezing, cat components, or Schrödinger-cat-type superpositions as multi-circle or multi-point patterns on the sphere (Liu et al., 2016).

The dynamical evolution of the Majorana constellation under time-dependent Hamiltonians encodes nonlinear interaction effects, with, for instance, stars migrating along specific great circles during Kerr-type evolutions, or forming "X-shaped" or multi-cyclic patterns at specific evolution times corresponding to cat states or phase transitions.

7. Outlook and Emerging Directions

The Majorana-Stellar representation stands as a unifying geometric language spanning high-spin quantum systems, quantum information processing, topological matter, quantum optics, and metrology. Its recent generalizations, such as to mixed states, multiconstellations, optical fields, and Grassmannian subspaces, continue to broaden its utility. The explicit connection between the spatial arrangement of stars and quantum features—such as geometric phase structure, entanglement, nonclassicality, and topological invariants—renders the MSR a central analytical and visualization device for contemporary quantum science (Roy et al., 3 Jul 2025, Chryssomalakos et al., 2021, Kam et al., 2020, Liu et al., 2014).