Majorana Representation in Quantum States

• Majorana representation is a geometric formalism that encodes quantum states into constellations of points on the Bloch sphere, revealing symmetry and entanglement.
• It constructs the state’s Majorana polynomial whose roots map to ‘stars’ that visualize entanglement, aiding invariant classification and analysis of quantum correlations.
• The framework extends to symmetric qubit and higher-dimensional systems, enabling insights into quantum dynamics, geometric phases, and error correction.

The Majorana representation, also called the Majorana stellar representation (MSR), is a geometric formalism that encodes a pure quantum state of finite Hilbert space dimension into a symmetric constellation of points—“Majorana stars”—on the Bloch sphere. Originally proposed by E. Majorana in 1932 for spin systems, the representation has become fundamental for analyzing quantum states with SU(2) symmetry, quantum correlations, entanglement, and more generally for mapping complex state spaces to tractable geometric descriptions relevant in quantum information, condensed matter, and mathematical physics.

1. Construction of the Majorana Representation

Given a pure spin-$j$ state $|\psi\rangle$ in the magnetic basis $|j,m\rangle$ ($m=-j,\ldots,j$), the state is written as

$|\psi\rangle = \sum_{m=-j}^{j} c_m |j,m\rangle.$

Majorana’s construction associates to $|\psi\rangle$ a homogeneous polynomial (the Majorana or stellar polynomial)

$$P(z) = \sum_{r=0}^{2j} a_r z^r, \quad a_{r} = (-1)^{2j-r} \sqrt{\frac{(2j)!}{r!(2j - r)!}}\, c_{m = r - j}.$$

The $2j$ roots $z_k$ of $P(z)$ are mapped to points on the Bloch sphere via inverse stereographic projection,

$$z_k = \tan(\theta_k/2) e^{i \phi_k}\; \Longleftrightarrow\; \theta_k = 2 \arctan|z_k|,\quad \phi_k = \arg(z_k),$$

thus generating a unique (up to phase and permutation) set of $2j$ “Majorana stars” $u_k$ on $S^2$ that fully determine $|\psi\rangle$ (Shokir, 2022, Liu et al., 2014).

This representation can be equivalently formulated for symmetric $n$-qubit states ($n=2j$), arbitrary $d$-level systems ($d-1$ stars), or generalized overcoherent states for other Lie algebras (Li et al., 2023, Liu et al., 2016). The correspondence extends to a factorized “wavefunction” in coherent-state parametrization:

$$\Psi(\theta,\phi) = \langle \theta, \phi | \psi \rangle \propto \prod_{k=1}^{2j} [\cos(\theta/2) e^{-i\phi/2} - z_k \sin(\theta/2) e^{i\phi/2}].$$

2. Properties and Generalizations

2.1 Covariance and Group Action

SU(2) rotations act as rigid rotations of the Majorana constellation:

$z_k \mapsto \frac{a\,z_k - b}{b^*\,z_k + a^*},$

where $(a,b) \in SU(2)$ parametrize the group element. Thus, state rotations correspond directly to geometric rotations of the stars (Shokir, 2022).

For higher symmetry groups, the coherent-state construction yields analogous mappings—e.g., for SU(1,1) or Heisenberg–Weyl, using the relevant ladder operators and reference vacua, followed by mapping complex roots to points on a non-compact analog of $S^2$ (e.g., hyperboloid, Bloch disk) (Liu et al., 2016).

2.2 Mixed States and Density Operators

MSR can be extended to mixed states by expressing a density matrix $\rho$ as a homogeneous bi-degree $(N,N)$ polynomial $p_\rho(z)$, corresponding to a family of $2s$ sub-constellations each living on a sphere with radius $$w_\sigma = \sqrt{\sum_{\mu=-\sigma}^{\sigma} |\rho_{\sigma\mu}|^2}$$ ($\sigma=1,\ldots,2s$). Each block inherits antipodal symmetry from Hermiticity and transforms covariantly under SU(2) (Serrano-Ensástiga et al., 2019).

Partial traces correspond to applying a linear differential operator $L$ to $p_\rho(z)$; tracing out a spin-$1/2$ amounts to

$$p_{\mathrm{Tr}_1 \rho}(z) = L[p_\rho(z)] = \frac{1}{N^2} \partial_{z_a} \partial^{z^a} p_\rho(z).$$

Constellations are inherited up to rescaling (Serrano-Ensástiga et al., 2019).

3. Applications in Quantum Information and Many-Body Physics

3.1 Entanglement Structure

In symmetric multiqubit states ($n$-qubit), the degeneracy pattern and geometric arrangement of Majorana stars encode SLOCC entanglement class: separable states have all points coincident, Dicke states correspond to two distinct stars, and maximally distinct constellations map to generalized GHZ states (Devi et al., 2011, Aulbach et al., 2010).

For the three-qubit case, the three-tangle (genuine tripartite entanglement) is expressible as a symmetric function of the chordal distances between Majorana stars, rendering SLOCC invariants manifestly geometric after appropriate symmetrization (Kam et al., 2019). In the case of $n=4,5,\ldots$, symmetrization is in general possible only for select SLOCC classes (Kam et al., 2019).

Entanglement measures such as concurrence and three-tangle are directly related to inter-star distances and relative arrangements (Liu et al., 2014, Shokir, 2022). The number of distinct Majorana stars classifies the entanglement type: $n_s=1$ (separable), $n_s=2$ (W-type), $n_s=n$ (GHZ-type), etc.

3.2 Geometric Phases and Berry Curvature

The Berry phase for an adiabatically cycled spin-$s$ state is expressible in terms of the solid angles subtended by the trajectories of the Majorana stars:

$$\gamma = -\frac{1}{2} \sum_{k=1}^{2s} \Omega[u_k] + \gamma_C,$$

where $\Omega[u_k] = \oint(1-\cos\theta_k)\,d\phi_k$ and $\gamma_C$ is a pairwise correlation term (Liu et al., 2014). This geometric characterization carries over to arbitrary $d$-level (finite-dimensional) systems—the Berry phase is then a sum over $d-1$ Majorana-star solid angles, with an explicit correction term for inter-star correlations.

The argument of weak or modular values for $N$-level systems factorizes into a sum of $N-1$ single-qubit Pancharatnam phases (solid angles), highlighting the topological structure of geometric quantum effects (Cormann et al., 2016).

4. Extension to Composite and Mixed-Spin Systems

For Hilbert spaces constructed as tensor products, such as spin-$s \otimes \frac{1}{2}$, the MSR generalizes via total angular momentum coupling:

• The space decomposes into two blocks with total spin $j=s+\frac{1}{2}$ and $j=s-\frac{1}{2}$, each admitting its own Majorana representation,
• An arbitrary pure state is a "pseudo-spin-$\frac{1}{2}$" superposition of these two blocks,
• The total number of Majorana stars is $(2s+1)+(2s-1)+1=4s+1$ (1904.02462).

This formalism enables geometric descriptions of inter-sector coherence, sectoral transitions, and entanglement oscillations in coupled systems.

5. Computational and Algebraic Aspects

5.1 Explicit Construction for Arbitrary $d$

For an arbitrary pure state $|a\rangle$ in $\mathbb{C}^d$, the Majorana mapping is:

• $$|a\rangle = \alpha\,T(v_1 \otimes \cdots \otimes v_{d-1})$$ for $v_k \in \mathbb{C}^2$,
• The roots $p_k$ of the degree-$d-1$ Majorana polynomial $$g(z) = \sum_{j=0}^{d-1}(-1)^j\binom{d-1}{j}a_j z^{d-1-j}$$ determine the points via $v_j=(1,p_j)^{\text{T}}$ (Li et al., 2023).

Inner products between two such states $|a\rangle$ and $|b\rangle$ can be analytically computed as permanents of the matrix of single-qubit overlaps:

$$\langle a|b \rangle = \frac{1}{(d-1)!} \mathrm{per}\left[\langle v_i | w_j \rangle\right]_{i,j=1}^{d-1}$$

where $v_i$, $w_j$ are the qubit vectors associated to $|a\rangle$, $|b\rangle$ (Li et al., 2023).

5.2 Algorithmic Summary

For symmetric $N$-qubit or spin-$j$ states:

1. Construct the Majorana polynomial,
2. Find its $N=2j$ (possibly degenerate) roots $z_i$,
3. Map each $z_i$ to $(\theta_i, \phi_i)$ on $S^2$,
4. The geometric configuration encodes the entanglement and dynamical properties (Devi et al., 2011, Aulbach et al., 2010).

For mixed states, decompose the density matrix, compute the corresponding Majorana-polynomial in each block, and read off the antipodal-symmetric constellations and radii.

6. Physical and Mathematical Significance

• Visualization: By encoding all amplitudes and phases of a high-dimensional state into a few points on $S^2$, the Majorana representation gives an immediate geometric picture of state evolution, coherence, and entanglement (Shokir, 2022, Aulbach et al., 2010).
• Invariant Classification: MSR enables invariant classification of symmetric quantum states, natural characterization of SLOCC classes, and direct geometric interpretation of quantum error correction codes, spherical designs, and anticoherence (Aulbach et al., 2010, Devi et al., 2011).
• Geometric Dynamics: In driven or dissipative systems (e.g., STIRAP protocols, nonlinear evolution), the motion and deformation of the Majorana constellation dynamically encode the nonadiabaticity, decoherence, or topological characteristics of state evolution (Dogra et al., 2020, Liu et al., 2016).
• Group Theory and Algebra: The concept extends beyond quantum mechanics into the theory of Majorana algebras, which underpins the axiomatic construction of objects like the Griess algebra and the Monster group in the context of axial and fusion laws, with MSR providing the explicit module structure for group representations (Franchi et al., 2020).

7. Examples and Canonical Configurations

Small-$d$ examples illustrate the graphical meaning:

• Qubit ($d=2$): One star, classical Bloch sphere.
• Qutrit ($d=3$): Two stars, full state space is parametrized by their joint location modulo SO(3) (Dogra et al., 2017, Dogra et al., 2020).
• Spin-3/2 or 4-level: Three stars, with permutation symmetry. Maximally entangled and symmetric states correspond to Platonic-solid point arrangements on $S^2$ (Aulbach et al., 2010).

Canonical states provide instructive reference constellations, and analytic mapping between operational gates and star configurations is available for explicit finite $d$ values (Dogra et al., 2017).

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The Majorana representation provides a unifying geometric and algebraic framework for the study of finite-dimensional pure and mixed quantum states, their symmetries, correlations, and dynamics, with broad implications in quantum information, many-body physics, and the theory of non-associative algebras (Shokir, 2022, Liu et al., 2014, 1904.02462, Liu et al., 2016, Serrano-Ensástiga et al., 2019, Devi et al., 2011, Aulbach et al., 2010).