# What is the Majorana stellar representation?

One can geometrically visualize spin-1/2 states using the Bloch sphere. A natural question then is: "Can one geometrically visualize spin-$$s$$ states using a similar object to the Bloch sphere?" It seems like the answer is yes due to something called the Majorana stellar representation.

But, what is the Majorana stellar representation?

The Majorana stellar representation is a way to geometrically visualize pure spin-s states. In essence, the Majorana stellar representation 1) establishes a bijection between states of Hilbert space and complex polynomials, and 2) establishes a bijection between the set of roots of said complex polynomials and sets of points on $$S^2$$. Let us be more precise.

Let $$\mathcal{H}_s$$ be the Hilbert space of a spin-s system. Consider an arbitrary (pure) state $$\lvert \psi \rangle \in \mathcal{H}_s$$. Write this state in the eigenbasis of $$S_z$$: $$\lvert \psi \rangle \rightarrow \sum_{m = -s}^s\lambda_m\lvert s, m\rangle. \tag{1}$$

Then, we can construct the so-called Majorana polynomial associated with $$\lvert \psi \rangle$$ as $$p_\psi(Z) = \sum_{m=-s}^s\sqrt{\begin{pmatrix} 2s \\ s-m\end{pmatrix}} \lambda_m Z^{s+m} \tag{2}$$ where $$\begin{pmatrix} 2s \\ s-m\end{pmatrix}$$ is $$2s$$ choose $$(s-m)$$. The complex roots $$\{\zeta_k\}$$ of $$p_\psi$$ uniquely specify $$p_\psi$$ up to scale (and hence, each ray of Hilbert space corresponds to an equivalence class of polynomials related to one another by multiplication of a complex scalar $$c \in \mathbb{C}$$). Thus, we have a bijection between the states $$\lvert \psi \rangle$$ and complex polynomials $$p_\psi$$.

Then, we define the constellation of $$\lvert \psi \rangle$$, denoted $$C_\psi$$, by the set of points of $$S^2$$, each individual point called stars, obtained via stereographic projection of the set of roots $$\{\zeta_k\}$$ lying in the complex plane to $$S^2$$. Concretely, this stereographic projection is defined by $$\zeta_k=\tan(\theta_k/2)e^{i\phi_k} \mapsto (\theta, \phi)_k \in S^2.$$

Hence, we have a bijection between sets of roots $$\{\zeta_k\}$$ and sets of points on $$S^2$$ $$\{(\theta, \phi)_k\}$$. Overall then, we have a bijection from $$\mathcal{H}_s$$ into sets of points on $$S^2$$ as we set out to show.

One may now think of $$S^2$$ in familiar Euclidean space $$\mathbb{R}^3$$ and geometrically visualize pure spin-s states.

[1] E. Serrano-Ensástiga and D. Braun, Majorana representation for mixed states, https://doi.org/10.1103/PhysRevA.101.022332

Silly Goose's answer already nicely explains the usual way to define the stellar representation via polynomials. However, the nature of the polynomials here remains somewhat mysterious: Why should this projection of roots of polynomials onto the Bloch sphere yield a nice representations of spin states? How does one come up with that polynomial? Here's how one might come up with this formulation:

It's a standard result that all irreducible representations of $$\mathrm{SU}(2)$$ can be built as the symmetric tensors of the fundamental representation $$\mathbf{2}$$, i.e. $$\mathbf{2}^{\otimes 2s} = (\mathbf{2s+1}) \oplus \text{other stuff}$$ - the representation of spin $$s$$ is the symmetric tensors on $$\mathbf{2}$$. So for any spin-$$s$$ state $$\lvert \psi\rangle_s$$ we may expand it in the tensor basis $$\lvert s_1,\dots s_{2s}\rangle$$ as $$\lvert \psi\rangle_s = \sum_{s_1 = -1/2}^{1/2}\dots \sum_{s_{2s}=-1/2}^{1/2} \lambda_{s_1\dots s_{2s}} \lvert s_1,\dots,s_{2s}\rangle \tag{1}$$ in terms of tensor products $$\lvert s_1,\dots,s_{2s}\rangle = \lvert s_1\rangle\otimes\dots\otimes \lvert s_{2s}\rangle$$ of eigenstates of $$S_z$$. The $$\lambda_{s_1,\dots,s_{2s}}$$ is symmetric in all indices.

Now recall that the ordinary Bloch sphere representation works like this: For any state $$\lvert \psi\rangle = \lambda_{-1/2} \lvert -1/2\rangle + \lambda_{1/2}\lvert 1/2\rangle$$ we have that $$\lambda_{-1/2}^\ast \lambda_{-1/2} + \lambda_{1/2}^\ast \lambda_{1/2} = 1$$ and we can multiply by a global phase to make $$\lambda_{-1/2}$$ real such that $$\lambda_{-1/2} = \cos(\theta/2)$$ and $$\lambda_{1/2} = \mathrm{e}^{\mathrm{i}\phi}\sin(\theta/2)$$. The corresponding point on the Bloch sphere is the point with spherical coordinates $$(\theta,\phi)$$.

The stereographic projection onto the plane $$\mathbb{R}^2\cong \mathbb{C}$$ maps this point to the complex number $$\zeta = \tan(\theta/2) \mathrm{e}^{\mathrm{i}\phi} = \frac{\lambda_{1/2}}{\lambda_{-1/2}},$$ which of course is the root of the polynomial $$\lambda_{1/2} - \lambda_{-1/2}z$$.

Now in eq. (1) we have $$2s+1$$ of the $$\lambda_{s_1,\dots,s_{2s}}$$ and so $$2s$$ non-trivial ratios $$\rho_{s_1,\dots,s_{2s}} = \frac{\lambda_{s_1,\dots,s_{2s}}}{\lambda_{-1/2,\dots,-1/2}}$$, each of which maps to a point on the Bloch sphere. Since everything here is symmetric in the indices $$s_i$$, we might just as well index the $$\rho$$ via a single index $$j$$ from $$1$$ to $$2s$$ that counts how many of the $$s_i$$ differ from $$-1/2$$. So we have $$2s$$ complex numbers $$\rho_j$$, but if you just map these to the Bloch sphere, you find they behave weirdly under rotations acting on the original $$\lvert \psi\rangle_s$$: A rotation by some angle $$\alpha$$ rotates each $$\rho_j$$ by the angle $$j\alpha$$, i.e. each of the $$\rho_j$$ acts as if it is the product of $$j$$ complex numbers behaving the normal way under rotation (all statements here after mapping the numbers to the Bloch sphere).

So we would actually like to find some numbers $$\zeta_i$$ such that $$\rho_j$$ is the sum of product of $$j$$ of the $$\zeta_i$$. Taking into account the symmetrization, one can work out that this leads to a system of equations between the $$\zeta_i$$ and the $$\rho_j$$ that looks exactly like Vieta's formulae for the relation between coefficients $$\rho_j$$ of a polynomial and its roots $$\zeta_i$$. This way one also immediately sees that the roots $$\zeta_i$$ really are the correct visualization on the Bloch sphere because each of them behaves the way one intuitively expects under rotation - which is entirely non-obvious if one just starts by writing down the polynomial.