# Math 401, Fall 2025: Thesis notes, S2, Majorana stellar representation of quantum states

## Majorana stellar representation of quantum states

> [!TIP]
>
> A helpful resource is [Geometry of Quantum states](https://www.cambridge.org/core/books/geometry-of-quantum-states/46B62FE3F9DA6E0B4EDDAE653F61ED8C) Section 4.4 and Chapter 7.

Vectors in $\mathbb{C}^{n+1}$ can be represented by a set of $n$ degree polynomials.

$$
\vec{Z}=(Z_1,\cdots,Z_n)\sim w(z)=Z_0+Z_1z+\cdots+Z_nz^n
$$

If $Z_0\neq 0$, then we can rescale the polynomial to make $Z_0=1$.

Therefore, points in $\mathbb{C}P^{n}$ will be one-to-one corresponding to the set of $n$ degree polynomials with $n$ complex roots.

$$
Z_0+Z_1z+\cdots+Z_nz^n=0=Z_0(z-z_1)(z-z_2)\cdots(z-z_n)
$$

If $Z_0=0$, then count $\infty$ as root.

Using stereographic projection of each root we can get a unordered collection of $S^2$. Example: $\mathbb{C}P=S^2$, $\mathbb{C}p^2=S^2\times S^2\setminus S_2$ where $S_2$ is symmetric group.

> [!NOTE]
>
> TODO: Check more definition from different area of mathematics (algebraic geometry, complex analysis, etc.) of the Majorana stellar representation of quantum states.
>
> Read Chapter 5 and 6 of [Geometry of Quantum states](https://www.cambridge.org/core/books/geometry-of-quantum-states/46B62FE3F9DA6E0B4EDDAE653F61ED8C) for more details.