Two possible representations of the four-dimensional Hilbert space are the 4D column vectors or the 2x2 Hermitian matrices. For example, consider the two following representations of an orthonormal basis:
$$\underbrace{\left\{\begin{bmatrix}1\\0\\0\\0\end{bmatrix},\begin{bmatrix}0\\1\\0\\0\end{bmatrix},\begin{bmatrix}0\\0\\1\\0\end{bmatrix},\begin{bmatrix}0\\0\\0\\1\end{bmatrix}\right\}}_\text{4D column vector representation}\quad\text{and}\quad\underbrace{\left\{\frac{1}{\sqrt{2}}\begin{bmatrix}1&0\\0&1\end{bmatrix},\frac{1}{\sqrt{2}}\begin{bmatrix}0&1\\1&0\end{bmatrix},\frac{1}{\sqrt{2}}\begin{bmatrix}0&-i\\i&0\end{bmatrix},\frac{1}{\sqrt{2}}\begin{bmatrix}1&0\\0&-1\end{bmatrix}\right\}}_{2\times2\text{ Hermitian matrix representation}}$$
Where an appropriate inner product for the matrix representation can be shown to be defined by $\langle \boldsymbol A|\boldsymbol B\rangle\equiv\operatorname{tr}\left(\boldsymbol A^\dagger\boldsymbol B\right)$.
I was wondering whether there are any situations where it would be helpful or be more intuitive to represent the state kets as matrices such as above?
To clarify, I know that we often write states as density matrices ($|\psi\rangle\langle\psi|$ - example pure state) to allow for mixed states. I am not interested in this but interested in matrix representations of kets.
