# How to decompose a density matrix of a mixed ensemble into a sum of pure ensembles [closed]

I'm trying to solve a problem where I am given a few matrices and asked to determine if they could be density matrices or not and if they are if they represent pure or mixed ensembles. In the case of mixed ensembles, I should find a decomposition in terms of a sum of pure ensembles. The matrix I'm having trouble with is this one

$$\rho = \left[\begin{array}{ccc}\frac{1}{2} & 0 & \frac{1}{4} \\ 0 & \frac{1}{4} & 0 \\ \frac{1}{4} & 0 & \frac{1}{4}\end{array}\right]$$

I know it's a mixed ensemble density matrix because $\mathrm{Tr}(\rho^2)<1$, but how can I decompose if I don't even know how big is this sum? I mean, any number of pure states may compose a mixed ensemble since they do not need to be orthogonal. How can I approach this?

That said, the diagonalization does seem to give you some awkward-looking vectors, i.e. it tells you to decompose as \begin{align} \rho &= \frac{3+\sqrt{5}}{8} \frac{1}{1+\varphi^2} \begin{pmatrix}\varphi\\0\\1\end{pmatrix}\begin{pmatrix}\varphi&0&1\end{pmatrix} \\ & \quad+ \frac{3-\sqrt{5}}{8} \frac{\varphi^2}{1+\varphi^2} \begin{pmatrix}-\varphi^{-1}\\0\\1\end{pmatrix}\begin{pmatrix}-\varphi^{-1}&0&1\end{pmatrix} \\ & \quad + \frac14 \begin{pmatrix}0\\1\\0\end{pmatrix}\begin{pmatrix}0&1&0\end{pmatrix} ,\end{align} where $\varphi$ is the golden ratio, which is pretty awkward, particularly when you can just notice that \begin{align} \rho = \frac14 \begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix} +\frac14 \begin{pmatrix}1&0&1\\0&0&0\\1&0&1\end{pmatrix} +\frac14 \begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix} \end{align} and take it from there. As I said, if you're being asked to provide an ensemble decomposition then all you need to do is provide one that works.