Question: Is there a parameterization of a a general qutrit 3-level system similar to: $$\rho = \begin{bmatrix} p_1 & r_{12}e^{-i\phi_{12}} & r_{13}e^{-i\phi_{13}}\\ \cdot & p_2 & r_{23}e^{-i\phi_{23}}\\ \cdot & \cdot & 1-p_1-p_2 \end{bmatrix} \tag{1.1}$$ With the constraints: $$\begin{align} & 0\leq p_j \\ & p_1+p_2\leq1 \\ & 0\leq r_{ij}\leq \sqrt{p_ip_j} \\ & 0 \leq \phi_{ij} < 2\pi \end{align} \tag{1.2}$$ ?
Today I learned a 3-level system can be parameterized, from Gell-Mann matrices, as:
$$ \rho = \begin{bmatrix} \frac13+a_3+\frac{a_8}{\sqrt 3} & a_1-ia_2 &a_4-ia_5 \\ \cdot & \frac13-a_3+\frac{a_8}{\sqrt 3} & a_6-ia_7 \\ \cdot &\cdot & \frac13-\frac{2a_8}{\sqrt 3} \end{bmatrix} \tag{2.1}$$ Subject to the constraints: $$\begin{align} & 0\leq \frac13\pm a_3+\frac{a_8}{\sqrt 3}\leq1 \\ & 0\leq \frac13-\frac{2a_8}{\sqrt 3}\leq1 \\ & \mathbf a\cdot \mathbf a\leq\frac13 \\ & 0 \leq \det(\rho) \end{align} \tag{2.2}$$
Disclaimer: I know parameterization $(1)$ is not correct, hence I am asking if there is anything similar. In particular, my problem with the Gell-Mann parameterization $(2)$ is that I have to compute a determinant each time I draw 8 random numbers... I would like something faster to generate and validate.
Background: I am doing numerical simulations, where I need to quickly generate a large number of random 3-level density operators. I was using parameterization $(2)$, with which you can easily draw the eight degrees of freedom of the qutrit.
However, the parameterization is wrong since it allows non-physical states, such as: $$\rho = \begin{bmatrix} p_1 &\sqrt{p_1 p_2} &\sqrt{p_1 p_3}\\ \cdot &p_2 &\sqrt{p_2 p_3}e^{-i\pi}\\ \cdot &\cdot &p_3 \end{bmatrix}$$ which is not positive semidefinite.