I've been trying to decompose a $3\times3$ density matrix with 3-dimensional Pauli matrices but it doesn't work for all matrices.
For example, the density matrix of the state $|0\rangle + |1\rangle + |2\rangle$ can be decomposed by obtaining the coefficients of the equation $$\rho = a_{X}X + a_{Y}Y + a_{Z}Z + a_{V}V + a_{X^2}X^2 + a_{Y^2}Y^2 + a_{Z^2}Z^2 + a_{V^2}V^2 + a_{I}I $$ using trace, e.g., $a_X = {\rm tr}(\rho X)/3 $. Here, $Y=XZ, V=XZ^2$, and $X$ and $Z$ are $3 \times 3$ Pauli matrices.
On the other hand, when I tried to decompose the state $|0\rangle + |2\rangle$ using the above equation, the result was not the same as $|0\rangle + |2\rangle$. Is it impossible to decompose a high-dimensional matrix using Pauli matrices?
