It is known that an arbitrary state (ie. a density matrix) for a qubit can be written as a linear combination of the Pauli matrices
$\rho = \frac{1}{2} (I + r_x \sigma_x + r_y \sigma_y + r_z \sigma_z)$,
where the real numbers $r_x$, $r_y$ and $r_z$ are coordinates of a point in the unit ball.
Can one write a similar decomposition for an arbitrary state in the case of a pair of qubits (ie. a bipartite quantum system), using matrices like $\sigma_x \otimes \sigma_x$, and so on?
