In my QM lecture it was claimed that if you have a system with degrees freedom $\vec{s}$ and its surroundings which have degrees of freedom $\vec{u}$ then every density matrix for the combined system can be expressed as
$$ \hat\rho = \sum_{\vec{u}} \sum_{\vec{s}} p_{\vec{u}, \vec{s}} \left|\vec{u}, \vec{s} \right> \left< \vec{u}, \vec{s}\right|$$
(I suppose that the sum is should range over orthonormal bases $\{\left| u \right> \}$ and $\{\left| s \right> \}$.)
To me it seems that this is not general enough to express all possible density matrices. Is that right or am I missing something?