What is the general form of a classical-quantum state?

In the literature, one comes across the following situation: Alice holds two registers $X$ and $A$ and it is given that $X$ is a classical register.

What is the most general way to write down Alice’s state? Is it just $sigma_{XA} = sum_i p_i vert iranglelangle ivert_X otimesrho^i_A$, with each $rho^i_A$ being a quantum state (positive semi-definite and trace one matrix)?

Sorry that this is a yes/no question because if yes, then there is not much to add. But if not, what would be the most general way to write Alice’s state?