# 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 $$\sigma_{XA} = \sum_i p_i \vert i\rangle\langle i\vert_X \otimes\rho^i_A$$, 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?
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