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As in the title, is a classical bipartite state on $AA'$ given by

$$\sum_{ij} p_{ij} \vert i\rangle\langle i\vert_A \otimes \vert j\rangle\langle j\vert_{A'}$$

with $\sum_{ij}p_{ij} = 1$.

In particular, is the second index $j$ even necessary or can this be replaced by $$\sum_{i} p_{i} \vert i\rangle\langle i\vert_A \otimes \vert i\rangle\langle i\vert_{A'}$$

with $\sum_i p_i = 1$ instead? If yes, how does one show this and if not, what exactly does the second state represent (some kind of special classical-classical state)?

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A classical state $$ \rho_X = \sum_x p_x |x\rangle \langle x | $$ is meant to represent a random variable $X$ which takes a value $x$ with probability $p_x$.

If we have two random variables $X$ and $Y$ with a joint distribution $p_{xy}$ then we can represent it by the bipartite classical state $$ \rho_{XY} = \sum_{xy} p_{xy} |x\rangle \langle x | \otimes |y\rangle \langle y |. $$ Note that this is consistent with the marginal states representing the marginal random variables!

The second state you mention in the question would represented the joint random variable $XY$ where $p_{xy} = 0$ whenever $x \neq y$.

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