Let $\rho$ be a state (positive trace class operator with trace equal to $1$) on $H\otimes H$, where $H$ is a separable Hilbert space (not necessarly finite dimensional space). We say $\rho$ is countable separable if $\rho=\sum_{i=1}^{\infty}\alpha_i\otimes\beta_i $, where $\alpha_i,\beta_i$ are states on $H$. Let $\{e_n\}$ be an orthonormal basis for $H$ and let $P_n$ be the orthogonal projection on to the space of span $\{e_i:1\leq i\leq n\}$.

Is the following two statements are equivalent:

(i) $P_n\rho P_n$ is countable separable, for all $n$. (ii) $\rho$ is countable separable.

Clearly, $(ii)\Rightarrow (i)$. What about the converse?