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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 necessarily finite dimensional). We say that $\rho$ is countable separable if $\rho=\sum_{i=1}^{\infty}\lambda_i\alpha_i\otimes\beta_i $, where $\alpha_i,\beta_i$ are states on $H$ and $\lambda_i\geq 0$ such that $\sum_{i=1}^{\infty}\lambda_i=1$.
Let $\{e_n\}$ be an orthonormal basis for $H$ and let $P_n$ be the orthogonal projection on to the space of span $\{e_i:1\leq i\leq n\}$. Are the following two statements equivalent?

1. $P_n\rho P_n$ is countable separable, for all $n$.
2. $\rho$ is countable separable.

Clearly, $1 \implies 2$. What about the converse?

Let $\rho$ be a positive trace class operator on $H\otimes H$, where $H$ is a separable Hilbert space (not necessarily finite dimensional). We say that $\rho$ is countable separable if $\rho=\sum_{i=1}^{\infty}\lambda_i\alpha_i\otimes\beta_i $, where $\alpha_i,\beta_i$ are states ( positive trace class operators with trace equal to $1$ ) on $H$ and $\lambda_i\geq 0$ such that $\sum_{i=1}^{\infty}\lambda_i< \infty$.
Let $\{e_n\}$ be an orthonormal basis for $H$ and let $P_n$ be the orthogonal projection on to the space of span $\{e_i:1\leq i\leq n\}$. Are the following two statements equivalent?

1. $(I_H\otimes P_n)\rho (I_H\otimes P_n^*)$ is countable separable, for all $n$.
2. $\rho$ is countable separable.

Clearly, $2 \implies 1$. What about the converse?

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}\lambda_i\alpha_i\otimes\beta_i $, where $\alpha_i,\beta_i$ are states on $H$ and $\lambda_i\geq 0$ such that $\sum_{i=1}^{\infty}\lambda_i=1$. 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?

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 necessarily finite dimensional). We say that $\rho$ is countable separable if $\rho=\sum_{i=1}^{\infty}\lambda_i\alpha_i\otimes\beta_i $, where $\alpha_i,\beta_i$ are states on $H$ and $\lambda_i\geq 0$ such that $\sum_{i=1}^{\infty}\lambda_i=1$. 
Let $\{e_n\}$ be an orthonormal basis for $H$ and let $P_n$ be the orthogonal projection on to the space of span $\{e_i:1\leq i\leq n\}$. Are the following two statements equivalent?

1. $P_n\rho P_n$ is countable separable, for all $n$.
2. $\rho$ is countable separable.

Clearly, $1 \implies 2$. What about the converse?

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}\lambda_i\alpha_i\otimes\beta_i $, where $\alpha_i,\beta_i$ are states on $H$ and $\lambda_i\geq 0$ such that $\sum_{i=1}^{\infty}\lambda_i=1$,. 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?

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}\lambda_i\alpha_i\otimes\beta_i $, where $\alpha_i,\beta_i$ are states on $H$ and $\lambda_i\geq 0$ such that $\sum_{i=1}^{\infty}\lambda_i=1$. 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?