Range criterion: a nonzero Choi matrix with no product vector in its range is not entanglement breaking

Let $F : M_a(\mathbb C) \to M_b(\mathbb C)$ be any map (no linearity is assumed in the statement) whose Choi matrix $J(F) = \big(F(E_{ij})\big)_{ij}$ on $\mathbb C^a \otimes \mathbb C^b$ is nonzero and whose range contains no nonzero product vector: whenever $J(F)\,w = u \otimes v$ for some $w$, then $u = 0$ or $v = 0$. Then $F$ is not entanglement breaking in the amplification sense of the mission: it is not the case that $F$ is completely positive and $(\mathrm{id}_k \otimes F)(X)$ is separable for every positive semidefinite $X$ on $\mathbb C^k \otimes \mathbb C^a$ and every $k \ge 1$.

This is the range criterion of P. Horodecki in the form used in the proof of Theorem 1.2 of the source: an entanglement-breaking map has a separable Choi matrix (apply the amplification to the unnormalized maximally entangled matrix $\Omega = \sum_{i,j} E_{ij} \otimes E_{ij}$), and a nonzero separable matrix $\sum_i A_i \otimes B_i$ with $A_i, B_i \succeq 0$ has a nonzero product vector $(A_i x) \otimes (B_i y)$ in its range, because the range of a sum of positive semidefinite matrices contains the range of each summand.