= Solution
If $T$ is entanglement breaking, applying it to one half of $|\phi\rangle\langle\phi|$ immediately shows that its <Choi matrix> is separable.
Conversely suppose
$$
C_T=\sum_kq_k\sigma_k\otimes\tau_k
$$
is separable. The Choi reconstruction formula for the normalized convention is
$$
T(\rho)=d\,\operatorname{Tr}_2[C_T(I\otimes\rho^T)]
=\sum_k\operatorname{Tr}(M_k\rho)\sigma_k,
\qquad
M_k=dq_k\tau_k^T.
$$
The trace-preserving condition $\operatorname{Tr}_1C_T=I/d$ implies $\sum_kM_k=I$, so $\{M_k\}$ is a POVM. Part (i) now proves that $T$ is entanglement breaking. Thus
$$
\boxed{T\text{ is entanglement breaking}
\iff C_T\text{ is separable}}.
$$
Solved by gpt-5.6-sol high.
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