= Solution
Use two precise finite-dimensional results. The <positive-map separability criterion> says that a state on $\mathbb C^2\otimes\mathbb C^3$ is separable exactly when $(\operatorname{id}_2\otimes\Phi)(\sigma)\geq0$ for every <positive linear map> $\Phi:M_3(\mathbb C)\to M_2(\mathbb C)$. The <Størmer-Woronowicz decomposability theorem> says that every such map is a <decomposable positive map>, so
$$
\Phi=\Phi_1+\Phi_2\circ T,
$$
with $\Phi_1,\Phi_2$ <completely positive maps> and $T$ the <matrix transpose> on $M_3$.
If $\sigma$ has <positive partial transpose>, both $\sigma$ and $\sigma^{T_B}$ are positive. Hence
$$
(\operatorname{id}_2\otimes\Phi)(\sigma)
=(\operatorname{id}_2\otimes\Phi_1)(\sigma)
+(\operatorname{id}_2\otimes\Phi_2)(\sigma^{T_B})\geq0.
$$
The <positive-map separability criterion> now proves that $\sigma$ is a <separable quantum state>. The dimension-specific decomposability theorem is essential: the conclusion does not extend to arbitrary bipartite dimensions.
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