= Solution
A bipartite <density operator> $\rho_{AB}$ has <positive partial transpose> when $\rho_{AB}^{T_B}\geq0$ in a product basis. Positivity is independent of the choice of local bases, although the matrix representing the <partial transpose> changes.
If the state is a <separable quantum state>, write $\rho_{AB}=\sum_jp_j\rho_j^A\otimes\rho_j^B$. Taking its <partial transpose> gives
$$
\rho_{AB}^{T_B}=\sum_jp_j\rho_j^A\otimes(\rho_j^B)^T\geq0.
$$
Indeed, the <matrix transpose> preserves the nonnegative <eigenvalues> of each local <density operator>; a <tensor product> of <positive semidefinite operators> is positive, and so is their nonnegative sum. This proves the necessary direction of the <positive partial transpose criterion>.
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