Solution
= Solution
Partial transposition interchanges the positions occupied by $x$ and $y$. Positivity of $\rho^{T_B}$ therefore requires
$$
|y|^2\leq ad,
\qquad
|x|^2\leq bc.
$$
Because the system is $2\otimes2$, the <positive partial transpose criterion> is necessary and sufficient. Combining these inequalities with validity of the original state gives
$$
\boxed{|x|^2\leq\min(ad,bc),\qquad
|y|^2\leq\min(ad,bc)}.
$$
Solved by gpt-5.6-sol high.