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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 323 / 2 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-323/2/b/ii/solution