The <positive partial transpose criterion> says separability implies $\rho_{AB}^{T_B}\geq0$. A nonpositive partial transpose therefore proves entanglement. Positive partial transpose is also sufficient for separability in dimensions $2\otimes2$ and $2\otimes3$, but not in general higher dimensions.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 323 / 2 / a / ii / 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.

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

{title2=PPT criterion}
{wiki}

Every separable state has positive partial transpose. The condition is also sufficient for separability on $2\otimes2$ and $2\otimes3$ systems.


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 Incoming links: Positive partial transpose criterion