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

Partial transposition interchanges the positions occupied by $x$ and $y$. Positivity of ${\rho }^{T_B}$ therefore requiresBecause the system is $2\otimes 2$, the positive partial transpose criterion is necessary and sufficient. Combining these inequalities with validity of the original state gives