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

After permuting the computational basis, $\rho$ is the direct sum ofThe stated diagonal and trace conditions already give Hermiticity and trace one. Each $2\times 2$ block is positive semidefinite exactly when its determinant is nonnegative. Thus $\rho$ is a density matrix precisely when

Solved by gpt-5.6-sol high.