Past exam of the introduction to quantum information course of the University of Oxford / 2015 / 2 / b / Solution

Expand the given pure state in the ordered computational basis $00,01,10,11$:The coefficients have squared magnitudes $1/3,1/6,1/3,1/6$, summing to one. The density matrix is the outer product $v{v}^{†}$, not the diagonal matrix of these four probabilities: its off-diagonal entries retain quantum coherence. ThereforeThe outer product is a Hermitian matrix and is positive semidefinite: $u^{†}\rho u=|v^{†}u{|}^2\ge 0$. Its trace is one, and ${\rho }^2=\rho$, as required for a pure state.