Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 58 / 4 / a / Solution

Use the PDF's Bell state labels throughout this question: its ${\Phi }_{-}$ is the antisymmetric spin singlet state, whereas ${\Psi }_{\pm }$ are the even-parity states. WriteExpanding the antisymmetric combination cancels its $00$ and $11$ components and givesThe two normalized orthogonal vectors are the columns of a unitary matrix $U$. Its determinant $a_0{b}_1-a_1{b}_0$ has modulus one because $\det (U^{†}U)=|\det U{|}^2=1$. ThereforeThis proves collective-unitary covariance of the two-qubit singlet: a common basis change alters only its global phase, and leaves its density projector unchanged.