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

Choose the phase gateIt is unitary. Acting on both qubits leaves $|00⟩$ unchanged and multiplies $|11⟩$ by $i^2=-1$, so $S\otimes S$ sends ${\Psi }_{+}$ to ${\Psi }_{-}$ with exactly the requested phase.

Using the Hadamard gate $H$, chooseIt is unitary and symmetric, andFor any matrix $U$, applying $U\otimes U$ to $(|00⟩+|11⟩)/\sqrt{2}$ gives the coefficient matrix $U{U}^T/\sqrt{2}$. Here $U_2{U}_2^T=U_2^2=X$, henceBoth displayed transformations are exact, not merely equal up to global phase. These common local unitaries permute the triplet Bell projectors while preserving the singlet projector by the preceding determinant identity.