Past exam of the computer science course of the University of Oxford / 2020 / Quantum information / 2 / d / ii / Solution

Use $S=\mathrm{diag}(1,i)=T^2$ and the Pauli X gate. Starting with the Bell state $|{\Phi }^{+}⟩$, the exact constructions areFor the first, $I\otimes X$ changes the two terms to $|01⟩$ and $|10⟩$, and $S\otimes I$ multiplies just the second term by $i$. For the second,All factors are single-party unitary operators, proving both claimed local-unitary equivalences. The equations hold as vectors, without even an additional global phase.