Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 10C (Quantum Information and Computation)

By the Born rule,andEach is the expectation of the projector onto the span of the two displayed matching-label product states.

In the $A$ basis, the transformations whose columns are the new basis vectors areThus $U_C$ is a rotation through $-\pi /3$, while $U_B$ is a real orthogonal reflection, equivalently a rotation with one basis-vector phase reversed.

For any real orthogonal matrix $U$ with columns $|u_j⟩$,Applying this to $U_B$ and $U_C$ and normalizing proves

For $|\psi ⟩=|00⟩$, independence of the product state givesFor the remaining pair,Therefore

For the real Bell state $|{\varphi }^{+}⟩$, real one-qubit vectors satisfyFor each adjacent pair $(A,B)$, $(B,C)$, and $(C,A)$, the two same-label overlaps have magnitude $1/2$. Hence every matching-label probability isIt follows that