Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 10E (Quantum Information and Computation)

Write $|{\Phi }^{+}⟩=2^{-1/2}{\sum }_{j=0}^1|j⟩|j⟩$. Then

Sinceapplication of $X$ gives $\cos \theta |0⟩-\sin \theta |1⟩$, and subsequent application of $Z$ givesThus $ZX|{\psi }_2⟩=|{\psi }_1⟩$.

The matrix $U_{\theta }$ is real orthogonal. Part (a) givesExpanding the left side in the computational basis yields

Using the decomposition from part (ii), $U_{\theta }^{-1}$ maps Alice's $|{\psi }_1⟩$ to $|0⟩$ and $|{\psi }_2⟩$ to $|1⟩$. Alice obtains $0$ or $1$, each with probability $1/2$. Conditional on outcome $0$, Bob has $|{\psi }_1⟩$; conditional on outcome $1$, Bob has $|{\psi }_2⟩$.

Alice performs the operation and measurement from part (iii) and sends the one-bit outcome to Bob. If the bit is $0$, Bob does nothing. If it is $1$, Bob applies $ZX$, which maps $|{\psi }_2⟩$ to $|{\psi }_1⟩$ by part (i). In either case Bob finishes with the known target state $|{\psi }_1⟩$, using one shared Bell pair and one classical bit.