Past exam of the laser science and quantum information processing course of the University of Oxford / 2015 / 8 / b

In superdense coding, Alice and Bob initially share $|{\Psi }^{+}{⟩}_{AB}$. Alice selects the same one-qubit Pauli gate $I,Z,X,XZ$ according to her message and sends her single encoded qubit to Bob. He now has both halves and measures in the Bell basis. For the chosen assignment,which reaches two bits for the uniform distribution. A circuit implementation first converts the resource convention with $X_B$, then applies ${\mathrm{CNOT}}_{A\to B}$ and $H_A$, and measures both in the computational basis. The four outcomes are distinct; relabeling them restores the selected message order.

Before Alice's transmission, Bob's reduced density operator is $I/2$ for every choice of her local operation. Thus distributing the message-independent entangled resource has transmitted no information about the subsequently chosen message. It is nevertheless a real physical resource: the two-bit rate counts the one additional transmitted qubit after entanglement has been shared, not a free initial distribution. The deterministic conditional probabilities above establish perfect correlation rather than merely matching output and input entropies.