Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 53 / 4 / B / Protocol 2 / Solution

Impossible to implement as written: Bob is never given the basis string. Step 7 requires him to know which incoming qubits were acted on by a Hadamard gate, but the announcement contains only the check positions. Since $H^{-1}=H$, knowing that a transformation is its own inverse does not identify which qubits need it.

Bob cannot recover the missing information from the qubits alone. For each encoded Bell pair, his reduced density matrix isfor either $b=0$ or $b=1$. His entire received register has the same density matrix $I/2^{2n}$ for every string $b$, so every local measurement in quantum mechanics has statistics independent of $b$. Nor can one fixed unitary undo both possibilities: preserving $|{\Phi }^{+}⟩$ requires that unitary to be proportional to $I$, whereas decoding $(I\otimes H)|{\Phi }^{+}⟩$ requires it to be proportional to $H$. Supplying the missing basis announcement after receipt would repair the protocol and make it the third one, but that communication is absent from the printed sequence.