Past exam of the computer science course of the University of Oxford / 2021 / Quantum information / 1 / c / ii

Reverse communication is also possible. Prepare Alice in $|+{⟩}_A$. Bob encodes $m=0$ as $|+{⟩}_B$ and $m=1$ as $|-{⟩}_B$. Since $X|\pm ⟩=\pm |\pm ⟩$, phase kickback giveswhere $|(-{)}^0⟩=|+⟩$ and $|(-{)}^1⟩=|-⟩$. Alice measures in the Hadamard basis: $+$ means zero and $-$ means one, deterministically.

The target's Pauli X gate eigenvalue becomes a relative phase on the control. This is the operational content of CNOT control reversal in the Hadamard basis; the computational basis control arrow alone does not determine every possible direction of communication.