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Prepare the three-vertex graph-state wire: start all three qubits in $|+⟩$ and apply Controlled-Z gates on edges $1-2$ and $2-3$. In this paper's equatorial qubit measurement convention the basis vectors can be chosen asThe given identity means that measuring an input qubit at angle $\theta$, with outcome $r$, teleports its state to the next qubit as $X^{r}U(\theta )|\psi ⟩$. Each outcome has probability $1/2$, since $X^{r}U(\theta )$ is unitary. This is one-bit teleportation with a known Pauli frame.

Measure vertex 1 at angle $\alpha$ and record $r$. Then measure vertex 2 at angle $\gamma =(-1{)}^r\beta$, taken modulo $2\pi$, and record $s$. The controlled-Z on edge $2-3$ commutes with the first measurement, so the two teleportation steps can be applied successively. The remaining normalized state isFor $r=0$ this is $X^{s}U(\beta )U(\alpha )|+⟩$. For $r=1$, the relation proved above gives $U(-\beta )X=e^{i\beta }ZU(\beta )$. Thus in every branch the remaining state isThe scalar is a global phase and has no effect on probabilities.

Finally measure vertex 3 in the computational basis, obtaining $z$, and reportThe $Z^r$ byproduct changes only computational-basis phases; $X^s$ changes the output label by $s$. Hence the corrected bit has exactly the ideal circuit's probability distribution in every branch, not merely on average. The procedure uses only single-qubit measurements after preparing the graph state, adaptive classical angle selection, and deterministic classical output processing.

Alice holds the first two qubits and Bob the third of a three-vertex graph-state wire. Alice measures at $\alpha$ and $(-1{)}^r\beta$, with outcomes $r,s$, and sends these two bits. Bob applies $Z^r{X}^s$, the inverse of the known Pauli frame $X^s{Z}^r$, obtaining $U(\beta )U(\alpha )|+⟩$ up to global phase. Bob need not know the angles. Every branch succeeds, with branch probability $1/4$, and no postselection is needed. Alice's measurements and Bob's corrections constitute local operations and classical communication on the previously shared entangled resource.