 Two-bit remote state preparation on a graph-state path

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.