GHZ-assisted quantum teleportation ($|{\chi }_{s,t,c}{⟩}_B=X^t{Z}^{s\oplus c}|\chi ⟩$)

Share a GHZ state on $A,B,C$ and let the unknown input $Q$ be $|\chi ⟩$. Alice applies CNOT from $Q$ to $A$, then a Hadamard gate to $Q$, and measures $Q,A$ computationally, giving $s,t$. Charlie measures $C$ in the Hadamard basis, giving $c$. Bob's conditional state isup to a global phase. Alice's two bits and Charlie's one bit therefore specify Bob's inverse correction $Z^{s\oplus c}{X}^t$.

To prove the phase rule, write $|\chi ⟩=\alpha |0⟩+\beta |1⟩$. After Alice's result, $BC$ is $\alpha |tt⟩+(-1{)}^s\beta |(1\oplus t)(1\oplus t)⟩$. Taking Charlie's Hadamard basis partial inner product multiplies its second term relative to its first by $(-1{)}^c$. The remaining common phase is $(-1{)}^{ct}$, which is physically irrelevant. All eight result triples have probability $1/8$. Charlie's record localizes a known-phase Bell pair, allowing the standard Bell-basis teleportation identity to complete the transfer.