= GHZ-assisted quantum teleportation
{c}
{title2=$|\chi_{s,t,c}\rangle_B=X^tZ^{s\oplus c}|\chi\rangle$}
Share a <GHZ state> on $A,B,C$ and let the unknown input $Q$ be $|\chi\rangle$. 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 is
$$
X^tZ^{s\oplus c}|\chi\rangle,
$$
up 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 $|\chi\rangle=\alpha|0\rangle+\beta|1\rangle$. After Alice's result, $BC$ is $\alpha|tt\rangle+(-1)^s\beta|(1\oplus t)(1\oplus t)\rangle$. 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.
Back to article page
