Controlled-NOT gate

The controlled-NOT gate maps $|a,b⟩$ to $|a,b\oplus a⟩$.

Writing a CNOT gate as $P_0\otimes I+P_1\otimes X$ gives $I\otimes Q_{+}+Z\otimes Q_{-}$, where $Q_{\pm }=(I\pm X)/2$. In the Hadamard basis, $Z$ flips the sign label and $X$ is diagonal. Thus the same gate matrix reverses its control and target roles under the two local basis changes.

A CNOT gate can correlate either the original control's computational label with a computational pointer, or the original target's sign-basis label with a sign-basis pointer. For the second interpretation prepare the original control in $|+⟩$ and read it in the Hadamard basis. Its two pointer states become $|+⟩$ and $|-⟩$, measuring $X$ on the original target. An apparatus role therefore depends on the ready state and readout as well as the measurement interaction.

The SWAP gate maps $|x⟩|y⟩$ to $|y⟩|x⟩$.

Applying controlled-NOT gates with directions $1\to 2$, $2\to 1$, and $1\to 2$ mapsand therefore implements the SWAP gate.

The three-qubit GHZ state is

For $n\ge 2$, apply a Hadamard gate to one qubit, then controlled-NOTs from it to all others. Replacing each controlled-NOT gate by a target Hadamard, a Controlled-Z gate, and another target Hadamard uses $1+3(n-1)$ allowed gates. Each one-qubit reduced state is $I/2$, proving entanglement across each qubit-versus-rest bipartition.

For $|\alpha ⟩=a|0⟩+b|1⟩$, a Bell measurement on $|\alpha ⟩$ and the first leg of a GHZ state leaves the other two legs in $a|00⟩\pm b|11⟩$, after applying $X$ to both legs for the two $\Psi$ outcomes.

Starting from $|000⟩$, apply a Hadamard gate to the first qubit and then controlled-NOT gates from the first qubit to each of the other two qubits.

If two senders share a GHZ state with a receiver, one sender can encode two bits with $X^x{Z}^z$ and the other one bit with $X^b$. Sending both qubits to the receiver produces one of eight orthogonal GHZ-basis states, so a joint measurement recovers all three bits.

The GHZ theorem uses perfect multipartite quantum correlations to contradict local predetermined values without a statistical inequality. For the three-qubit GHZ state, three certain Pauli-product relations force a fourth product to have the opposite sign from its quantum-mechanical value.