CNOT Gate

Overview

Single-qubit gates can rotate and phase-shift individual qubits, but they cannot create correlations between qubits.

The CNOT gate (Controlled-NOT) is the simplest and most important two-qubit gate because it introduces a new capability:

• conditional action: “do something to one qubit depending on the value of another.”

That conditional action is the doorway to entanglement, the kind of correlation that cannot be explained as “each qubit has its own independent state.”

Intuition

A two-qubit computational basis state looks like

$$|ab\rangle\quad\text{where}\quad a,b\in\{0,1\}.$$

Think of the first qubit as a condition and the second as the thing you act on.

• The control qubit decides whether anything happens.
• The target qubit is the one that gets changed.

CNOT follows a simple rule:

• If the control is $|0\rangle$, do nothing to the target.
• If the control is $|1\rangle$, apply X to the target (flip it).

So it is literally “NOT the target, controlled by the control.”

Why does this matter for entanglement?

Because it can turn a superposition in the control into correlation across both qubits. A gate that acts on only one qubit cannot do that—the second qubit would stay independent.

This ties directly back to Foundations:

• You learned tensor products and that a general two-qubit state is $$|\psi\rangle = \alpha|00\rangle + \beta|01\rangle + \gamma|10\rangle + \delta|11\rangle.$$
• You also learned that some states cannot be written as $|\phi\rangle\otimes|\chi\rangle$. Those are entangled.

CNOT is a standard “entanglement maker.”

Formal Description

Action on computational basis states

Let the first qubit be the control and the second be the target. Then CNOT acts as:

$$\text{CNOT}|00\rangle = |00\rangle,$$ $$\text{CNOT}|01\rangle = |01\rangle,$$ $$\text{CNOT}|10\rangle = |11\rangle,$$ $$\text{CNOT}|11\rangle = |10\rangle.$$

Read these as:

• when the first bit is 0, the second bit is unchanged;
• when the first bit is 1, the second bit is flipped.

What happens to amplitudes

Because quantum gates are linear, knowing the action on basis states determines the action on any superposition.

If

$$|\psi\rangle = \alpha|00\rangle + \beta|01\rangle + \gamma|10\rangle + \delta|11\rangle,$$

then applying CNOT gives

$$\text{CNOT}|\psi\rangle = \alpha|00\rangle + \beta|01\rangle + \gamma|11\rangle + \delta|10\rangle.$$

Notice what changed:

• the amplitudes on $|10\rangle$ and $|11\rangle$ are swapped, because those are the cases where the control is 1.

Matrix form exists (it’s a 4×4 unitary), but for intuition, the basis-action rule above is usually the clearest way to understand CNOT.

Worked Example

A canonical entanglement example starts with a product state:

$$|+\rangle\otimes|0\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\otimes|0\rangle.$$

Expand it:

$$|+0\rangle = \tfrac{1}{\sqrt{2}}(|00\rangle + |10\rangle).$$

Now apply CNOT (control = first qubit, target = second qubit):

• $\text{CNOT}|00\rangle = |00\rangle$
• $\text{CNOT}|10\rangle = |11\rangle$

So

$$\text{CNOT}|+0\rangle = \tfrac{1}{\sqrt{2}}(|00\rangle + |11\rangle) = |\Phi^+\rangle.$$

This is a Bell state. Interpretation:

• If you measure both qubits in the computational basis, you get perfectly correlated outcomes: 00 or 11 with probability 1/2 each.
• The state is not a product state; it cannot be written as $|\phi\rangle\otimes|\chi\rangle$.

This is entanglement in an operational sense: the joint outcomes have correlations that come from a single shared state, not from independent qubits.

Turtle Tip

Common Pitfalls

Quick Check

What’s Next

CNOT is one specific controlled operation: “apply X to the target when the control is 1.” Next we generalize the idea to controlled gates like CZ and controlled-$U$ (CU): the same conditional logic, but with different target operations—including phase-based entanglement.