How the CNOT Gate Turns Qubits Into Entangled Pairs

Multi-qubit gates, like the CNOT gate, are fundamental to quantum computing for creating entanglement, where qubit states are correlated and cannot be described independently. The CNOT gate's conditional operation, flipping a target qubit only if the control qubit is in the |1⟩ state, is the key mechanism for generating these entangled states, such as Bell and GHZ states.

Daron Krikorian

October 16, 2026 · 5 min read

Multi-qubit gates, chiefly the CNOT gate, are the mechanism by which quantum computers create entanglement: correlated states of two or more qubits that cannot be described as independent parts. The recipe is short. Put one qubit into a superposition with a Hadamard gate, then use the CNOT's conditional rule — flip the target only if the control reads 1 — to weld the qubits together. Two qubits treated this way yield a Bell state; three or more yield a GHZ state. Understanding that one conditional operation is the key to seeing why entanglement is a computational resource rather than a curiosity.

The Foundation of Quantum Computing: Multi-Qubit Gates

Single-qubit gates can rotate an individual qubit, but they cannot make one qubit's state depend on another's. That dependence is exactly what multi-qubit gates provide. Microsoft's quantum education materials describe multi-qubit gates as playing an important role across quantum computing, with the CNOT gate serving as the standard tool for producing Bell states — maximally entangled two-qubit states, also known as Einstein-Podolsky-Rosen states, in which the state of one qubit is entirely dependent on the state of the other.

IBM's Quantum Learning course frames the same idea from the state side: an entangled state is one that cannot be represented as a tensor product of individual quantum states. In other words, no matter how you try to factor the joint state into a description of qubit A and a separate description of qubit B, it cannot be done. Gates that act on more than one qubit at a time are the only way to reach such states, which is why they sit at the center of every quantum circuit that does serious work.

The CNOT Gate: A Key to Entanglement

The CNOT gate (written CX in many contexts) takes two inputs: a control qubit and a target qubit. Its rule is conditional. If the control qubit is in the |1⟩ state, the gate flips the target; if the control is |0⟩, the target is left untouched. A formal verification hosted at the University of Chicago states this plainly as a lemma: CNOT applied to |1,1⟩ yields |1,0⟩, while applied to a superposed control paired with |0⟩ it yields the equal combination of |00⟩ and |11⟩.

That second case is where the magic hides. When the control is in a definite state, the CNOT behaves like an ordinary classical conditional. But when the control is in superposition — in both |0⟩ and |1⟩ at once — the gate applies both of its behaviors simultaneously. The target flips for the |1⟩ branch of the superposition and stays put for the |0⟩ branch. The two qubits emerge correlated across both branches, and the joint state can no longer be split apart. The conditional operation is the mechanism of entanglement generation.

How a CNOT Gate Creates a Bell State

The standard circuit for entangling two qubits uses just two gates. The following table traces both qubits step by step from a definite starting state to the maximally entangled Bell state |Φ⁺⟩, showing how each gate's rule transforms the system.

Tracing two qubits from |00⟩ to the entangled Bell state |Φ⁺⟩ = (1/√2)(|00⟩ + |11⟩)
Step	Control qubit (qubit 1)	Target qubit (qubit 2)	Gate operation	Resulting system state
Initialize	\|0⟩	\|0⟩	Both qubits set to the ground state	\|00⟩
Step 1	Superposition of \|0⟩ and \|1⟩	\|0⟩	Hadamard on qubit 1 creates an equal superposition	(1/√2)(\|00⟩ + \|10⟩)
Step 2	Superposition of \|0⟩ and \|1⟩	Flipped only in the \|1⟩ branch	CNOT with qubit 1 as control, qubit 2 as target: flip the target if the control is 1	(1/√2)(\|00⟩ + \|11⟩) — the Bell state \|Φ⁺⟩

Notice what the CNOT did in step 2. In the |0⟩ branch of the control's superposition, the target stayed |0⟩. In the |1⟩ branch, the target flipped to |1⟩. The two branches became |00⟩ and |11⟩, perfectly correlated. Microsoft's documentation describes exactly this procedure: initialize two qubits to |00⟩, and the algorithm proceeds from there; IBM's course implements it as a Hadamard on qubit 0 followed by a CNOT from qubit 0 to qubit 1. The University of Chicago formalization proves the same circuit correct: CNOT applied after a Hadamard on the first qubit produces the Bell state by definition.

The result is inseparable. Measuring either qubit yields 0 or 1 with equal probability, but the outcome of the second qubit is fully determined by the first: both read 0, or both read 1. IBM's course notes that when this circuit runs on real hardware, |00⟩ and |11⟩ dominate the observed results, with a few other outcomes attributable to noise and qubit decoherence — a reminder that the ideal math describes a target, not a noiseless machine.