Gates, circuits and entanglement

A circuit is a sequence of operations

A quantum circuit shows what to prepare, which operations to apply, and what to measure. We read its operations from left to right. A horizontal line follows a qubit through time; it does not mean an electrical wire physically carrying that qubit between boxes. A gate symbol represents a controlled operation. In the ideal model, the gates in this section are reversible: another suitable gate can undo them before measurement.

The X gate exchanges |0⟩ and |1⟩. Applied to |0⟩ it gives |1⟩; applied to |1⟩ it gives |0⟩. Applying X twice returns either state to its starting point. The Hadamard gate, written H, maps |0⟩ to |+⟩ and |1⟩ to |−⟩. It also maps |+⟩ back to |0⟩ and |−⟩ back to |1⟩. Therefore H followed immediately by H restores the original state in the ideal model. It is not equivalent to two independent coin tosses.

A controlled gate acts on two qubits

The controlled-NOT gate, or CNOT, has a control qubit and a target qubit. On computational-basis inputs, it flips the target exactly when the control is 1. For the notation in this lesson, the first digit is the control and the second is the target: |00⟩ stays |00⟩, |01⟩ stays |01⟩, |10⟩ becomes |11⟩, and |11⟩ becomes |10⟩. Other software may display bit strings in a different order, so always check its convention.

CNOT is a coherent operation, not an instruction to measure the control and then decide what to do. If the input contains a superposition, apply the same mapping to each term and retain their amplitudes. This is how a gate can produce entanglement. A computational-basis truth table is a useful first step, but tracking a superposition explains behaviour that the table alone does not show.

Create a Bell pair

Prepare two qubits in |00⟩. Apply H to the first, producing (|00⟩ + |10⟩)/√2. Then apply CNOT with the first qubit as control. The |00⟩ term stays put and the |10⟩ term becomes |11⟩. The result is (|00⟩ + |11⟩)/√2, often called the Bell state |Φ+⟩. A pair in this state is a Bell pair. This derivation describes an ideal circuit, not a claim that physical gates are flawless.

The pair has a joint state that cannot be expressed as a separate pure state for the first qubit multiplied by a separate pure state for the second. That is entanglement for this pure-state example. If both qubits are measured in the computational basis, 00 and 11 each occur with probability 1/2. Neither party can predict their individual result in advance, although their results agree when compared.

Correlations need careful interpretation

Agreement alone is not proof of entanglement. Two people could share a classical random bit and each write it down, producing the same 00/11 statistics. Distinguishing that classical mixture from the Bell pair requires additional, appropriate measurements. For example, applying H to both qubits before computational-basis measurement gives perfectly agreeing outcomes for the ideal |Φ+⟩ state. The 50/50 classical mixture of |00⟩ and |11⟩ instead gives all four outcomes equally often in that setting. Rigorous experimental verification must also account for noise and the assumptions of its test.

Entanglement does not send a chosen message by itself. If Alice and Bob share a Bell pair, Alice cannot choose which random measurement outcome she gets. Bob’s local outcome statistics do not reveal which local measurement Alice chose when her results are not communicated. They need an ordinary classical channel to compare their records or use measurement outcomes in a communication protocol. Entanglement therefore provides no faster-than-light messaging.

Why a quantum network cannot use an ordinary copy command

The no-cloning theorem rules out a universal physical operation that perfectly copies every arbitrary unknown quantum state while leaving the original intact. It does not forbid preparing a known state repeatedly. It also does not forbid copying a promised computational-basis value using CNOT and a target initially in |0⟩. The restriction concerns a copier that works for all possible input states.

To see the problem, try that CNOT copier with control |+⟩ and target |0⟩. It creates a Bell pair, not two independent |+⟩ states. Measuring both in the computational basis gives only 00 or 11; two independent plus states would also allow 01 and 10. Quantum communication must respect this difference. Later sections use entanglement and classical messages to transfer information without making a forbidden extra copy.

Common misconceptions

A small glossary

Gate

A specified operation on one or more qubits; the ideal gates used here are reversible.

CNOT

A two-qubit gate that flips the target on basis inputs when its control is 1, extended coherently to superpositions.

Entanglement

Quantum correlations that cannot be represented as a mixture of independently specified local states; for a pure pair, its state cannot be factored into separate single-qubit states.

Bell pair

Two qubits sharing one of the maximally entangled two-qubit Bell states.

No-signalling

Local actions on one share of entanglement cannot alone communicate a chosen message to the other party.