Create maximal entanglement with just two gates
How do you create a pair of qubits that are perfectly correlated, even though each one individually looks completely random?
Bell states are the simplest entangled states and the foundation of quantum information. They were proposed by physicist John Bell in 1964 to test whether quantum mechanics could be explained by hidden local variables. The answer was no — and Bell states became the cornerstone of quantum teleportation, superdense coding, quantum key distribution, and error correction. If you understand Bell states, you understand the core resource that makes quantum computing different from classical computing.
Start with two qubits in |00⟩. Apply Hadamard to the first qubit to put it in superposition: now the state is (|0⟩+|1⟩)/√2 ⊗ |0⟩. Then apply CNOT with the first qubit as control. The |0⟩ branch leaves the target alone, giving |00⟩. The |1⟩ branch flips the target, giving |11⟩. The result is (|00⟩+|11⟩)/√2 — a state where the two qubits are perfectly correlated but individually random. This is entanglement in its purest form.
Entanglement is created by applying a two-qubit gate (CNOT) to a qubit that is already in superposition. If the control qubit were definite (|0⟩ or |1⟩), CNOT would just do a classical operation. Superposition + controlled interaction = entanglement.
Both qubits start in the |0⟩ state. The system is completely unentangled.
Hadamard puts qubit 0 into an equal superposition. The two qubits are still independent — no entanglement yet.
CNOT creates entanglement. The |00⟩ term stays |00⟩ (control is 0, target unchanged). The |10⟩ term becomes |11⟩ (control is 1, target flips). Now the two qubits are linked.
You get 00 or 11, each with 50% probability. You never see 01 or 10. The outcomes are perfectly correlated even though each individual outcome is random.
Measuring 00 or 11 with equal probability (never 01 or 10) confirms the Bell state. The perfect correlation cannot be explained by each qubit independently deciding its value.
The Bell state |Φ⁺⟩ = (|00⟩+|11⟩)/√2 is a maximally entangled state of two qubits. It cannot be factored as |ψ_A⟩⊗|ψ_B⟩, which is the mathematical definition of entanglement. The reduced density matrix of each qubit is ρ = I/2 (maximally mixed), meaning each qubit alone carries zero information about the measurement outcome. All the information is in the correlations. There are four Bell states, forming an orthonormal basis for the two-qubit Hilbert space: |Φ±⟩ = (|00⟩±|11⟩)/√2 and |Ψ±⟩ = (|01⟩±|10⟩)/√2.
Classically, you could prepare two correlated coins (both heads or both tails) by looking at one and setting the other. But the coins had definite values all along.
In a Bell state, neither qubit has a definite value before measurement. The correlation exists without either qubit having a pre-determined outcome. This is what Bell’s theorem proves cannot be classical.
Bell states do not allow faster-than-light communication. Each qubit’s local measurement outcome is completely random — the correlations only become visible when you compare results, which requires classical communication.
Bell states are the starting point for quantum teleportation (transferring a state using entanglement + classical bits) and superdense coding (sending 2 classical bits using 1 qubit + entanglement).
In a Bell state |Φ⁺⟩, you measure qubit 0 and get |1⟩. What is the state of qubit 1?