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Two qubits, one shared answer
A Bell state is the simplest example of entanglement. Each qubit on its own gives a random 0 or 1, yet when both are measured in the same basis the results are perfectly correlated. For |Φ⁺⟩ they always match in both Z and X. It takes only two gates to build one, and the effect holds no matter how far apart the qubits travel. It still can't carry a message: each side alone just sees random bits, and the match only shows up when the two compare notes over an ordinary channel.
Each pair is a fresh measurement of |Φ⁺⟩. Individually random, always in agreement.
01 · Refresher
Two-qubit states need four amplitudes
From page 01: two qubits have four basis states, |00⟩ |01⟩ |10⟩ |11⟩. Some two-qubit states can be split into "qubit A is this, qubit B is that." For example, |0⟩⊗|+⟩. A Bell state can't be split that way at all.
02 · The core idea
Build a Bell state with H and CNOT
Put the first qubit into superposition with a Hadamard, then use it as the control for a CNOT. The target gets flipped "in the |1⟩ branch only," so the two qubits become correlated branch by branch.
Try it: choose the input and step through
Each of the four inputs produces a different Bell state. The vector shows all four amplitudes after each gate.
03 · Measuring the pair
Random on its own, perfect together
Alice takes one qubit and Bob takes the other. Each can measure in the Z basis (0 or 1) or the X basis (+ or −, by applying H first). Run many shots and compare their results.
Try it: measure the Bell state built above
Start with both in Z. Then try putting both in X, and finally use different bases.
04 · The full set
The four Bell states
Click a state to see its formula and which input creates it.
05 · Security connection
Entanglement as a security tool
The E91 protocol (Artur Ekert, 1991) distributes Bell pairs to Alice and Bob. When they measure in matching bases, their results agree and become shared key bits. An eavesdropper who intercepts and measures a qubit disturbs the entanglement, and the correlations weaken in a way Alice and Bob can test statistically with a Bell test (CHSH).