Entanglement
H then CX turns |00⟩ into the Bell state (|00⟩+|11⟩)/√2: each bit alone is a fair coin, yet the two bits agree on every shot, and simple arithmetic shows no pair of independent per-qubit descriptions can reproduce this. The correlation is a property of the joint state — it cannot be used to send a message.
Build the state first, by hand. Start at 00 with amplitude 1. Apply H to q0: amplitudes become 0.7071 on 00 and 0.7071 on 01. Apply CX with control q0, target q1: the 00 component has control 0 and stays; the 01 component has control 1, so q1 flips and it becomes 11. Final amplitudes: 0.7071 on 00, 0.7071 on 11, zero elsewhere.
Squares: 0.5 and 0.5. So the prediction is: half the shots read 00, half read 11, and 01 and 10 never appear. This state, (|00⟩ + |11⟩)/√2, is called a Bell state, and it is entangled — a joint state that cannot be split into a separate description for each qubit. The next section proves that with arithmetic rather than assertion.
You’ve read the opening of chapter 11 — 7 more sections follow, with worked examples and circuits you can run on the page. A free account unlocks every chapter of every course (paid plans aren’t live yet — early readers get everything free).