You can photocopy this page a thousand times. You cannot photocopy an unknown qubit even once. That single impossibility is the bedrock under teleportation, quantum money and unbreakable keys — so it’s worth seeing exactly where the copier jams.
Build a machine that copies |0⟩ perfectly, and copies |1⟩ perfectly. Reasonable. Now feed it |+⟩ — a state that is half each. A real copy would be two independent |+⟩ qubits. What the machine actually spits out is something else entirely: the two halves come out welded together, entangled, not copied. The copier didn’t break a rule of engineering — it ran into a rule of the universe.
Picture a photocopier tuned to duplicate solid colors — pure red, pure blue — flawlessly. Hand it a sheet that's a 50/50 swirl of the two and ask for two identical swirls. It can't. The best it manages is to weld the two output sheets together so they always show the same random shade — linked, never independent copies. The machine isn't broken; duplicating an unknown blend was never on the menu.
Slide the input from a pole toward the equator. The left grid is what an honest copy |ψ⟩|ψ⟩ should look like; the right is what the best linear machine actually produces. At the poles they match perfectly. Anywhere in between, the fidelity falls off a cliff.
Suppose a copier U exists with U|0⟩|0⟩ = |0⟩|0⟩ and U|1⟩|0⟩ = |1⟩|1⟩. Gates are linear, so feeding it a superposition forces:
That’s the whole theorem. The cross terms αβ|01⟩ and αβ|10⟩ are exactly the cells lit on the left grid and dark on the right — the copy the machine can never fill in.
“No-cloning means quantum computers can’t copy anything.” They copy known states freely — if you know it’s |+⟩, just build two. What you can’t do is duplicate an unknown state handed to you, because pinning it down would mean measuring it, which destroys it.
Far from a nuisance, this is the feature that makes the rest possible. Teleportation must destroy the original (no surviving copy). An eavesdropper can’t silently photocopy your key in transit. Quantum repeaters, two chapters on, can’t simply amplify a signal the classical way — they have to work around this very law.