No-Cloning Theorem & Quantum Repeaters

This simulator runs two connected pieces of real quantum-communication physics side by side. The first is a literal cloning attempt: an unknown qubit state is fed through a CNOT gate with a blank ancilla — the simplest circuit that could conceivably copy it — and the simulator computes the actual output fidelity, showing it fall below 1 for any genuine superposition exactly as the no-cloning theorem requires. The second panel follows the practical consequence: because an unknown quantum state can never be copied to boost a fading signal the way a classical repeater amplifies light, a direct fiber link's transmission decays exponentially with distance, while a chain of quantum-repeater stations using entanglement swapping — never cloning, only combining independently-generated short-range entangled pairs — keeps far more of its fidelity over the same distance. Tune the input state angle to watch the cloning fidelity formula in real time, then switch to the distance view and adjust fiber loss, repeater spacing and swap efficiency to see why long-haul quantum networks are built as repeater chains rather than amplifiers.