No-Cloning & Quantum Repeaters — Fidelity vs Distance

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Input state angle θ 30°

|ψ⟩ = cos θ |0⟩ + sin θ |1⟩. θ=0° or 90° is a basis state (classical bit); θ=45° is an equal superposition — the hardest case to copy.

Copy fidelity F

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No-cloning bound

F < 1

P(measure 0)

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P(measure 1)

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Why cloning fails

A "naive cloner" runs a CNOT gate: qubit |ψ⟩ = cos θ|0⟩ + sin θ|1⟩ as control, a blank ancilla |0⟩ as target. CNOT sends |0⟩|0⟩→|00⟩ and |1⟩|0⟩→|11⟩, so it copies a classical bit (θ=0 or 90°) perfectly. But fed a superposition it produces an entangled state, not two independent copies:

CNOT(|ψ⟩⊗|0⟩) = cosθ|00⟩ + sinθ|11⟩  ≠  |ψ⟩ ⊗ |ψ⟩

Tracing out either qubit leaves a mixed state ρ = cos²θ|0⟩⟨0| + sin²θ|1⟩⟨1|, whose overlap with the original gives the real copy fidelity:

F(θ) = ⟨ψ|ρ|ψ⟩ = cos⁴θ + sin⁴θ

F = 1 only at θ = 0°/90° (orthogonal basis states — classical information can be copied). At θ = 45° it drops to F = 0.5, no better than a random guess. This is the no-cloning theorem in action: any unitary process that perfectly copies two non-orthogonal states would have to preserve their inner product while also driving it to zero, which is impossible.

Why repeaters, not amplifiers

A classical optical amplifier restores a weak signal by reading and reproducing it — copying. No-cloning forbids exactly that for an unknown quantum state, so a direct quantum link only ever gets weaker with distance, following the real fiber-attenuation law:

T_direct(L) = 10^(−αL/10)

A quantum repeater chain never tries to copy the state. It splits the link into short segments, generates entanglement independently across each (heralded, then held in memory), and joins them with entanglement swapping — a measurement-based operation that consumes two short-range Bell pairs and produces one long-range pair, without ever cloning anything:

T_repeater(L) ≈ T_seg · p_swap^(N−1),  N = ⌈L / d_seg⌉, T_seg = 10^(−α·d_seg/10)

Because the fiber loss only ever applies over the short segment length d_seg, the repeater curve decays far more slowly with L than the direct exponential — the whole reason real long-haul quantum networks are being built as repeater chains instead of quantum amplifiers.

No-Cloning Theorem & Quantum Repeaters