The no-cloning theorem (Wootters & Zurek, 1982) says no physical process can take an unknown pure state |ψ⟩ and produce two perfect copies. Proof sketch: a hypothetical cloning unitary U with U|ψ⟩|0⟩ = |ψ⟩|ψ⟩ must preserve inner products (unitarity), but ⟨ψ|φ⟩ ≠ ⟨ψ|φ⟩² unless the states are identical or orthogonal — contradiction for any generic pair.
Fidelity of two pure qubit states:
F(n1, n2) = (1 + n1·n2) / 2 (n = Bloch vector)
Measure along random axis m, Born rule:
P(+m) = (1 + n0·m) / 2
clone state = ±m (whichever outcome occurred)
⟨F⟩ over all random m = 2/3 ← exact, derivable in closed form
Optimal universal cloner (Bužek–Hillery, 1996):
clone Bloch vector = (2/3) n0 (shrunk, now a mixed state)
F = (1 + 2/3) / 2 = 5/6 ≈ 0.833 ← best possible for 1→2 cloning
- θ / φ sliders — set the true (unknown to the "cloner") Bloch vector of the source qubit.
- Measure & Resend — each trial projects the source onto a fresh random measurement axis, collapses it via the Born rule, and "clones" the outcome; run many trials and watch the average fidelity converge to 2/3 no matter what θ, φ you chose.
- Optimal Cloner — the best possible cloning machine allowed by quantum mechanics: deterministic, no measurement, but the output is necessarily a mixed state with a shrunken Bloch vector, capped at 5/6 fidelity.
- The scattered dots are the collapse outcomes accumulated over your trials — they cluster near the source direction but spread across the whole sphere, visualising exactly why no fixed strategy can do better than 2/3.
- Drag the Bloch disk to rotate the view — the sphere is drawn with a fixed-tilt orthographic projection, so its outline stays a perfect circle from any angle.
Real-world relevance: this ceiling is what makes quantum key distribution (BB84) secure — an eavesdropper intercepting a qubit cannot clone it and forward an undetectable perfect copy, so any measurement attempt introduces statistically detectable errors.