Quantum No-Cloning Theorem

Unknown Qubit State

Polar angle θ 60°

Azimuth φ 30°

Cloning Strategy

An eavesdropper measures the qubit along a random basis, then prepares two "clones" in the outcome they got — the classic intercept-resend attack.

Live Readouts

Last clone fidelity

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Trials run

Average fidelity ⟨F⟩

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Theory bound

0.667

No single-shot measurement can beat an average fidelity of 2/3 when cloning a completely unknown qubit — that ceiling holds no matter which basis the eavesdropper guesses.

How it works

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.

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.

Source state (unknown qubit) Latest clone Trial history (collapse outcomes)

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