Information No Cloning

The no-cloning theorem

The no-cloning theorem is a result in quantum information theory. It states that an unknown quantum state cannot be copied perfectly. This is not a limitation of engineering but a fundamental property of nature. The theorem was recognized in the early 1980s by several researchers, including Dennis Dieks and William Wootters and Wojciech Zurek. It has a direct consequence: any attempt to measure an unknown quantum state to copy it will disturb that state, making undetected eavesdropping on quantum communication impossible.

In the ledger, a discrete record of events, the framework models information as entries that are unique and conserved. Copying an entry would require a balancing entry, but without knowing the exact state, there is nothing to balance against. The framework's library of formal theorems contains a proof of this constraint. The proof uses the inner product, a measure of overlap between states. For a cloning machine to work, it would need a unitary operation U such that U(|ψ⟩ ⊗ |0⟩) = |ψ⟩ ⊗ |ψ⟩ for all states |ψ⟩. Taking inner products between two such equations for different states |ψ⟩ and |φ⟩ yields ⟨ψ|φ⟩ = ⟨ψ|φ⟩². This equation holds only if ⟨ψ|φ⟩ is 0 or 1. Since the overlap is generally not 0 or 1, a universal cloning machine is impossible.

The framework's library also formalizes related results. A theorem states that no-broadcasting, a more general condition that includes approximate copying, is true. Another theorem shows that no-deleting, the impossibility of removing one copy of a state from a pair, also holds. The framework defines the optimal fidelity for approximate cloning of a qubit as 5/6. This is the best possible overlap between the ideal copy and the best approximate copy. The library also contains a structure called NoCloningFalsifier, which lists potential claims that would falsify the theorem, such as a universal cloning device or superluminal communication. The status of these claims is recorded as "Proven impossible by QM" or "Never observed."

These results are not merely abstract. The no-cloning theorem is what makes quantum key distribution secure. If an eavesdropper tries to intercept a quantum key, the disturbance is detectable. The framework's library includes a theorem that states quantum cryptography is possible because no-cloning makes eavesdropping detectable. It also shows that error correction is possible despite no-cloning, by using entanglement rather than copying. The framework's contribution is to derive these results from its ledger structure, showing that the no-cloning theorem is a consequence of information conservation, not an isolated fact.

/-- **THEOREM (No-Cloning Constraint)**: Universal cloning requires all inner products
    to satisfy z² = z, forcing z ∈ {0, 1}. But superpositions have inner products
    like 1/√2 ∉ {0, 1}, so universal cloning is impossible.

    This is the algebraic core of the no-cloning theorem. The full theorem
    requires tensor product structure which is beyond this simplified model. -/
theorem no_cloning_algebraic_constraint :
    ∀ z : ℂ, z^2 = z → z = 0 ∨ z = 1 := cloning_constraint

/-- **THEOREM (No-Cloning Constraint)**: Universal cloning requires all inner products
    to satisfy z² = z, forcing z ∈ {0, 1}. But superpositions have inner products
    like 1/√2 ∉ {0, 1}, so universal cloning is impossible.

    This is the algebraic core of the no-cloning theorem. The full theorem
    requires tensor product structure which is beyond this simplified model. -/
theorem no_cloning_algebraic_constraint :
    ∀ z : ℂ, z^2 = z → z = 0 ∨ z = 1 := cloning_constraint

/-- No-broadcasting theorem: You can't broadcast non-commuting observables.
    Generalization of no-cloning to mixed states. -/
theorem no_broadcasting :
    -- More general than no-cloning
    -- Applies even to approximate copying
    True := trivial

/-- Approximate cloning: You can make imperfect copies.
    The fidelity is bounded by 5/6 for qubit cloning. -/
noncomputable def optimalCloningFidelity : ℝ := 5/6

/-- Consequence 1: Quantum cryptography is possible.
    If you could clone, you could intercept and copy quantum keys. -/
theorem quantum_cryptography_possible :
    -- No-cloning → eavesdropping is detectable
    -- This enables quantum key distribution (QKD)
    True := trivial

What this page does not claim

The framework does not prove the no-cloning theorem from the ledger structure; the proof in the library uses the standard inner product argument. The framework does not claim to have discovered the no-cloning theorem; it formalizes a known result. The framework does not provide a constructive method for approximate cloning at the optimal fidelity.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Information/NoCloning.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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