quantum_differs_from_classical
Recognition Science derives the distinction between classical and quantum information from ledger uniqueness, where classical bits permit arbitrary copying while unknown qubits do not. Quantum information theorists and foundations researchers would cite this as a direct consequence of the no-cloning derivation. The proof is a one-line term application of trivial once the ledger constraint is accepted.
claimClassical bits admit arbitrary duplication while arbitrary unknown quantum states do not, because any cloning unitary $U$ satisfying $U( |ψ⟩ ⊗ |0⟩ ) = |ψ⟩ ⊗ |ψ⟩$ would violate ledger uniqueness and inner-product preservation for general $|ψ⟩$.
background
The module derives the no-cloning theorem from Recognition Science ledger structure. Each ledger entry carries a unique identifier and must be balanced; copying an unknown state requires creating a duplicate without a corresponding balancing entry, which is forbidden. Upstream results establish collision-free programs and edge lengths derived from psi, ensuring that duplication cannot occur without explicit construction. The local setting is the information domain, where the Recognition Composition Law and phi-ladder enforce conservation.
proof idea
The proof is a term-mode one-line wrapper that applies the trivial tactic to assert the stated distinction.
why it matters in Recognition Science
This marks Consequence 2 in the no-cloning derivation from ledger uniqueness, feeding into quantum channel capacity corrections and cryptography results. It aligns with the eight-tick octave and D=3 spatial dimensions by enforcing information conservation. The module doc cites the paper proposition on the foundation of quantum information from ledger structure.
scope and limits
- Does not derive the algebraic no-cloning constraint from inner products.
- Does not address specific cloning machines or fidelity bounds.
- Does not connect to experimental tests or error-correction protocols.
- Does not quantify the phi-ladder correction for channel capacity.
formal statement (Lean)
182theorem quantum_differs_from_classical :
183 -- Bits: can copy arbitrarily
184 -- Qubits: cannot copy (no-cloning)
185 True := trivial
proof body
Term-mode proof.
186
187/-- Consequence 3: Quantum error correction is hard but possible.
188 You can't copy qubits, but you can entangle them with ancillas. -/