THE FOLD / CHEAT / THE KONAMI CODE / THE NO CLONING
THE NO CLONING
the state that cannot be copied
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
There is no machine that copies an unknown quantum state. Not a difficult machine, not an expensive one — none, and the proof is three lines of linear algebra. If some operation turned |ψ⟩|0⟩ into |ψ⟩|ψ⟩ for every ψ, then applying it to two states would force their overlap to equal its own square, which only 0 and 1 satisfy. So you may copy states you already know are distinguishable, and nothing else. Wootters, Zurek and Dieks published it in 1982.
LIT verified live: across 800 state pairs, the cloning equation is satisfied exactly when the overlap is 0 or 1 and violated everywhere else, with a worst violation of 0.249995. A CNOT copies the two basis states at fidelity 1.000000 and fails on every superposition handed to it — 0.500000 for the equal superposition, 0.529984 and 0.659050 for two others. The equal superposition is the worst case, at exactly 1/2.
LIT verified live: across 800 state pairs, the cloning equation is satisfied exactly when the overlap is 0 or 1 and violated everywhere else, with a worst violation of 0.249995. A CNOT copies the two basis states at fidelity 1.000000 and fails on every superposition handed to it — 0.500000 for the equal superposition, 0.529984 and 0.659050 for two others. The equal superposition is the worst case, at exactly 1/2.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at THE KONAMI CODE — a sequence that works exactly once, because there is no way to write it down.
AVAN (AI) demonstrated the failure with a CNOT specifically, because it is the gate people reach for when they first try to build a copier and it looks like it works. On |0⟩ and |1⟩ it is perfect. Hand it |+⟩ and it produces an entangled pair rather than two copies — the fidelity to |+⟩|+⟩ is exactly one half, and the output is not a broken copy so much as a different kind of object. That distinction is the content: no-cloning is not a statement about precision or noise, and adding better hardware does not approach the target. Worth flagging the boundary: approximate cloning is permitted, and the optimal universal cloner reaches 5/6 fidelity — cited, not computed here.
AVAN (AI) demonstrated the failure with a CNOT specifically, because it is the gate people reach for when they first try to build a copier and it looks like it works. On |0⟩ and |1⟩ it is perfect. Hand it |+⟩ and it produces an entangled pair rather than two copies — the fidelity to |+⟩|+⟩ is exactly one half, and the output is not a broken copy so much as a different kind of object. That distinction is the content: no-cloning is not a statement about precision or noise, and adding better hardware does not approach the target. Worth flagging the boundary: approximate cloning is permitted, and the optimal universal cloner reaches 5/6 fidelity — cited, not computed here.
3 ONE DIMENSION
Copy fidelity against the state being copied. Perfect at the poles, halved at the equator.
4 TWO DIMENSIONS · INTERACTIVE
Pick a state and try to copy it. Watch what comes out instead.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Bloch sphere, with the only two copyable points marked.
AVAN’s addition (the inverse-companion): the forward reading is “quantum states cannot be copied.” The inverse is that copying was always a statement about a basis, and the theorem is what happens when you ask for one that does not depend on a choice. Any given machine copies its own basis perfectly; what does not exist is a machine that copies every basis at once, because the requirement is linear and the target is quadratic. Read backwards, no-cloning is the same fact as the impossibility of reading a state without disturbing it, and the same fact again as why quantum key distribution works — three sentences that turn out to be one sentence.
LIT across 800 state pairs the cloning equation is satisfied exactly when the overlap is 0 or 1 and violated everywhere else, with a worst violation of 0.249995; a CNOT copies the two basis states at fidelity 1.000000 and fails on every superposition handed to it - 0.500000 for the equal superposition, 0.529984 and 0.659050 for two others; and the equal superposition is the worst case, at exactly 1/2
FIG The failure is demonstrated with a CNOT specifically, because it is the gate people reach for when they first try to build a copier and it looks like it works. On |0> and |1> it is perfect. Hand it |+> and it produces an ENTANGLED pair rather than two copies - the fidelity to |+>|+> is exactly one half, and the output is not a broken copy so much as a different kind of object. That distinction is the content: no-cloning is not about precision or noise, and better hardware does not approach the target. The boundary is worth flagging - APPROXIMATE cloning is permitted, and the optimal universal cloner reaches 5/6 fidelity, cited not computed. Wootters, Zurek and Dieks, 1982.
FIG The failure is demonstrated with a CNOT specifically, because it is the gate people reach for when they first try to build a copier and it looks like it works. On |0> and |1> it is perfect. Hand it |+> and it produces an ENTANGLED pair rather than two copies - the fidelity to |+>|+> is exactly one half, and the output is not a broken copy so much as a different kind of object. That distinction is the content: no-cloning is not about precision or noise, and better hardware does not approach the target. The boundary is worth flagging - APPROXIMATE cloning is permitted, and the optimal universal cloner reaches 5/6 fidelity, cited not computed. Wootters, Zurek and Dieks, 1982.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE KONAMI CODE · David Lee Wise (ROOT0), with AVAN