Manuscript I@2 · I.tex · theorem · 85631fa7

Theorem: Semantic no-cloning

Metatheory of the specified calculus › Path maps and circuit adequacy · lines 891–914 · proof included or linked

I@2#thm:nocloning

No closed term $c : Q ⊸ Q \otimes Q$ returns two copies of every pure qubit input.

Proof. By Lemma lem:path, its averaged output map $Φ$ is linear on density operators. A universal cloner would satisfy

Φ (∣ φ ⟩ ⟨ φ ∣) = ∣ φ ⟩ ⟨ φ ∣^{\otimes 2} .

But

I /2 = \frac{1}{2} (∣ 0 ⟩ ⟨ 0 ∣ + ∣ 1 ⟩ ⟨ 1 ∣) = \frac{1}{2} (∣ + ⟩ ⟨ + ∣ + ∣ - ⟩ ⟨ - ∣) .

Linearity gives two expressions for $Φ (I /2)$ . Their diagonal entries at $∣ 01 ⟩$ are respectively $0$ and $1/4$ , a contradiction.

Displayed formulas in this passage

unlabeled · line 901
$Φ (∣ φ ⟩ ⟨ φ ∣) = ∣ φ ⟩ ⟨ φ ∣^{\otimes 2} .$
unlabeled · line 906
$I /2 = \frac{1}{2} (∣ 0 ⟩ ⟨ 0 ∣ + ∣ 1 ⟩ ⟨ 1 ∣) = \frac{1}{2} (∣ + ⟩ ⟨ + ∣ + ∣ - ⟩ ⟨ - ∣) .$