Manuscript Hawk03-A@1 · Hawk03-A.tex · proposition · 9a109c12

Proposition: A unitary can cause complete proxy departure

Purifier distillation and intervention dependence · lines 1110–1133 · proof included or linked

Hawk03-A@1#prop:unitarydeparture

For $d = 2$ , take

∣ β ⟩ = p_{0} ∣ 00 ⟩ + p_{1} ∣ 11 ⟩, p_{0}, p_{1} > 0.

The exterior unitary $U_{F D} (X_{F} \otimes I_{D}) U_{F D}^{†}$ preserves the message channel but changes the original proxy-pass probability to zero.

Proof. In logical coordinates the unitary acts only as $X_{A}$ . Its overlap with the original pair is

⟨ β ∣ (X_{A} \otimes I_{B}) ∣ β ⟩ = Tr (τ_{A} X) = 0.

It does not act on the logical message, so the message decoder is unchanged.

Displayed formulas in this passage

unlabeled · line 1113
$∣ β ⟩ = p_{0} ∣ 00 ⟩ + p_{1} ∣ 11 ⟩, p_{0}, p_{1} > 0.$
unlabeled · line 1127
$⟨ β ∣ (X_{A} \otimes I_{B}) ∣ β ⟩ = Tr (τ_{A} X) = 0.$

Nearby in this section

section: Purifier distillation and intervention dependence
proposition: Proposition: Distillation changes the original proxy
remark: Remark: No intervention-independent vacuum claim
section: Purifier distillation and intervention dependence