Manuscript X-K-ST@1 · X-K-ST.tex · proposition · 93c39c0f

Proposition: No universal return from the destructive output

One destructive clock and its exact limitations · lines 558–577 · proof included or linked

X-K-ST@1#prop:noreturn

The clock channel has no left inverse on all clock inputs. No fixed continuation supplied only with that output and input-independent new resources can universally restore the consumed unknown clock.

Proof. For a fixed surviving state $ρ_{S}$ , the two inputs

∣ ψ_{+} ⟩ ⟨ ψ_{+} ∣ \otimes ρ_{S}, ∣ ψ_{-} ⟩ ⟨ ψ_{-} ∣ \otimes ρ_{S}, ∣ ψ_{\pm} ⟩ = \frac{∣ τ _{0} ⟩ \pm ∣ τ _{1} ⟩}{2},

are different and have identical outputs. Every fixed continuation gives identical outputs on identical inputs, so it cannot restore both.

Displayed formulas in this passage

unlabeled · line 568
$∣ ψ_{+} ⟩ ⟨ ψ_{+} ∣ \otimes ρ_{S}, ∣ ψ_{-} ⟩ ⟨ ψ_{-} ∣ \otimes ρ_{S}, ∣ ψ_{\pm} ⟩ = \frac{∣ τ _{0} ⟩ \pm ∣ τ _{1} ⟩}{2},$

Proposition: No universal return from the destructive output

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