Manuscript X-Enter-ST@1 · X-Enter-ST.tex · proposition · f76acc1f

Proposition: No universal quantum return

Stage D: complete instruments for supplied evaluations › No acquisition and no universal clock return · lines 1022–1041 · proof included or linked

X-Enter-ST@1#prop:noreturn

For $d_{C} \geq 2$ , $M$ has no channel left inverse on its full clock–residual input space.

Proof. With any fixed residual $ω_{A}$ , the distinct clock states

∣ χ_{\pm} ⟩ = \frac{∣ τ _{0} ⟩ \pm ∣ τ _{1} ⟩}{2}

give the same output

\frac{1}{2} (∣ 0, q_{0} ⟩ ⟨ 0, q_{0} ∣_{R} + ∣ 1, q_{1} ⟩ ⟨ 1, q_{1} ∣_{R}) \otimes ω_{A} .

No left inverse can send this one output to both inputs.

Displayed formulas in this passage

unlabeled · line 1031
$∣ χ_{\pm} ⟩ = \frac{∣ τ _{0} ⟩ \pm ∣ τ _{1} ⟩}{2}$
unlabeled · line 1036
$\frac{1}{2} (∣ 0, q_{0} ⟩ ⟨ 0, q_{0} ∣_{R} + ∣ 1, q_{1} ⟩ ⟨ 1, q_{1} ∣_{R}) \otimes ω_{A} .$