Manuscript I-B@1 · I-B-v1.tex · proposition · 5357688f

Proposition: No universal quantum return

What destructive clock reading does and does not do › Consumption, nonreturn, and the surviving record · lines 787–809 · proof included or linked

I-B@1#prop:no-return

For $D \geq 2$ , $M_{C}$ has no channel left inverse on its full $C \otimes X$ input space.

Proof. Let

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

and fix any matter state $η_{X}$ . The distinct inputs

∣ χ_{+} ⟩ ⟨ χ_{+} ∣ \otimes η_{X}, ∣ χ_{-} ⟩ ⟨ χ_{-} ∣ \otimes η_{X}

both have output

\frac{1}{2} (∣ 0 ⟩ ⟨ 0 ∣_{R} + ∣ 1 ⟩ ⟨ 1 ∣_{R}) \otimes η_{X} .

A left inverse would send one output to two different inputs.

Displayed formulas in this passage

unlabeled · line 795
$∣ χ_{\pm} ⟩ = \frac{∣ τ _{0} ⟩ \pm ∣ τ _{1} ⟩}{2},$
unlabeled · line 800
$∣ χ_{+} ⟩ ⟨ χ_{+} ∣ \otimes η_{X}, ∣ χ_{-} ⟩ ⟨ χ_{-} ∣ \otimes η_{X}$
unlabeled · line 805
$\frac{1}{2} (∣ 0 ⟩ ⟨ 0 ∣_{R} + ∣ 1 ⟩ ⟨ 1 ∣_{R}) \otimes η_{X} .$