Manuscript X-A-conv-ST@1 · X-A-conv-ST.tex · proposition · 8ca47088

Proposition: No universal clock return

Complete instruments for supplied evaluations › What the instrument does not acquire · lines 1732–1749 · proof included or linked

X-A-conv-ST@1#prop:noreturn

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

Proof. With a fixed residual $ω_{A}$ , the distinct clock inputs

\frac{∣ τ _{0} ⟩ + ∣ τ _{1} ⟩}{2}, \frac{∣ τ _{0} ⟩ - ∣ τ _{1} ⟩}{2}

give the same recorded output. A left inverse could not return both different inputs from that one output.

Displayed formulas in this passage

unlabeled · line 1741
$\frac{∣ τ _{0} ⟩ + ∣ τ _{1} ⟩}{2}, \frac{∣ τ _{0} ⟩ - ∣ τ _{1} ⟩}{2}$