Manuscript En09-A@1 · En09-A.tex · proposition · eafec9de

Proposition: No universal quantum return from the readout

The philosophy of Enter as an interface discipline › Consumption is an access statement · lines 1027–1050 · proof included or linked

En09-A@1#prop:no-return

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For $d \geq 2$ , $M_{C}$ has no channel left inverse on its full clock–system input space. In particular, with a fixed system preparation, its output cannot universally return an arbitrary unknown quantum clock state.

Proof. Let

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

and let $η_{S}$ be any fixed system state. The distinct inputs $∣ χ_{+} ⟩ ⟨ χ_{+} ∣ \otimes η_{S}$ and $∣ χ_{-} ⟩ ⟨ χ_{-} ∣ \otimes η_{S}$ have the same output:

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

A left inverse would have to map this single output to both inputs, which is impossible.

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unlabeled · line 1037
$∣ χ_{\pm} ⟩ = \frac{∣ τ _{0} ⟩ \pm ∣ τ _{1} ⟩}{2},$
unlabeled · line 1045
$\frac{1}{2} (∣ 0 ⟩ ⟨ 0 ∣_{R} + ∣ 1 ⟩ ⟨ 1 ∣_{R}) \otimes η_{S} .$

Proposition: No universal quantum return from the readout

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