Manuscript X-J-ST@1 · X-J-ST.tex · theorem · 8f0d1a36

Theorem: Complete signal instrument

Section C: complete destructive instruments and the positive gap › A failure-retaining amplitude instrument · lines 1206–1234 · proof included or linked

X-J-ST@1#thm:signal, X-J-ST@1#eq:signaldrop

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The instrument (eq:Mplus)–(eq:MF) is complete. It satisfies, for every operator and every untouched reference $E$ ,

Tr_{R} [(M \otimes id_{E}) (Z)] = Tr_{Q} Z .

The measured signal name $Q$ is consumed. The quantum output consists of $A_{fin}$ , with its conditional reference correlations.

Proof. The Kraus maps are

1 - r (⟨ 1 ∣ \otimes I), 1 - r (⟨ 0 ∣ \otimes I), r (⟨ 0 ∣ \otimes I), r (⟨ 1 ∣ \otimes I),

with the last two assigned the common outcome $F$ . Their adjoint products sum to the identity. Their sum after discarding the record is the partial trace over $Q$ . Both statements remain true after tensoring with $I_{E}$ . The output types give consumption.

Displayed formulas in this passage

X-J-ST@1#eq:signaldrop · line 1211
$Tr_{R} [(M \otimes id_{E}) (Z)] = Tr_{Q} Z .$
unlabeled · line 1222
$1 - r (⟨ 1 ∣ \otimes I), 1 - r (⟨ 0 ∣ \otimes I), r (⟨ 0 ∣ \otimes I), r (⟨ 1 ∣ \otimes I),$

Theorem: Complete signal instrument

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