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

A failure-retaining amplitude instrument

Section C: complete destructive instruments and the positive gap › A failure-retaining amplitude instrument · lines 1181–1205

X-J-ST@1#eq:Mplus, X-J-ST@1#eq:Mminus, X-J-ST@1#eq:MF

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Let $Q ≅ C^{2}$ be the signal register, and let $A_{fin}$ contain every unmeasured finite system. Fix $0 \leq r \leq 1$ . On operators on $Q \otimes A_{fin}$ , define

M_{+} (Z) M_{-} (Z) M_{F} (Z) = (1 - r) (⟨ 1 ∣ \otimes I) Z (∣ 1 ⟩ \otimes I), = (1 - r) (⟨ 0 ∣ \otimes I) Z (∣ 0 ⟩ \otimes I), = r Tr_{Q} Z .

The complete output is

M (Z) = y \in {F, +, -} \sum ∣ y ⟩ ⟨ y ∣_{R} \otimes M_{y} (Z) .

The failure branch is retained. Its two internal destructive outcomes are summed, not postselected. No raw failure bit is exported at this declared boundary.

Displayed formulas in this passage

X-J-ST@1#eq:Mplus · line 1187
$M_{+} (Z) M_{-} (Z) M_{F} (Z) = (1 - r) (⟨ 1 ∣ \otimes I) Z (∣ 1 ⟩ \otimes I), = (1 - r) (⟨ 0 ∣ \otimes I) Z (∣ 0 ⟩ \otimes I), = r Tr_{Q} Z .$
X-J-ST@1#eq:Mminus · line 1187
$M_{+} (Z) M_{-} (Z) M_{F} (Z) = (1 - r) (⟨ 1 ∣ \otimes I) Z (∣ 1 ⟩ \otimes I), = (1 - r) (⟨ 0 ∣ \otimes I) Z (∣ 0 ⟩ \otimes I), = r Tr_{Q} Z .$
X-J-ST@1#eq:MF · line 1187
$M_{+} (Z) M_{-} (Z) M_{F} (Z) = (1 - r) (⟨ 1 ∣ \otimes I) Z (∣ 1 ⟩ \otimes I), = (1 - r) (⟨ 0 ∣ \otimes I) Z (∣ 0 ⟩ \otimes I), = r Tr_{Q} Z .$
unlabeled · line 1198
$M (Z) = y \in {F, +, -} \sum ∣ y ⟩ ⟨ y ∣_{R} \otimes M_{y} (Z) .$

A failure-retaining amplitude instrument

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