Manuscript I-B-VP@3 · I-B-VP-v3.tex · proposition · 93c04eb4

Proposition: Complete destructive report instrument

Finite reporting, evidence grades, and integrity · lines 2736–2766 · proof included or linked

I-B-VP@3#prop:instrument

Let $C ≅ C^{D}$ , $D \geq 2$ , have calibrated orthonormal basis ${∣ τ_{j} ⟩}$ , and let $A$ contain all surviving finite registers. Install finite words $w_{j}$ , retaining $j$ even when words coincide. Put

D_{j} = ⟨ τ_{j} ∣_{C} \otimes I_{A}, M (X) = j \sum ∣ j, w_{j} ⟩ ⟨ j, w_{j} ∣_{R} \otimes D_{j} X D_{j}^{†} .

Then $M$ is completely positive and trace preserving,

Tr_{R} M (X) = Tr_{C} X,

and the identities persist with an untouched finite reference. There is no output quantum $C$ -register.

Proof. The Kraus maps are $L_{j} = ∣ j, w_{j} ⟩ \otimes D_{j}$ , and

j \sum L_{j}^{†} L_{j} = j \sum ∣ τ_{j} ⟩ ⟨ τ_{j} ∣ \otimes I_{A} = I .

Writing $X = \sum_{j k} ∣ τ_{j} ⟩ ⟨ τ_{k} ∣ \otimes X_{j k}$ , the output blocks are $X_{j j}$ , whose sum is $Tr_{C} X$ . The proof is unchanged with a reference factor.

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unlabeled · line 2743
$D_{j} = ⟨ τ_{j} ∣_{C} \otimes I_{A}, M (X) = j \sum ∣ j, w_{j} ⟩ ⟨ j, w_{j} ∣_{R} \otimes D_{j} X D_{j}^{†} .$
unlabeled · line 2748
$Tr_{R} M (X) = Tr_{C} X,$
unlabeled · line 2758
$j \sum L_{j}^{†} L_{j} = j \sum ∣ τ_{j} ⟩ ⟨ τ_{j} ∣ \otimes I_{A} = I .$

Proposition: Complete destructive report instrument

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