Manuscript I-B-G@2 · I-B-G-v2.tex · theorem · 3b4d700f

Theorem: Finite supplied-report instrument

A finite Enter catalogue of supplied evaluations › Complete destructive reporting · lines 2104–2143 · proof included or linked

I-B-G@2#thm:instrument, I-B-G@2#eq:instrument

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Let $C ≅ C^{D}$ , $D = 2^{q} \geq 2$ , have a calibrated orthonormal basis ${∣ τ_{j} ⟩}$ . Let $A$ contain every surviving finite register. Install finite report words $w_{j}$ , with explicit unused and failure labels, and put

D_{j} = ⟨ τ_{j} ∣_{C} \otimes I_{A} .

Then

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

is completely positive and trace preserving and obeys

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

These identities hold with every untouched finite reference. The quantum output contains $A$ , not $C$ .

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

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

Expanding

X = j, k \sum ∣ τ_{j} ⟩ ⟨ τ_{k} ∣ \otimes X_{j k}

shows that the $j$ -th output block is $X_{j j}$ . Summing those blocks gives the partial trace. Tensoring the calculation with a reference identity proves the complete statements.

Displayed formulas in this passage

I-B-G@2#eq:instrument · line 2115
$M (X) = j \sum ∣ j, w_{j} ⟩ ⟨ j, w_{j} ∣_{R} \otimes D_{j} X D_{j}^{†}$
unlabeled · line 2111
$D_{j} = ⟨ τ_{j} ∣_{C} \otimes I_{A} .$
unlabeled · line 2121
$Tr_{R} M (X) = Tr_{C} X .$
unlabeled · line 2131
$j \sum L_{j}^{†} L_{j} = j \sum ∣ τ_{j} ⟩ ⟨ τ_{j} ∣ \otimes I_{A} = I_{C A} .$
unlabeled · line 2136
$X = j, k \sum ∣ τ_{j} ⟩ ⟨ τ_{k} ∣ \otimes X_{j k}$

Theorem: Finite supplied-report instrument

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