Manuscript X-A-conv-ST@1 · X-A-conv-ST.tex · theorem · 8ca47088

Theorem: Complete supplied-report instrument

Complete instruments for supplied evaluations › The destructive task clock · lines 1650–1682 · proof included or linked

X-A-conv-ST@1#thm:instrument, X-A-conv-ST@1#eq:discard

Equation (eq:clock) is CPTP and satisfies

Tr_{R} M_{C} (Z) = Tr_{C} Z

on all operators, including every untouched-reference extension.

On a positive-probability branch, the normalized quantum output is its actual conditional $A$ -state. No normalized state is assigned to a zero-probability branch. The returned quantum interface contains $A$ , not $C$ .

Proof. The recorded Kraus maps are

L_{j} = ∣ j, q_{j} ⟩_{R} \otimes D_{j}, j \sum L_{j}^{†} L_{j} = I_{C A} .

This proves complete positivity and trace preservation. Expand

Z = j, l \sum ∣ τ_{j} ⟩ ⟨ τ_{l} ∣_{C} \otimes Z_{j l} .

Its $j$ -th output block is $Z_{j j}$ , and summing gives $Tr_{C} Z$ . The same calculation with $A \otimes E$ proves every reference extension and the conditional block statement. The codomain specifies consumption of $C$ .

Displayed formulas in this passage

X-A-conv-ST@1#eq:discard · line 1653
$Tr_{R} M_{C} (Z) = Tr_{C} Z$
unlabeled · line 1668
$L_{j} = ∣ j, q_{j} ⟩_{R} \otimes D_{j}, j \sum L_{j}^{†} L_{j} = I_{C A} .$
unlabeled · line 1674
$Z = j, l \sum ∣ τ_{j} ⟩ ⟨ τ_{l} ∣_{C} \otimes Z_{j l} .$

Theorem: Complete supplied-report instrument

Displayed formulas in this passage

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