Manuscript X-Enter-ST@1 · X-Enter-ST.tex · theorem · f76acc1f

Theorem: Complete supplied-evaluation instrument

Stage D: complete instruments for supplied evaluations › The destructive task clock · lines 912–954 · proof included or linked

X-Enter-ST@1#thm:clock, X-Enter-ST@1#eq:drop

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The map (eq:clockinstrument) is CPTP and

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

on every input operator. Both identities hold after tensoring with $id_{E}$ for every untouched reference.

For a basis-task input,

(M \otimes id_{E}) (∣ τ_{j} ⟩ ⟨ τ_{j} ∣_{C} \otimes ρ_{A E}) = ∣ j, q_{j} ⟩ ⟨ j, q_{j} ∣_{R} \otimes ρ_{A E} .

For a general correlated input the returned blocks are the actual subnormalized conditional operators $(D_{j} \otimes I_{E}) X (D_{j}^{†} \otimes I_{E})$ .

The returned quantum interface contains $A$ , not $C$ .

Proof. The recorded Kraus maps are

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

This proves complete positivity and trace preservation.

Expand

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

The $j$ -th output block is $X_{j j}$ . Summing these blocks gives $Tr_{C} X$ , proving (eq:drop). The same expansion with $A \otimes E$ proves every reference identity. The basis-input statement is its special case. The quantum codomain specifies consumption of $C$ .

Displayed formulas in this passage

X-Enter-ST@1#eq:drop · line 915
$Tr_{R} M (X) = Tr_{C} X$
unlabeled · line 923
$(M \otimes id_{E}) (∣ τ_{j} ⟩ ⟨ τ_{j} ∣_{C} \otimes ρ_{A E}) = ∣ j, q_{j} ⟩ ⟨ j, q_{j} ∣_{R} \otimes ρ_{A E} .$
unlabeled · line 937
$K_{j} = ∣ j, q_{j} ⟩_{R} \otimes D_{j}, j \sum K_{j}^{†} K_{j} = j \sum ∣ τ_{j} ⟩ ⟨ τ_{j} ∣_{C} \otimes I_{A} = I .$
unlabeled · line 945
$X = j, l \sum ∣ τ_{j} ⟩ ⟨ τ_{l} ∣_{C} \otimes X_{j l} .$

Theorem: Complete supplied-evaluation instrument

Displayed formulas in this passage

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