Manuscript X-H-ST@1 · X-H-ST.tex · theorem · 943aa27e

Theorem: Complete supplied-residual instrument

Complete destructive instruments for the two dictionaries › The classical residual-report instrument · lines 1491–1537 · proof included or linked

X-H-ST@1#thm:instrument, X-H-ST@1#eq:drop

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The map (eq:instrument) is CPTP and

Tr_{R} M_{cl} (X) = Tr_{C} X .

The identities hold on all input operators and after adjoining any untouched reference $E$ .

For clock preparation $∣ τ_{j} ⟩ ⟨ τ_{j} ∣_{C} \otimes ρ_{A E}$ , the output is

∣ j, q_{j} ⟩ ⟨ j, q_{j} ∣_{R} \otimes ρ_{A E} .

For an arbitrary correlated input, the returned quantum block is the subnormalized conditional block $D_{j} X D_{j}^{†}$ , not an assumed unchanged conditional state.

The clock factor $C$ is consumed from the returned quantum interface.

Proof. A Kraus family is

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 $X = \sum_{j, l} ∣ τ_{j} ⟩ ⟨ τ_{l} ∣ \otimes X_{j l}$ . The $j$ -block in (eq:instrument) is $X_{j j}$ ; their sum is $Tr_{C} X$ . The same expansion with $A \otimes E$ proves the reference identities and the basis-input formula.

The quantum codomain is $A$ , not $C \otimes A$ . The measured clock names are therefore unavailable as quantum inputs to a continuation through this interface.

Displayed formulas in this passage

X-H-ST@1#eq:drop · line 1494
$Tr_{R} M_{cl} (X) = Tr_{C} X .$
unlabeled · line 1503
$∣ j, q_{j} ⟩ ⟨ j, q_{j} ∣_{R} \otimes ρ_{A E} .$
unlabeled · line 1517
$K_{j} = ∣ j, q_{j} ⟩_{R} \otimes D_{j}, j \sum K_{j}^{†} K_{j} = j \sum ∣ τ_{j} ⟩ ⟨ τ_{j} ∣_{C} \otimes I_{A} = I .$

Theorem: Complete supplied-residual instrument

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