Manuscript X-A-ST@1 · X-A-ST.tex · theorem · d9ccbde1

Theorem: Clock instrument

A finite Enter interface for supplied evaluations › Complete destructive-clock instrument · lines 1937–1962 · proof included or linked

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

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The map

M_{C} (X) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} X D_{k}^{†} = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C} [(P_{k} \otimes I) X (P_{k} \otimes I)]

is completely positive and trace preserving. It obeys

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

on all operators, including every untouched-reference extension. Its quantum output contains $S$ , not $C$ .

Proof. The recorded Kraus operators are $∣ k ⟩_{R} \otimes D_{k}$ ; their squared moduli sum to $I_{C S}$ . Writing $X = \sum_{k, ℓ} ∣ τ_{k} ⟩ ⟨ τ_{ℓ} ∣ \otimes X_{k ℓ}$ identifies each output block with $X_{k k}$ . Summation proves the discard identity. The same calculation includes any reference factor.

Displayed formulas in this passage

X-A-ST@1#eq:clock · line 1940
$M_{C} (X) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} X D_{k}^{†} = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C} [(P_{k} \otimes I) X (P_{k} \otimes I)]$
unlabeled · line 1948
$Tr_{R} M_{C} (X) = Tr_{C} X$

Theorem: Clock instrument

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