Manuscript X-D-ST@1 · X-D-ST.tex · theorem · e0cf8fdb

Theorem: Complete clock instrument

Finite Enter encoding of supplied limiting evaluations › Complete destructive-clock channel · lines 2182–2216 · proof included or linked

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

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The channel

M (X) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} X D_{k}^{†} = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C_{clk}} [(P_{k} \otimes I_{S}) X (P_{k} \otimes I_{S})]

is CPTP and satisfies

Tr_{R} M (X) = Tr_{C_{clk}} X .

The identities hold on all operators after adjoining any untouched reference. The measured clock names are consumed; no quantum clock factor is returned.

Proof. The recorded Kraus maps are $∣ k ⟩_{R} \otimes D_{k}$ . Their squared moduli sum to

k \sum P_{k} \otimes I_{S} = I_{C_{clk} S} .

Expand an arbitrary operator as

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

The $k$ -th output block and the indicated projected partial trace both equal $X_{k k}$ . Summing proves the discard identity. The same calculation includes any reference factor.

Displayed formulas in this passage

X-D-ST@1#eq:clock · line 2185
$M (X) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} X D_{k}^{†} = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C_{clk}} [(P_{k} \otimes I_{S}) X (P_{k} \otimes I_{S})]$
unlabeled · line 2194
$Tr_{R} M (X) = Tr_{C_{clk}} X .$
unlabeled · line 2205
$k \sum P_{k} \otimes I_{S} = I_{C_{clk} S} .$
unlabeled · line 2209
$X = k, ℓ \sum ∣ τ_{k} ⟩ ⟨ τ_{ℓ} ∣ \otimes X_{k ℓ} .$

Theorem: Complete clock instrument

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