Manuscript X-K-ST@1 · X-K-ST.tex · theorem · 93c39c0f

Theorem: Complete destructive-clock identity

One destructive clock and its exact limitations · lines 510–552 · proof included or linked

X-K-ST@1#thm:oneclock, X-K-ST@1#eq:drop

Equation (eq:oneclock) is CPTP and equals

k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C} [(P_{k} \otimes I_{S}) X (P_{k} \otimes I_{S})]

on every input operator. For every reference extension,

Tr_{R} [(M_{C} \otimes id_{E}) (X)] = Tr_{C} X .

For a state $ρ$ , the $k$ -block is

σ_{k} = D_{k} ρ D_{k}^{†}, p_{k} = Tr σ_{k} .

Its normalized conditional state is $σ_{k} / p_{k}$ only when $p_{k} > 0$ . No normalized state is assigned to a zero block.

The output quantum context contains $S$ , not the measured names in $C$ .

Proof. The recorded Kraus maps are

∣ k ⟩_{R} \otimes D_{k}, k \sum D_{k}^{†} D_{k} = k \sum P_{k} \otimes I_{S} = I_{C S} .

This proves complete positivity and trace preservation. Expand

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

Both formulas return the $k k$ -block. Their sum is $Tr_{C} X$ , proving (eq:drop). Replacing $I_{S}$ by $I_{S} \otimes I_{E}$ proves the reference statements. The output type gives consumption.

Displayed formulas in this passage

X-K-ST@1#eq:drop · line 518
$Tr_{R} [(M_{C} \otimes id_{E}) (X)] = Tr_{C} X .$
unlabeled · line 513
$k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C} [(P_{k} \otimes I_{S}) X (P_{k} \otimes I_{S})]$
unlabeled · line 524
$σ_{k} = D_{k} ρ D_{k}^{†}, p_{k} = Tr σ_{k} .$
unlabeled · line 537
$∣ k ⟩_{R} \otimes D_{k}, k \sum D_{k}^{†} D_{k} = k \sum P_{k} \otimes I_{S} = I_{C S} .$
unlabeled · line 545
$X = k, l \sum ∣ τ_{k} ⟩ ⟨ τ_{l} ∣ \otimes X_{k l} .$

Theorem: Complete destructive-clock identity

Displayed formulas in this passage

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