Manuscript X-C-ST@1 · X-C-ST.tex · theorem · b679954e

Theorem: Complete clock instrument

Finite installed encoding and the clock instrument › Complete destructive readout · lines 2217–2255 · proof included or linked

X-C-ST@1#thm:instrument, X-C-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_{S}) X (P_{k} \otimes I_{S})]

is CPTP and satisfies

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

All identities hold after adjoining any untouched reference. The clock names are consumed; there is no quantum $C$ -output.

Proof. The recorded Kraus maps are $∣ k ⟩_{R} \otimes D_{k}$ , and

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

This proves complete positivity and trace preservation.

For an arbitrary input operator,

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

the $k$ -th output block is $X_{k k}$ . It is also the indicated clock partial trace. Summing the blocks proves the discard identity. The calculation is unchanged with a reference tensor factor retained.

Displayed formulas in this passage

X-C-ST@1#eq:clock · line 2220
$M_{C} (X) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} X D_{k}^{†} = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C} [(P_{k} \otimes I_{S}) X (P_{k} \otimes I_{S})]$
unlabeled · line 2228
$Tr_{R} M_{C} (X) = Tr_{C} X .$
unlabeled · line 2239
$k \sum (∣ k ⟩_{R} \otimes D_{k})^{†} (∣ k ⟩_{R} \otimes D_{k}) = k \sum P_{k} \otimes I_{S} = I_{C S} .$
unlabeled · line 2247
$X = k, ℓ \sum ∣ τ_{k} ⟩ ⟨ τ_{ℓ} ∣ \otimes X_{k ℓ},$

Theorem: Complete clock instrument

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