Manuscript En07-A@1 · En07-A.tex · lemma · 44229ce9

Lemma: Erasure-interface identity and coherent dilation

Enter, Drop, and the complete apparatus ledger › Destructive partial readout · lines 690–737 · proof included or linked

En07-A@1#lem:erasure-interface, En07-A@1#eq:drop, En07-A@1#eq:read-dilation

For a finite encoded residual $B$ , destructive readout followed by discard equals partial trace:

(id_{QR} \otimes (Drop \circ Enter)) (ρ_{QR B}) = Tr_{B} ρ_{QR B} .

Consequently,

(id_{R} \otimes (Drop \circ Enter)) V = N .

For the instrument (eq:instrument), the map

W_{read} ∣ ψ ⟩ = x \sum ∣ x ⟩_{K} ∣ x ⟩_{E} \otimes M_{x} ∣ ψ ⟩

is an isometry. Tracing $E$ gives the observable instrument. Its complete coherent output $K E T$ preserves the input, whereas the classical interface alone generally does not.

Proof. The diagonal $B$ -block sum is exactly partial trace:

x \sum (I_{QR} \otimes ⟨ x ∣) ρ_{QR B} (I_{QR} \otimes ∣ x ⟩) = Tr_{B} ρ_{QR B} .

This identity holds for every joint state and therefore with arbitrary references.

For the dilation,

W_{read}^{†} W_{read} = x \sum M_{x}^{†} M_{x} = I .

Tracing the orthogonal $E$ -labels removes cross-outcome terms and gives (eq:instrument). Complete restoration follows from Theorem thm:global.

Displayed formulas in this passage

En07-A@1#eq:drop · line 694
$(id_{QR} \otimes (Drop \circ Enter)) (ρ_{QR B}) = Tr_{B} ρ_{QR B} .$
En07-A@1#eq:read-dilation · line 705
$W_{read} ∣ ψ ⟩ = x \sum ∣ x ⟩_{K} ∣ x ⟩_{E} \otimes M_{x} ∣ ψ ⟩$
unlabeled · line 700
$(id_{R} \otimes (Drop \circ Enter)) V = N .$
unlabeled · line 719
$x \sum (I_{QR} \otimes ⟨ x ∣) ρ_{QR B} (I_{QR} \otimes ∣ x ⟩) = Tr_{B} ρ_{QR B} .$
unlabeled · line 729
$W_{read}^{†} W_{read} = x \sum M_{x}^{†} M_{x} = I .$

Lemma: Erasure-interface identity and coherent dilation

Displayed formulas in this passage

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