Manuscript Hawk04-C@1 · Hawk04-C-v1.tex · theorem · 94d46352

Theorem: Residual readout-and-drop is partial trace

Row 5: the final-burst and no-remnant analogy › The additional terminal program · lines 1071–1114 · proof included or linked

Hawk04-C@1#thm:drop, Hawk04-C@1#eq:residual-drop, Hawk04-C@1#eq:forget-channel, Hawk04-C@1#eq:forget-accessible

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For every state on $Q \otimes R \otimes B$ ,

(id_{QR} \otimes dropBit_{K}) (id_{QR} \otimes M_{B}) (ρ) = Tr_{B} ρ,

where the identities in the displayed composition act on the untouched factors. Equivalently, as a channel from $R \otimes B$ to $R$ ,

P_{forget B} = dropBit_{K} \circ Enter_{B} = Tr_{B} .

Consequently,

P_{forget B} V = N .

The radiation-only recovery error is exactly the pre-existing $ε_{rec} (N)$ .

Proof. The left side of (eq:residual-drop) is

z \sum (I_{QR} \otimes ⟨ z ∣_{B}) ρ (I_{QR} \otimes ∣ z ⟩_{B}),

which is the defining partial trace. Substitute $ρ = (id_{Q} \otimes V) (X)$ to obtain (eq:forget-accessible) with every reference. The optimization defining recovery error depends only on the unchanged channel $N$ , so its value is unchanged.

Displayed formulas in this passage

Hawk04-C@1#eq:residual-drop · line 1074
$(id_{QR} \otimes dropBit_{K}) (id_{QR} \otimes M_{B}) (ρ) = Tr_{B} ρ,$
Hawk04-C@1#eq:forget-channel · line 1084
$P_{forget B} = dropBit_{K} \circ Enter_{B} = Tr_{B} .$
Hawk04-C@1#eq:forget-accessible · line 1093
$P_{forget B} V = N .$
unlabeled · line 1103
$z \sum (I_{QR} \otimes ⟨ z ∣_{B}) ρ (I_{QR} \otimes ∣ z ⟩_{B}),$

Theorem: Residual readout-and-drop is partial trace

Displayed formulas in this passage

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