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

Proposition: A complete dilation of readout and forgetting

Row 5: the final-burst and no-remnant analogy › A coherent implementation does not annihilate information · lines 1124–1158 · proof included or linked

Hawk04-C@1#prop:dilation, Hawk04-C@1#eq:readout-dilation, Hawk04-C@1#eq:forget-dilation

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For a basis ${∣ z ⟩}$ of $B$ , the map

W ∣ z ⟩_{B} = ∣ z ⟩_{K} ∣ z ⟩_{E}

is an isometry. Tracing $E$ gives the destructive classical readout channel. If discarding $K$ is represented by transferring it to an inaccessible store $F$ , the resulting complete isometry can be written

J ∣ z ⟩_{B} = ∣ z ⟩_{E} ∣ z ⟩_{F} .

Tracing $E \otimes F$ gives $Tr_{B}$ , while joint $E F$ access restores the original code on $B$ .

Proof. Distinct basis vectors in either formula have orthogonal images, so $W^{†} W = J^{†} J = I_{B}$ . For $X = \sum_{z, w} X_{z w} ∣ z ⟩ ⟨ w ∣$ ,

Tr_{E} (W X W^{†}) = z \sum X_{z z} ∣ z ⟩ ⟨ z ∣_{K} .

Likewise, tracing both outputs of $J X J^{†}$ gives $\sum_{z} X_{z z} = Tr X$ . For a joint state on $QR B$ , the same calculation gives $Tr_{B}$ while leaving $QR$ untouched. The complete left inverse follows from Theorem thm:global, applied to $J$ .

Displayed formulas in this passage

Hawk04-C@1#eq:readout-dilation · line 1127
$W ∣ z ⟩_{B} = ∣ z ⟩_{K} ∣ z ⟩_{E}$
Hawk04-C@1#eq:forget-dilation · line 1135
$J ∣ z ⟩_{B} = ∣ z ⟩_{E} ∣ z ⟩_{F} .$
unlabeled · line 1147
$Tr_{E} (W X W^{†}) = z \sum X_{z z} ∣ z ⟩ ⟨ z ∣_{K} .$

Proposition: A complete dilation of readout and forgetting

Displayed formulas in this passage

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