Manuscript En07-C@1 · En07-C.tex · lemma · 3f7e9ebb

Lemma: Enter followed by Drop equals partial trace

Destructive Enter, forgetting, and classical records · lines 687–725 · proof included or linked

En07-C@1#lem:erasure-interface, En07-C@1#eq:enter-drop, En07-C@1#eq:enter-dilation

On all operators, including all referenced operators,

Drop_{Z} \circ Enter_{B} = Tr_{B} .

A complete coherent dilation is the isometry

W (∣ r ⟩_{R} ∣ z ⟩_{B}) = ∣ r ⟩_{R} ∣ z ⟩_{Z} ∣ z ⟩_{F} .

It satisfies

W^{†} W = I_{R B}, Tr_{F} Ad_{W} = Enter_{B} .

Proof. Write

X = z, z^{'} \sum X_{z z^{'}} \otimes ∣ z ⟩ ⟨ z^{'} ∣_{B} .

Then

Enter_{B} (X) = z \sum X_{z z} \otimes ∣ z ⟩ ⟨ z ∣_{Z}, Drop_{Z} Enter_{B} (X) = z \sum X_{z z} = Tr_{B} X .

The same block expansion applies with an untouched reference. The vectors on the right side of (eq:enter-dilation) preserve inner products of basis vectors, so $W$ is an isometry. Tracing $F$ removes terms with $z \neq = z^{'}$ and gives (eq:enter).

Displayed formulas in this passage

En07-C@1#eq:enter-drop · line 690
$Drop_{Z} \circ Enter_{B} = Tr_{B} .$
En07-C@1#eq:enter-dilation · line 695
$W (∣ r ⟩_{R} ∣ z ⟩_{B}) = ∣ r ⟩_{R} ∣ z ⟩_{Z} ∣ z ⟩_{F} .$
unlabeled · line 702
$W^{†} W = I_{R B}, Tr_{F} Ad_{W} = Enter_{B} .$
unlabeled · line 711
$X = z, z^{'} \sum X_{z z^{'}} \otimes ∣ z ⟩ ⟨ z^{'} ∣_{B} .$
unlabeled · line 715
$Enter_{B} (X) = z \sum X_{z z} \otimes ∣ z ⟩ ⟨ z ∣_{Z}, Drop_{Z} Enter_{B} (X) = z \sum X_{z z} = Tr_{B} X .$

Lemma: Enter followed by Drop equals partial trace

Displayed formulas in this passage

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