Manuscript En07-B@1 · En07-B.tex · lemma · 45dff4a1

Lemma: Readout-and-drop equals partial trace

Enter, forgetting, and surviving quantum registers · lines 618–655 · proof included or linked

En07-B@1#lem:erasure-interface

For every operator $X$ on $A \otimes B$ ,

(id_{A} \otimes Drop_{K}) (id_{A} \otimes Enter_{B}) (X) = Tr_{B} X .

The equality holds with arbitrary external references. The coherent readout map

W_{B} : B ⟶ K \otimes E, W_{B} ∣ j ⟩ = ∣ j ⟩_{K} ∣ j ⟩_{E}

is an isometry and dilates $Enter_{B}$ .

Proof. Expand

X = j, k \sum X_{j k} \otimes ∣ j ⟩ ⟨ k ∣_{B} .

Enter retains the terms $X_{j j}$ and attaches orthogonal classical labels. Dropping the labels gives $\sum_{j} X_{j j} = Tr_{B} X$ .

Also $W_{B}^{†} W_{B} = I_{B}$ , and

(id_{A} \otimes W_{B}) X (id_{A} \otimes W_{B}^{†}) = j, k \sum X_{j k} \otimes ∣ j ⟩ ⟨ k ∣_{K} \otimes ∣ j ⟩ ⟨ k ∣_{E} .

Tracing $E$ gives Enter; tracing $K \otimes E$ gives $Tr_{B} X$ . A reference can be included in $A$ , so the proof already contains every referenced version.

Displayed formulas in this passage

unlabeled · line 621
$(id_{A} \otimes Drop_{K}) (id_{A} \otimes Enter_{B}) (X) = Tr_{B} X .$
unlabeled · line 629
$W_{B} : B ⟶ K \otimes E, W_{B} ∣ j ⟩ = ∣ j ⟩_{K} ∣ j ⟩_{E}$
unlabeled · line 638
$X = j, k \sum X_{j k} \otimes ∣ j ⟩ ⟨ k ∣_{B} .$
unlabeled · line 646
$(id_{A} \otimes W_{B}) X (id_{A} \otimes W_{B}^{†}) = j, k \sum X_{j k} \otimes ∣ j ⟩ ⟨ k ∣_{K} \otimes ∣ j ⟩ ⟨ k ∣_{E} .$

Lemma: Readout-and-drop equals partial trace

Displayed formulas in this passage

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