Manuscript Em03-H@8 · Em03-H-v8.tex · section · 91e9e3bd

Exact signed extraction and its reached space

Exact signed extraction and its reached space · lines 801–817

Set

W_{st} x = V_{st} x \otimes ∣ \emptyset ⟩_{R}, Z_{u} = R_{R} U_{ext, m} (u) W_{st} .

The full-code channels used below are

E_{u} (E_{α β}) = Tr_{B} ∣ Z_{u} α ⟩ ⟨ Z_{u} β ∣, E_{u}^{c} (E_{α β}) = Tr_{R} ∣ Z_{u} α ⟩ ⟨ Z_{u} β ∣ .

For even inputs, an additional branch phase cancels in the accessible channel. It must not be omitted on general parity-coherent inputs.

Displayed formulas in this passage

unlabeled · line 804
$W_{st} x = V_{st} x \otimes ∣ \emptyset ⟩_{R}, Z_{u} = R_{R} U_{ext, m} (u) W_{st} .$
unlabeled · line 809
$E_{u} (E_{α β}) = Tr_{B} ∣ Z_{u} α ⟩ ⟨ Z_{u} β ∣, E_{u}^{c} (E_{α β}) = Tr_{R} ∣ Z_{u} α ⟩ ⟨ Z_{u} β ∣ .$