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

Complete restoration and exact outgoing recovery

Complete restoration and exact outgoing recovery · lines 477–498

En07-C@1#eq:choi-definition

This complete-output restoration is also documented in CheungEn07A,CheungEn07B,CheungEn04A,CheungHawk04A. It does not give a decoder on $R$ alone.

Fix a code basis ${∣ i ⟩}_{i = 1}^{d}$ and let

E_{ij} = ∣ i ⟩ ⟨ j ∣, ∣ Φ_{d} ⟩ = \frac{1}{d} i = 1 \sum^{d} ∣ i ⟩_{Q} ∣ i ⟩_{C} .

The normalized Choi state of a channel $Φ$ with input $C$ is

J_{Φ} = (id_{Q} \otimes Φ) (∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣) = \frac{1}{d} i, j \sum E_{ij}^{Q} \otimes Φ (E_{ij}) .

Displayed formulas in this passage

En07-C@1#eq:choi-definition · line 490
$J_{Φ} = (id_{Q} \otimes Φ) (∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣) = \frac{1}{d} i, j \sum E_{ij}^{Q} \otimes Φ (E_{ij}) .$
unlabeled · line 484
$E_{ij} = ∣ i ⟩ ⟨ j ∣, ∣ Φ_{d} ⟩ = \frac{1}{d} i = 1 \sum^{d} ∣ i ⟩_{Q} ∣ i ⟩_{C} .$