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

Local preservation and exact full-code recovery

Local preservation and exact full-code recovery · lines 376–396

The domain of $L_{V}$ is $R \otimes B$ . It is a complete-access quantum decoder, not generally a decoder on $R$ . Its off-range completion does not make it a unitary inverse on all of $R \otimes B$ . Compare the corresponding restore maps in CheungHawk04A (Theorem thm:global) and CheungEn07A (Theorem thm:global).

Fix a code basis and let

E_{ij} = ∣ i ⟩ ⟨ j ∣, J_{n} (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

unlabeled · line 388
$E_{ij} = ∣ i ⟩ ⟨ j ∣, J_{n} (Φ) = (id_{Q} \otimes Φ) (∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣) = \frac{1}{d} i, j \sum E_{ij}^{Q} \otimes Φ (E_{ij}) .$

Nearby in this section

theorem: Theorem: Complete-output restoration
theorem: Theorem: Exact complete-complement criterion
section: Local preservation and exact full-code recovery
corollary: Corollary: The precise no-hiding conclusion
corollary: Corollary: A strict finite-dimensional obstruction
remark: Remark: The information target matters