Manuscript Oai280-A@1 · Oai280-A.tex · section · 27038823

Complete channels and their complements

Stage 3: fusion, observable survival, and exact recovery › Complete channels and their complements · lines 918–945

Let $C ≅ C^{d}$ , $d \geq 2$ , and let

W : C ⟶ R \otimes B

be a finite-dimensional isometry. Define

N (X) = Tr_{B} (W X W^{†}), C (X) = Tr_{R} (W X W^{†}) .

The complete $B$ includes all unreturned registers and preparation purifiers.

Use

d_{tr} (ρ, η) = \frac{1}{2} ∥ ρ - η ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥Φ - Ψ ∥_{⋄} .

The diamond norm retains arbitrary external references Watrous. For a decoder $D : B (R) \to B (C)$ , put

e (D; N) = d_{⋄} (D N, id_{C}), ε_{rec} (N) = D in f e (D; N) .

Displayed formulas in this passage

unlabeled · line 921
$W : C ⟶ R \otimes B$
unlabeled · line 926
$N (X) = Tr_{B} (W X W^{†}), C (X) = Tr_{R} (W X W^{†}) .$
unlabeled · line 934
$d_{tr} (ρ, η) = \frac{1}{2} ∥ ρ - η ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥Φ - Ψ ∥_{⋄} .$
unlabeled · line 941
$e (D; N) = d_{⋄} (D N, id_{C}), ε_{rec} (N) = D in f e (D; N) .$