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

Proposition: Complete-complement and declared-decoder certificates

Approximate recovery and finite certificates › Dimension-explicit Choi certificates · lines 1146–1187 · proof included or linked

En07-B@1#prop:certificates

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Define

χ_{B} (V) = σ \in S (B) in f \frac{1}{2} ∥ J_{n} (E) - I_{Q} / d \otimes σ ∥_{1} .

Then

χ_{B} (V) \leq δ_{full} (V) \leq min {1, d χ_{B} (V)} .

For a declared decoder $D$ , define

r_{D} = \frac{1}{2} ∥ J_{n} (DN) - P_{Φ_{d}} ∥_{1} .

Then

r_{D} \leq ε (D; N) \leq min {1, d r_{D}} .

In particular,

ε_{rec} (N) \leq min {1, 2 d χ_{B} (V)} .

Proof. Apply Lemma lem:choi to $E$ and each replacement channel, whose normalized Choi operator is $I_{Q} / d \otimes σ$ , and take infima. Apply it also to $DN$ and $id_{C}$ . The last estimate follows from Theorem thm:approx.

Displayed formulas in this passage

unlabeled · line 1149
$χ_{B} (V) = σ \in S (B) in f \frac{1}{2} ∥ J_{n} (E) - I_{Q} / d \otimes σ ∥_{1} .$
unlabeled · line 1157
$χ_{B} (V) \leq δ_{full} (V) \leq min {1, d χ_{B} (V)} .$
unlabeled · line 1162
$r_{D} = \frac{1}{2} ∥ J_{n} (DN) - P_{Φ_{d}} ∥_{1} .$
unlabeled · line 1168
$r_{D} \leq ε (D; N) \leq min {1, d r_{D}} .$
unlabeled · line 1174
$ε_{rec} (N) \leq min {1, 2 d χ_{B} (V)} .$

Proposition: Complete-complement and declared-decoder certificates

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