Manuscript Oai290-A@1 · Oai290-A.tex · proposition · 7c6e104a

Proposition: A target-specific lower witness

A named observable is a different task › Observable-specific obstructions · lines 599–634 · proof included or linked

Oai290-A@1#prop:effectlower, Oai290-A@1#eq:effectlower, Oai290-A@1#eq:accesslower

For any two code states $ρ, σ$ , every boundary effect $G$ satisfies

η_{F} (G) \geq \frac{1}{2} max {0, ∣ Tr (F (ρ - σ)) ∣ - d_{tr} (N (ρ), N (σ))} .

For full-code recovery, every decoder satisfies

e (D N) \geq \frac{1}{2} max {0, d_{tr} (ρ, σ) - d_{tr} (N (ρ), N (σ))} .

Proof. Each target expectation differs from its reconstructed expectation by at most $η_{F} (G)$ . The difference between the two boundary-effect expectations is bounded by the boundary trace distance. The triangle inequality gives (eq:effectlower).

For (eq:accesslower), each decoded state is within $e (D N)$ of its ideal input. Apply the triangle inequality and contraction under $D$ .

Displayed formulas in this passage

Oai290-A@1#eq:effectlower · line 603
$η_{F} (G) \geq \frac{1}{2} max {0, ∣ Tr (F (ρ - σ)) ∣ - d_{tr} (N (ρ), N (σ))} .$
Oai290-A@1#eq:accesslower · line 613
$e (D N) \geq \frac{1}{2} max {0, d_{tr} (ρ, σ) - d_{tr} (N (ρ), N (σ))} .$

Nearby in this section

section: Observable-specific obstructions