Manuscript Hawk02-E@1 · Hawk02-E.tex · theorem · 4841592b

Theorem: Certified residual and recovery bounds

Complete-residual certificates and decoder certificates › The dimension factor · lines 1204–1253 · proof included or linked

Hawk02-E@1#thm:certificate, Hawk02-E@1#eq:residual-certificate, Hawk02-E@1#eq:recovery-certificate

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Fix $t \leq T$ , and compute the finite complementary channel $C_{t}$ from $V_{t}^{(m, Δ)}$ . Let $σ_{B}$ be any rigorously specified finite residual density operator and put

e_{J} (t) = \frac{1}{2} J_{n} (C_{t}) - τ_{Q} \otimes σ_{B}_{1} .

Embed $σ_{B}$ into the complete residual space, with the unused modes in vacuum. Then

δ_{full} (t) ε_{rec} (t) \leq min {1, d e_{J} (t) + b_{reg}}, \leq min {1, 2 [d e_{J} (t) + b_{reg}]} .

If a validated calculation provides $e_{J} (t) \leq e_{cert} (t)$ , that certified upper bound may replace $e_{J} (t)$ .

Proof. Theorem thm:regulator gives

d_{⋄} (C_{t}, C_{t}) \leq b_{reg} .

The triangle inequality and Lemma lem:choi give

d_{⋄} (C_{t}, T_{σ_{B}}) \leq b_{reg} + d e_{J} (t) .

Taking the replacement-state infimum proves (eq:residual-certificate). Equation (eq:KSW) proves (eq:recovery-certificate).

Displayed formulas in this passage

Hawk02-E@1#eq:residual-certificate · line 1222
$δ_{full} (t) ε_{rec} (t) \leq min {1, d e_{J} (t) + b_{reg}}, \leq min {1, 2 [d e_{J} (t) + b_{reg}]} .$
Hawk02-E@1#eq:recovery-certificate · line 1222
$δ_{full} (t) ε_{rec} (t) \leq min {1, d e_{J} (t) + b_{reg}}, \leq min {1, 2 [d e_{J} (t) + b_{reg}]} .$
unlabeled · line 1211
$e_{J} (t) = \frac{1}{2} J_{n} (C_{t}) - τ_{Q} \otimes σ_{B}_{1} .$
unlabeled · line 1241
$d_{⋄} (C_{t}, C_{t}) \leq b_{reg} .$
unlabeled · line 1245
$d_{⋄} (C_{t}, T_{σ_{B}}) \leq b_{reg} + d e_{J} (t) .$

Theorem: Certified residual and recovery bounds

Displayed formulas in this passage

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