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

Corollary: A complete-residual certificate

Quantitative certificates and the additional endpoint obligations › Complement proximity is an upper certificate · lines 1665–1700 · proof included or linked

Hawk04-C@1#cor:certificate, Hawk04-C@1#eq:certificate-leakage, Hawk04-C@1#eq:certificate-recovery

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Suppose a finite approximation $E$ has been placed in the same complete residual output space as $E$ by a specified isometric embedding, and a proved allowance gives

d_{⋄} (E, E) \leq b .

For a rigorously specified residual state $σ$ , put

e = \frac{1}{2} J_{n} (E) - I_{Q} / d \otimes σ_{1} .

Then

δ_{full} (E) ε_{rec} (N) \leq min {1, b + d e}, \leq min {1, 2 (b + d e)} .

Proof. Lemma lem:choi gives $d_{⋄} (E, T_{σ}) \leq d e$ . Add $b$ by the triangle inequality and take the replacement-state infimum. Equation (eq:ksw) proves the recovery bound.

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Hawk04-C@1#eq:certificate-leakage · line 1684
$δ_{full} (E) ε_{rec} (N) \leq min {1, b + d e}, \leq min {1, 2 (b + d e)} .$
Hawk04-C@1#eq:certificate-recovery · line 1684
$δ_{full} (E) ε_{rec} (N) \leq min {1, b + d e}, \leq min {1, 2 (b + d e)} .$
unlabeled · line 1670
$d_{⋄} (E, E) \leq b .$
unlabeled · line 1674
$e = \frac{1}{2} J_{n} (E) - I_{Q} / d \otimes σ_{1} .$

Corollary: A complete-residual certificate

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