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

Lemma: Residual distinguishability witnesses

Approximate recovery and finite certificates › Dimension-explicit Choi certificates · lines 1195–1222 · proof included or linked

En07-B@1#lem:witness

If two code states satisfy

x = d_{tr} (E (ρ_{0}), E (ρ_{1})),

then $δ_{full} (V) \geq x /2$ . If $O = O^{†}$ , $∥ O ∥ \leq 1$ , and

w = Tr (O E (ρ_{0} - ρ_{1})),

then $δ_{full} (V) \geq w /4$ .

Proof. For every $σ$ , the triangle inequality gives

x \leq d_{tr} (E (ρ_{0}), σ) + d_{tr} (σ, E (ρ_{1})) \leq 2 d_{⋄} (E, T_{σ}) .

Take the infimum. Also $w \leq ∥ E (ρ_{0} - ρ_{1}) ∥_{1} = 2 x$ .

Displayed formulas in this passage

unlabeled · line 1198
$x = d_{tr} (E (ρ_{0}), E (ρ_{1})),$
unlabeled · line 1203
$w = Tr (O E (ρ_{0} - ρ_{1})),$
unlabeled · line 1211
$x \leq d_{tr} (E (ρ_{0}), σ) + d_{tr} (σ, E (ρ_{1})) \leq 2 d_{⋄} (E, T_{σ}) .$