Manuscript En07-C@1 · En07-C.tex · lemma · 3f7e9ebb

Lemma: Residual distinguishability witnesses

Approximate complement certificates and finite tests › Finite matrix certificates · lines 1143–1177 · proof included or linked

En07-C@1#lem:witness

If two logical states satisfy

x = d_{tr} (E (ρ_{0}), E (ρ_{1})),

then

δ_{leak} (E) \geq x /2, ε_{rec} (N) \geq x^{2} /8.

If $O = O^{†}$ , $∥ O ∥_{\infty} \leq 1$ , and

w = ∣ Tr (O E (ρ_{0} - ρ_{1})) ∣,

then

δ_{leak} (E) \geq w /4, ε_{rec} (N) \geq w^{2} /32.

Proof. For every replacement state $σ$ ,

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

Minimize and use (eq:leak-lower). Trace duality gives $w \leq 2 x$ , proving the remaining statements.

Displayed formulas in this passage

unlabeled · line 1146
$x = d_{tr} (E (ρ_{0}), E (ρ_{1})),$
unlabeled · line 1150
$δ_{leak} (E) \geq x /2, ε_{rec} (N) \geq x^{2} /8.$
unlabeled · line 1156
$w = ∣ Tr (O E (ρ_{0} - ρ_{1})) ∣,$
unlabeled · line 1160
$δ_{leak} (E) \geq w /4, ε_{rec} (N) \geq w^{2} /32.$
unlabeled · line 1169
$x \leq d_{tr} (E (ρ_{0}), σ) + d_{tr} (σ, E (ρ_{1})) \leq 2 d_{⋄} (E, T_{σ}) .$