Manuscript Hawk01-B@1 · Hawk01-B.tex · section · cfa976f1

Single-reference diagnostics

Exact and approximate residual-leakage certificates › Single-reference diagnostics · lines 518–553

Hawk01-B@1#eq:Bellleakage, Hawk01-B@1#eq:MIconversion

Thus $F_{e} = 1$ proves exact channel identity, but approximate Bell fidelity alone does not give a dimension-independent worst-case guarantee.

For the complementary Bell state

ω^{S F} = (id_{S} \otimes C) (∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣),

put

μ = D_{tr} (ω^{S F}, \frac{I _{S}}{d} \otimes ω^{F}) .

Then (eq:Choiconversion) gives

δ_{leak} \leq min {1, d μ} .

If only its mutual information is known, quantum Pinsker's inequality in bit units gives

μ \leq \frac{ln 2}{2} I (S : F), δ_{leak} \leq min {1, d \frac{ln 2}{2} I (S : F)} .

These are diagnostic-to-channel conversions, not replacements for the diamond metric.

Displayed formulas in this passage

Hawk01-B@1#eq:Bellleakage · line 535
$δ_{leak} \leq min {1, d μ} .$
Hawk01-B@1#eq:MIconversion · line 541
$μ \leq \frac{ln 2}{2} I (S : F), δ_{leak} \leq min {1, d \frac{ln 2}{2} I (S : F)} .$
unlabeled · line 525
$ω^{S F} = (id_{S} \otimes C) (∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣),$
unlabeled · line 530
$μ = D_{tr} (ω^{S F}, \frac{I _{S}}{d} \otimes ω^{F}) .$