Manuscript Hawk05-C@1 · Hawk05-C.tex · section · ba0719f1

A precise ensemble comparison

Typicality and physical transfer require different allowances › A precise ensemble comparison · lines 767–794

Hawk05-C@1#eq:moment

Total variation is a strong sufficient assumption, not a claim that physical scrambling achieves it.

A weaker hypothesis suffices for a leading capacity law. For a fixed pure input, let

M_{2}^{μ} = E_{μ} ∣ v ⟩ ⟨ v ∣^{\otimes 2}, M_{2}^{μ} - \frac{I + F}{D ( D + 1 )}_{1} \leq χ .

The purity swap witness has operator norm one, so

E_{μ} Tr ρ_{m}^{2} \leq \frac{m + n}{mn + 1} + χ .

The proof of Proposition prop:deficits then gives

E_{μ} (lo g_{2} m - S (R)) \leq min {lo g_{2} m, lo g_{2} [m (\frac{m + n}{mn + 1} + χ)]} .

Two-copy information can control this leading deficit; it does not by itself establish either exact entropy mean.

Displayed formulas in this passage

Hawk05-C@1#eq:moment · line 785
$E_{μ} (lo g_{2} m - S (R)) \leq min {lo g_{2} m, lo g_{2} [m (\frac{m + n}{mn + 1} + χ)]} .$
unlabeled · line 774
$M_{2}^{μ} = E_{μ} ∣ v ⟩ ⟨ v ∣^{\otimes 2}, M_{2}^{μ} - \frac{I + F}{D ( D + 1 )}_{1} \leq χ .$
unlabeled · line 780
$E_{μ} Tr ρ_{m}^{2} \leq \frac{m + n}{mn + 1} + χ .$