Manuscript Oai265-A@2 · Oai265-A.tex · lemma · c0a34a45

Lemma: Elementary replica identities

Stage 1: audit of the five proof interfaces › Replica identities: exact algebra versus quantitative transport · lines 496–525 · proof included or linked

Oai265-A@2#lem:replica

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For a density matrix $ρ$ , $p > 0$ , and $n \geq 1$ ,

\frac{1}{n} lo g Tr [(ρ^{\otimes n})^{p}] = lo g Tr (ρ^{p}),

and

- \frac{d}{d p} lo g Tr (ρ^{p})_{p = 1} = S (ρ) .

For a bipartite pure vector with marginal $ρ_{X}$ ,

∥ (ρ_{X}^{(p - 1) /2} \otimes I) Ω ∥^{2} = Tr (ρ_{X}^{p}),

where inverse powers act only on the positive spectral support. Also,

I (X^{n} : Y^{n} ∣ Z^{n})_{ρ^{\otimes n}} = n I (X : Y ∣ Z)_{ρ} .

Proof. Diagonalize $ρ$ . Tensor-product eigenvalues multiply and entropies add. The derivative follows by differentiating the finite sum over positive eigenvalues. The vector-norm identity follows from the Schmidt decomposition. Substitution into the four entropy terms proves conditional-information additivity.

Displayed formulas in this passage

unlabeled · line 499
$\frac{1}{n} lo g Tr [(ρ^{\otimes n})^{p}] = lo g Tr (ρ^{p}),$
unlabeled · line 504
$- \frac{d}{d p} lo g Tr (ρ^{p})_{p = 1} = S (ρ) .$
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$∥ (ρ_{X}^{(p - 1) /2} \otimes I) Ω ∥^{2} = Tr (ρ_{X}^{p}),$
unlabeled · line 513
$I (X^{n} : Y^{n} ∣ Z^{n})_{ρ^{\otimes n}} = n I (X : Y ∣ Z)_{ρ} .$

Lemma: Elementary replica identities

Displayed formulas in this passage

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