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

Theorem: Nonuniform product-transfer calibration

Greybody weighting: what is calculable and what is declared › Probability-weighted transfer is a different exact proxy · lines 1345–1385 · proof included or linked

Hawk05-C@1#thm:product, Hawk05-C@1#eq:productentropy, Hawk05-C@1#eq:productrecovery

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For arbitrary input $ρ_{C}$ ,

S (R) = j = 1 \sum^{K} h_{2} (q_{j}) + S \sum w_{S} S (ρ_{C_{S}}) .

Its exact whole-product-code recovery error is

ε_{rec} (N) = 1 - j = 1 \prod^{K} [1 - q_{j} (1 - d_{j}^{- 2})] .

These are product-erasure formulas, not Haar Page formulas.

Proof. The accessible output is an orthogonal direct sum over flags $S$ , with weights $w_{S}$ and states $ρ_{C_{S}}$ . The flag entropy is the sum of the independent binary entropies, which proves (eq:productentropy).

Compress every occupied block and append $I_{C_{j}} / d_{j}$ on vacancy. The decoded product channel is

j ⨂ [(1 - q_{j}) id_{C_{j}} + q_{j} T_{I_{C_{j}} / d_{j}}] .

Write each replacement as a Weyl average. The identity-conjugation coefficient is $\prod_{j} [1 - q_{j} (1 - d_{j}^{- 2})]$ ; separating that term gives the upper bound (eq:productrecovery).

For the lower bound use the product Bell input. On flag $S$ , the missing references are independent maximally mixed states and no decoder can attain full Bell overlap larger than $\prod_{j \in / S} d_{j}^{- 2}$ . Average over flags. The resulting overlap bound is the same product coefficient, proving optimality.

Displayed formulas in this passage

Hawk05-C@1#eq:productentropy · line 1348
$S (R) = j = 1 \sum^{K} h_{2} (q_{j}) + S \sum w_{S} S (ρ_{C_{S}}) .$
Hawk05-C@1#eq:productrecovery · line 1353
$ε_{rec} (N) = 1 - j = 1 \prod^{K} [1 - q_{j} (1 - d_{j}^{- 2})] .$
unlabeled · line 1370
$j ⨂ [(1 - q_{j}) id_{C_{j}} + q_{j} T_{I_{C_{j}} / d_{j}}] .$

Theorem: Nonuniform product-transfer calibration

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