Manuscript VII-F-ST@1 · VII-F-ST.tex · proposition · 9eee7a44

Proposition: Exact fixed-table comparison

Observation and recovery separations › Fixed-table stability and its limit · lines 1555–1577 · proof included or linked

VII-F-ST@1#prop:table-stability

If the table probe (eq:table-probe) is replaced by one with probabilities $q_{a}$ , on the same code and payload interfaces, then

d_{⋄} (P_{q}, P_{q}) = a max ∣ q_{a} - q_{a} ∣.

Thus numerical approximation of this fixed finite table has a complete-channel limit.

Proof. Clock pinching gives diagonal blocks $X_{aa}$ , and the two binary-record coefficients differ by $\pm (q_{a} - q_{a})$ . The amplified trace-norm difference is at most $2 max_{a} ∣ q_{a} - q_{a} ∣ \sum_{a} ∥ X_{aa} ∥_{1}$ . Pinching is trace-norm contractive, giving the upper bound. Input $P_{a} \otimes ρ_{S}$ attains the bound at a maximizing index.

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unlabeled · line 1560
$d_{⋄} (P_{q}, P_{q}) = a max ∣ q_{a} - q_{a} ∣.$

Proposition: Exact fixed-table comparison

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