Manuscript En01-A@1 · En01-A.tex · theorem · 558b972b

Theorem: Visibility envelope and two-branch identity

Cross operators, fidelity, screening, and noisy architecture · lines 1373–1403 · proof included or linked

En01-A@1#thm:visibility, En01-A@1#eq:visibility-bound, En01-A@1#eq:two-branch

For every $J, S$ ,

δ_{J} (S) \leq r < r^{'} J -split \sum p_{r} p_{r^{'}} v_{r r^{'}} (S) .

If exactly two branches $r, r^{'}$ are supported, then

δ_{J} (S) = {p_{r} p_{r^{'}} v_{r r^{'}} (S), 0, D (r, r^{'}) \cap J \neq = \emptyset, D (r, r^{'}) \cap J = \emptyset.

The equality in the first line uses perfect-record structure.

Proof. Apply the triangle inequality to (eq:cross-expansion) and use $∥Γ + Γ^{†} ∥_{1} \leq 2∥Γ ∥_{1}$ . For two branches, an unsplit pair contributes nothing to $X_{J}$ . A split pair is either screened, making both sides zero, or has orthogonal supports as in Proposition prop:screening, giving equality.

Displayed formulas in this passage

En01-A@1#eq:visibility-bound · line 1376
$δ_{J} (S) \leq r < r^{'} J -split \sum p_{r} p_{r^{'}} v_{r r^{'}} (S) .$
En01-A@1#eq:two-branch · line 1383
$δ_{J} (S) = {p_{r} p_{r^{'}} v_{r r^{'}} (S), 0, D (r, r^{'}) \cap J \neq = \emptyset, D (r, r^{'}) \cap J = \emptyset.$

Theorem: Visibility envelope and two-branch identity

Displayed formulas in this passage

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