Manuscript Oai280-C@1 · Oai280-C.tex · proposition · 7d44b8c0

Proposition: When subsystem certification reduces to a qubit

Dimension-correct channel conversion › Mixed subsystem and algebra Choi targets · lines 1367–1385 · proof included or linked

Oai280-C@1#prop:spectatorblind

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If an independently established structural identity gives

Λ = Γ Tr_{S},

then

d_{⋄} (Λ, Tr_{S}) = d_{⋄} (Γ, id_{C_{L}}) .

Without that identity, tests with one fixed spectator do not establish the equality.

Proof. Composition by $Tr_{S}$ gives the upper bound. For the lower bound, embed an arbitrary $C_{L}$ input with a fixed state of $S$ , retaining its reference.

Displayed formulas in this passage

unlabeled · line 1370
$Λ = Γ Tr_{S},$
unlabeled · line 1374
$d_{⋄} (Λ, Tr_{S}) = d_{⋄} (Γ, id_{C_{L}}) .$

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section: Mixed subsystem and algebra Choi targets