Manuscript VII-E@1 · VII-E.tex · corollary · b713cf3c

Corollary: Exact compatibility with an untouched reference

Record compatibility, residuals, and composition › Untouched entangled references · lines 1234–1255 · proof included or linked

VII-E@1#cor:reference, VII-E@1#eq:reference-equality

Under eq:pullback, for every finite reference system $N$ ,

(M_{E} \circ T) \otimes id_{N} = M_{Π} \otimes id_{N} .

In particular, the joint record–reference outputs agree for entangled inputs.

Proof. For every operator $X$ ,

(M_{E} \circ T) (X) = w \sum ∣ w ⟩ ⟨ w ∣ Tr (T^{*} (E_{w}) X) = w \sum ∣ w ⟩ ⟨ w ∣ Tr (Π_{w} X) = M_{Π} (X) .

Equal linear maps remain equal after tensoring with an identity.

Displayed formulas in this passage

VII-E@1#eq:reference-equality · line 1237
$(M_{E} \circ T) \otimes id_{N} = M_{Π} \otimes id_{N} .$
unlabeled · line 1248
$(M_{E} \circ T) (X) = w \sum ∣ w ⟩ ⟨ w ∣ Tr (T^{*} (E_{w}) X) = w \sum ∣ w ⟩ ⟨ w ∣ Tr (Π_{w} X) = M_{Π} (X) .$