Manuscript VII-G@1 · VII-G.tex · proposition · 54f561e8

Proposition: Exact source–feedback readout equivalence

Implementation equivalence and represented readout › Immediate and coherent implementations that agree · lines 1031–1052 · proof included or linked

VII-G@1#prop:deferred, VII-G@1#eq:source-feedback-map

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Both implementations have selective maps

X ⟼ U_{Γ_{a} (g)} M_{a} X M_{a}^{†} U_{Γ_{a} (g)}^{†} .

They therefore agree on the full classical-record–matter output, including every entangled reference.

Proof. Projection onto pointer outcome $a$ gives

(⟨ a ∣_{A} \otimes I) C_{g} V_{g}^{src} = U_{Γ_{a} (g)} M_{a} .

This is the same branch operator as immediate readout followed by feedback. Equality of branch maps, and then tensoring with a reference, proves the conclusion.

Displayed formulas in this passage

VII-G@1#eq:source-feedback-map · line 1034
$X ⟼ U_{Γ_{a} (g)} M_{a} X M_{a}^{†} U_{Γ_{a} (g)}^{†} .$
unlabeled · line 1045
$(⟨ a ∣_{A} \otimes I) C_{g} V_{g}^{src} = U_{Γ_{a} (g)} M_{a} .$

Proposition: Exact source–feedback readout equivalence

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