Manuscript VIII-A@3 · VIII-A-v3.tex · lemma · 2398a747

Lemma: Common destructive observation

Completeness and observation invariance › Restricted immediate versus deferred readout · lines 1061–1092 · proof included or linked

VIII-A@3#A:lem:deferred, VIII-A@3#A:eq:deferred

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The isometry $V$ in A:eq:copy satisfies

Tr_{S} (V X V^{†}) = z \sum ∣ z ⟩ ⟨ z ∣_{A} \otimes M_{z} X M_{z}^{†} .

Let $Λ$ be a channel acting only on $B$ , and let $N_{A}$ destructively read $A$ into $R$ . Then, with the required identity extensions,

Tr_{S} N_{A} ((id_{S A} \otimes Λ) (V X V^{†})) = (id_{R} \otimes Λ) M_{S} (X) .

The identities include every untouched external reference.

Proof. Expand $X = \sum_{z, z^{'}} ∣ z ⟩ ⟨ z^{'} ∣_{S} \otimes X_{z z^{'}}$ . Coherent recording appends $∣ z ⟩ ⟨ z^{'} ∣_{A}$ . Tracing $S$ removes exactly the terms $z \neq = z^{'}$ , giving the first formula. Operations on $B$ act on the blocks $X_{z z^{'}}$ and commute with the indicated restriction and carrier readout. This proves A:eq:deferred. A reference is another untouched factor in the same calculation.

Displayed formulas in this passage

VIII-A@3#A:eq:deferred · line 1072
$Tr_{S} N_{A} ((id_{S A} \otimes Λ) (V X V^{†})) = (id_{R} \otimes Λ) M_{S} (X) .$
unlabeled · line 1064
$Tr_{S} (V X V^{†}) = z \sum ∣ z ⟩ ⟨ z ∣_{A} \otimes M_{z} X M_{z}^{†} .$

Lemma: Common destructive observation

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