Manuscript En09-A@1 · En09-A.tex · lemma · eafec9de

Lemma: Recorded density-operator semantics

Frozen native data and measurement semantics › The imported Enter characterization · lines 355–387 · proof included or linked

En09-A@1#lem:density-semantics, En09-A@1#eq:branch-map

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Under Hypothesis hyp:enter, destructive readout in the clock basis has unnormalized branch map

I_{k} (X) = D_{k} X D_{k}^{†}, D_{k} = ⟨ τ_{k} ∣_{C} \otimes I_{S} .

The same formula holds with every untouched reference.

Proof. For a pure input, multiplication of the normalized positive-branch residual projector by its probability gives

∥ D_{k} ∣ Ξ ⟩ ∥^{2} \frac{D _{k} ∣ Ξ ⟩ ⟨ Ξ ∣ D _{k} ^{†}}{∥ D _{k} ∣ Ξ ⟩ ∥ ^{2}} = D_{k} ∣ Ξ ⟩ ⟨ Ξ ∣ D_{k}^{†} .

A zero component gives the zero operator, without assigning a normalized residual.

For a mixed input, purify it with a reference $E$ , apply the pure formula with $D_{k} \otimes I_{E}$ , and take the required reduced view. The result is $D_{k} X D_{k}^{†}$ . This expression depends only on $X$ , so it is independent of the purification or ensemble presentation. Keeping rather than tracing a reference gives the same formula with its identity factor.

Displayed formulas in this passage

En09-A@1#eq:branch-map · line 359
$I_{k} (X) = D_{k} X D_{k}^{†}, D_{k} = ⟨ τ_{k} ∣_{C} \otimes I_{S} .$
unlabeled · line 371
$∥ D_{k} ∣ Ξ ⟩ ∥^{2} \frac{D _{k} ∣ Ξ ⟩ ⟨ Ξ ∣ D _{k} ^{†}}{∥ D _{k} ∣ Ξ ⟩ ∥ ^{2}} = D_{k} ∣ Ξ ⟩ ⟨ Ξ ∣ D_{k}^{†} .$

Lemma: Recorded density-operator semantics

Displayed formulas in this passage

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