Manuscript Em04-A@2 · Em04-A-v2.tex · theorem · f5b28abc

Theorem: Typed deferred readout

Finite local instruments and typed records · lines 989–1022 · proof included or linked

Em04-A@2#thm:deferral

For a measured factor $S$ , complement $B$ , and computational basis $∣ z ⟩_{S}$ , define

V_{rec} (∣ z ⟩_{S} ∣ b ⟩_{B}) = ∣ z ⟩_{S} ∣ z ⟩_{A} ∣ b ⟩_{B} .

Reading $A$ computationally and retaining $S$ gives

I^{nd} (M) = z \sum ∣ z ⟩ ⟨ z ∣_{R} \otimes P_{z} M P_{z}, P_{z} = ∣ z ⟩ ⟨ z ∣_{S} \otimes I_{B} .

Reading $A$ and discarding $S$ gives instead

M (M) = z \sum ∣ z ⟩ ⟨ z ∣_{R} \otimes (⟨ z ∣_{S} \otimes I_{B}) M (∣ z ⟩_{S} \otimes I_{B}) .

These identities hold with any external reference and with the same fixed continuation channel on $B$ in both experiments.

Proof. Expand $M = \sum_{z, w} ∣ z ⟩ ⟨ w ∣_{S} \otimes M_{z w}$ . Coherent recording gives $\sum_{z, w} ∣ z ⟩ ⟨ w ∣_{S} \otimes ∣ z ⟩ ⟨ w ∣_{A} \otimes M_{z w}$ . Reading $A$ removes the $z \neq = w$ terms and produces $\sum_{z} ∣ z ⟩ ⟨ z ∣_{R} \otimes ∣ z ⟩ ⟨ z ∣_{S} \otimes M_{z z}$ . Tracing out $S$ gives the destructive formula. A reference simply enlarges the untouched matrix coefficients. A fixed channel on $B$ replaces each $M_{z w}$ by its image.

Displayed formulas in this passage

unlabeled · line 993
$V_{rec} (∣ z ⟩_{S} ∣ b ⟩_{B}) = ∣ z ⟩_{S} ∣ z ⟩_{A} ∣ b ⟩_{B} .$
unlabeled · line 998
$I^{nd} (M) = z \sum ∣ z ⟩ ⟨ z ∣_{R} \otimes P_{z} M P_{z}, P_{z} = ∣ z ⟩ ⟨ z ∣_{S} \otimes I_{B} .$
unlabeled · line 1004
$M (M) = z \sum ∣ z ⟩ ⟨ z ∣_{R} \otimes (⟨ z ∣_{S} \otimes I_{B}) M (∣ z ⟩_{S} \otimes I_{B}) .$

Theorem: Typed deferred readout

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