Manuscript VII-E-ST@1 · VII-E-ST.tex · theorem · d28d9c5d

Theorem: Restricted deferred readout

Continuations, deferred readout, and spectator blindness · lines 621–659 · proof included or linked

VII-E-ST@1#thm:deferred, VII-E-ST@1#eq:deferred

Let ${K_{a}}$ be a complete Kraus family and ${U_{a}}$ unitaries on the surviving system. Immediate pointer readout followed by $U_{a}$ , and coherent controlled $U_{a}$ followed by pointer readout, implement the same recorded instrument:

X ⟼ a \sum ∣ a ⟩ ⟨ a ∣_{R} \otimes U_{a} K_{a} X K_{a}^{†} U_{a}^{†} .

This identity is complete under references.

For coherent copying of an original computational register, agreement with destructive readout holds only after both outputs are restricted to the same record–matter interface. A retained coherent carrier is not an additional common output.

Proof. The isometry

W ψ = a \sum ∣ a ⟩_{A} \otimes K_{a} ψ

satisfies $W^{†} W = I$ . Controlled feedback changes it to $\sum_{a} ∣ a ⟩_{A} \otimes U_{a} K_{a} ψ$ . Destructive readout of $A$ gives (eq:deferred). The immediate implementation has the same selective maps.

For computational copying, write $X = \sum_{z, z^{'}} ∣ z ⟩ ⟨ z^{'} ∣ \otimes X_{z z^{'}}$ . After coherent copying and label-preserving controlled operations, final readout and restriction to the common interface retain precisely the diagonal blocks $X_{z z}$ . If both copied measured registers are consumed, the duplicate classical outcome can be explicitly forgotten. No quantum return of a consumed register is involved.

Displayed formulas in this passage

VII-E-ST@1#eq:deferred · line 628
$X ⟼ a \sum ∣ a ⟩ ⟨ a ∣_{R} \otimes U_{a} K_{a} X K_{a}^{†} U_{a}^{†} .$
unlabeled · line 643
$W ψ = a \sum ∣ a ⟩_{A} \otimes K_{a} ψ$

Theorem: Restricted deferred readout

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