Manuscript En07-A@1 · En07-A.tex · section · 44229ce9

Destructive partial readout

Enter, Drop, and the complete apparatus ledger › Destructive partial readout · lines 662–689

En07-A@1#eq:instrument

Let $S$ be $s$ measured qubits and $T$ the surviving quantum register. Put

M_{x} = ⟨ x ∣_{S} \otimes I_{T}, x \in {0, 1}^{s} .

The observable instrument is

E_{S} (X) = x \sum ∣ x ⟩ ⟨ x ∣_{K} \otimes M_{x} X M_{x}^{†} .

The names in $S$ are consumed; $T$ survives.

For a pure input $ψ$ , the branch probability and positive-probability residual are

p_{x} = ∥ M_{x} ψ ∥^{2}, ψ_{x} = \frac{M _{x} ψ}{p _{x}} (p_{x} > 0) .

No normalized residual is assigned to a zero-probability branch. Complete-channel calculations use unnormalized branches and retain every outcome.

Displayed formulas in this passage

En07-A@1#eq:instrument · line 671
$E_{S} (X) = x \sum ∣ x ⟩ ⟨ x ∣_{K} \otimes M_{x} X M_{x}^{†} .$
unlabeled · line 666
$M_{x} = ⟨ x ∣_{S} \otimes I_{T}, x \in {0, 1}^{s} .$
unlabeled · line 681
$p_{x} = ∥ M_{x} ψ ∥^{2}, ψ_{x} = \frac{M _{x} ψ}{p _{x}} (p_{x} > 0) .$