Manuscript X-Enter-ST@1 · X-Enter-ST.tex · section · f76acc1f

Finite programs and complete selective maps

Stage C: the retained finite amplitude trial › Finite programs and complete selective maps · lines 584–608

X-Enter-ST@1#eq:pathmaps, X-Enter-ST@1#eq:record

A finite program uses fixed finite-register unitaries, fresh initialized registers, finite classical control, and destructive computational-basis readout. Every live quantum name is eventually consumed or returned. Failure outcomes are included.

For a complete finite history $h$ , compose its allocation isometries, gates, and measurement bras into a linear path map $K_{h}$ . Group histories by the reported classical output $y$ :

Φ_{y} (ρ) = h : out (h) = y \sum K_{h} ρ K_{h}^{†}, h \sum K_{h}^{†} K_{h} = I .

The complete recorded channel is

Rec_{Φ} (ρ) = y \sum ∣ y ⟩ ⟨ y ∣_{R} \otimes Φ_{y} (ρ) .

Displayed formulas in this passage

X-Enter-ST@1#eq:pathmaps · line 596
$Φ_{y} (ρ) = h : out (h) = y \sum K_{h} ρ K_{h}^{†}, h \sum K_{h}^{†} K_{h} = I .$
X-Enter-ST@1#eq:record · line 603
$Rec_{Φ} (ρ) = y \sum ∣ y ⟩ ⟨ y ∣_{R} \otimes Φ_{y} (ρ) .$

Finite programs and complete selective maps

Displayed formulas in this passage

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