Manuscript En02-C@1 · En02-C-v1.tex · section · 7549c6df

Preparation and success map

A constructive Hayden–Preskill decoder › Preparation and success map · lines 1098–1121

En02-C@1#eq:L

Allocate two $k$ -qubit registers $F, M$ and prepare $∣ Φ_{m} ⟩_{F M}$ . Apply

U^{*} : F E ⟶ C^{'} D^{'} .

Project $D D^{'}$ onto $∣ Φ_{δ} ⟩_{D D^{'}}$ . This projection can be implemented by inverse Bell preparation and destructive computational measurement, declaring success only when all $2 d$ outcomes are zero.

The corresponding linear map from $E D$ to $M C^{'}$ is

L = (⟨ Φ_{δ} ∣_{D D^{'}} \otimes I_{M C^{'}}) (I_{D} \otimes U_{F E}^{*}) (∣ Φ_{m} ⟩_{F M} \otimes I_{E D}) .

The successful output channel traces $C^{'}$ , operationally by destructive readout with unreported outcome.

Displayed formulas in this passage

En02-C@1#eq:L · line 1112
$L = (⟨ Φ_{δ} ∣_{D D^{'}} \otimes I_{M C^{'}}) (I_{D} \otimes U_{F E}^{*}) (∣ Φ_{m} ⟩_{F M} \otimes I_{E D}) .$
unlabeled · line 1103
$U^{*} : F E ⟶ C^{'} D^{'} .$