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

An exact algebraic recovery condition

A constructive Hayden–Preskill decoder › An exact algebraic recovery condition · lines 1122–1157

En02-C@1#eq:exactdecouple

The complementary channel is

N^{c} (X) = Tr_{D} (U (X \otimes I_{B} / β) U^{†}) .

Assume

N^{c} (X) = tr (X) \frac{I _{C}}{γ} .

This is a checkable equality on all message matrix units. It is a hypothesis, not a consequence of unitarity.

For each computational $C$ -label $a$ , define a Kraus map $E_{a} : M \to E D$ by

(E_{a})_{(j, ℓ), i} = \frac{1}{β} U_{(a, ℓ), (i, j)} .

Then

N (X) = a \sum E_{a} X E_{a}^{†}, E_{a}^{†} E_{a^{'}} = \frac{δ _{a a^{'}}}{γ} I_{M} .

The maps $V_{a} = γ E_{a}$ are orthogonal isometries. Let

S = a \sum V_{a} V_{a}^{†}

be their range projector in $E D$ .

Displayed formulas in this passage

En02-C@1#eq:exactdecouple · line 1131
$N^{c} (X) = tr (X) \frac{I _{C}}{γ} .$
unlabeled · line 1125
$N^{c} (X) = Tr_{D} (U (X \otimes I_{B} / β) U^{†}) .$
unlabeled · line 1140
$(E_{a})_{(j, ℓ), i} = \frac{1}{β} U_{(a, ℓ), (i, j)} .$
unlabeled · line 1146
$N (X) = a \sum E_{a} X E_{a}^{†}, E_{a}^{†} E_{a^{'}} = \frac{δ _{a a^{'}}}{γ} I_{M} .$
unlabeled · line 1153
$S = a \sum V_{a} V_{a}^{†}$