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

Theorem: Exact postselected recovery identity

A constructive Hayden–Preskill decoder › An exact algebraic recovery condition · lines 1158–1215 · proof included or linked

En02-C@1#thm:HPpost, En02-C@1#eq:Lnorm, En02-C@1#eq:HPsuccess

Under (eq:exactdecouple),

L^{†} L L N (X) L^{†} = \frac{1}{m ^{2}} S, = \frac{1}{m ^{2}} X_{M} \otimes \frac{I _{C^{'}}}{γ} .

The successful recovered map is therefore

Φ_{succ} N = \frac{1}{m ^{2}} id_{M} .

Its success probability is $p = m^{- 2} = 2^{- 2 k}$ , independent of every message-reference input.

Proof. From (eq:L),

L_{(i, a), (j, ℓ)} = \frac{1}{m δ} U_{(a, ℓ), (i, j)}^{*} = \frac{β}{m δ} (E_{a}^{†})_{i, (j, ℓ)} .

Thus

L^{†} L = \frac{β}{m δ} a \sum E_{a} E_{a}^{†} = \frac{β}{m δ γ} S = \frac{1}{m ^{2}} S,

using $m β = γ δ$ .

Likewise, the $a$ -th block of $L E_{a^{'}}$ is

\frac{β}{m δ} \frac{δ _{a a^{'}}}{γ} I_{M} .

Summing over $a^{'}$ proves

L N (X) L^{†} = \frac{β}{m δ γ ^{2}} X \otimes I_{C^{'}} = \frac{1}{m ^{2}} X \otimes \frac{I _{C^{'}}}{γ} .

All identities are linear, so tensoring with a reference preserves them.

Displayed formulas in this passage

En02-C@1#eq:Lnorm · line 1161
$L^{†} L L N (X) L^{†} = \frac{1}{m ^{2}} S, = \frac{1}{m ^{2}} X_{M} \otimes \frac{I _{C^{'}}}{γ} .$
En02-C@1#eq:HPsuccess · line 1161
$L^{†} L L N (X) L^{†} = \frac{1}{m ^{2}} S, = \frac{1}{m ^{2}} X_{M} \otimes \frac{I _{C^{'}}}{γ} .$
unlabeled · line 1170
$Φ_{succ} N = \frac{1}{m ^{2}} id_{M} .$
unlabeled · line 1179
$L_{(i, a), (j, ℓ)} = \frac{1}{m δ} U_{(a, ℓ), (i, j)}^{*} = \frac{β}{m δ} (E_{a}^{†})_{i, (j, ℓ)} .$
unlabeled · line 1189
$L^{†} L = \frac{β}{m δ} a \sum E_{a} E_{a}^{†} = \frac{β}{m δ γ} S = \frac{1}{m ^{2}} S,$
unlabeled · line 1201
$\frac{β}{m δ} \frac{δ _{a a^{'}}}{γ} I_{M} .$
unlabeled · line 1206
$L N (X) L^{†} = \frac{β}{m δ γ ^{2}} X \otimes I_{C^{'}} = \frac{1}{m ^{2}} X \otimes \frac{I _{C^{'}}}{γ} .$