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

A fully explicit one-message Hayden–Preskill example

C5: amplification and the single-copy ledger › A fully explicit one-message Hayden–Preskill example · lines 1438–1475

Take $k = b = c = d = 1$ and

U ∣ a ⟩_{M} ∣ j ⟩_{B} = ∣ j ⟩_{C} ∣ a ⟩_{D} .

Thus $U$ is SWAP, initially $B E$ is a Bell pair, and

N (ρ) = \frac{I _{E}}{2} \otimes ρ_{D} .

The complement is $I_{C} /2$ , so exact decoupling holds.

SWAP has a three-CNOT implementation. Also,

Z_{0} (2) = X_{1} X_{2} CZ_{1, 2} X_{1} X_{2},

which costs five gates and no ancilla. Therefore:

Programme	Peak	$G$ bound	$D$ bound	$n_{U^{*}}$	$n_{U^{T}}$
Conjugate postselection	4	7	8	1	0
One-iteration amplification	4	31	32	2	1
Direct recovery of this example	2	0	1	0	0

Postselection succeeds with probability $1/4$ ; one iteration is deterministic and exact. Direct recovery simply consumes $E$ and returns $D$ . The table prices the chosen published architecture, not an optimum for this deliberately simple evolution.

Displayed formulas in this passage

unlabeled · line 1441
$U ∣ a ⟩_{M} ∣ j ⟩_{B} = ∣ j ⟩_{C} ∣ a ⟩_{D} .$
unlabeled · line 1445
$N (ρ) = \frac{I _{E}}{2} \otimes ρ_{D} .$
unlabeled · line 1452
$Z_{0} (2) = X_{1} X_{2} CZ_{1, 2} X_{1} X_{2},$