Manuscript Hawk07-B@1 · Hawk07-B.tex · section · b9785538

One payload, one sampled device

Scaled preparation and a rederived recovery threshold › One payload, one sampled device · lines 459–501

Hawk07-B@1#eq:model

Declare

L ≃ C^{d}, H ≃ C^{h}, E ≃ C^{h}, F ≃ C^{f} .

Before the unknown payload is used, prepare the independent resource

∣ Φ_{h} ⟩_{H E} = h^{- 1/2} a = 1 \sum^{h} ∣ a ⟩_{H} ∣ a ⟩_{E}, ∣ 0 ⟩_{F} .

The vault $L H F$ has dimension $D = d h f$ . Sample one Haar unitary $U \in U (D)$ , retain its device description, and use that same $U$ at every cut.

At cut $k$ , regroup the vault as $P_{k} B_{k}$ , with

r_{k} = dim P_{k}, b_{k} = dim B_{k}, r_{k} b_{k} = D .

The receiver is the entire factor

R_{k} = P_{k} E, dim R_{k} = r_{k} h .

The reference $Q$ , possibly entangled with $L$ , is untouched. The complete isometry is

V_{k}^{U} ∣ ψ ⟩ = (U \otimes I_{E}) (∣ ψ ⟩_{L} ∣ Φ_{h} ⟩_{H E} ∣ 0 ⟩_{F}),

with the indicated output regrouping. Later release does not act on already released cells.

This is a finite entanglement-assisted recovery task, comparable in structure to Hayden–Preskill retrieval HaydenPreskill. It is not a model of its gravitational dynamics.

Displayed formulas in this passage

Hawk07-B@1#eq:model · line 488
$V_{k}^{U} ∣ ψ ⟩ = (U \otimes I_{E}) (∣ ψ ⟩_{L} ∣ Φ_{h} ⟩_{H E} ∣ 0 ⟩_{F}),$
unlabeled · line 462
$L ≃ C^{d}, H ≃ C^{h}, E ≃ C^{h}, F ≃ C^{f} .$
unlabeled · line 469
$∣ Φ_{h} ⟩_{H E} = h^{- 1/2} a = 1 \sum^{h} ∣ a ⟩_{H} ∣ a ⟩_{E}, ∣ 0 ⟩_{F} .$
unlabeled · line 479
$r_{k} = dim P_{k}, b_{k} = dim B_{k}, r_{k} b_{k} = D .$
unlabeled · line 483
$R_{k} = P_{k} E, dim R_{k} = r_{k} h .$