Manuscript Hawk04-B@2 · Hawk04-B.tex · section · ba3b0713

Preparation, insertion, and access

An independently specified regulated microscopic model › Preparation, insertion, and access · lines 1348–1414

Choose a normalized simultaneous spin eigenvector $∣ B_{s} ⟩$ of fixed source parity and let $β \geq 0$ . Prepare

∣ Ω_{S} ⟩ = Z_{s} (β)^{- 1/2} e^{- β H_{S} /2} ∣ B_{s} ⟩, Z_{s} (β) = ⟨ B_{s} ∣ e^{- β H_{S}} ∣ B_{s} ⟩ .

The bounds

e^{- β M_{N}} \leq Z_{s} (β) \leq 1

prove that the preparation is well defined. This is motivated by Euclidean-filtered SYK preparations KourkoulouMaldacena, without importing a gravitational dictionary.

The microscopic diary is a qubit encoded in the even-parity sector of two carrier modes:

J_{D} ∣ 0 ⟩ = ∣ 00 ⟩_{D}, J_{D} ∣ 1 ⟩ = ∣ 11 ⟩_{D} .

With a fixed occupation ordering, exchange those modes with two declared source modes by a fixed second-quantized fermionic swap:

W ∣ ψ ⟩ = U_{sw} (J_{D} ∣ ψ ⟩ \otimes ∣ Ω_{S} ⟩) .

The map $W : C \to H_{S} \otimes H_{D}$ is an isometry. It uses one diary-independent unitary. The displaced source state and its correlations remain in the carrier; that carrier is retained in the residual.

Choose orthonormal packet functions $p_{1}, \dots, p_{q} \in h$ , and set

r = span {p_{1}, \dots, p_{q}}, R = F_{s} (r),

B = H_{S} \otimes H_{D} \otimes F_{s} (r^{⊥}) .

With vacuum bath preparation and a fixed regrouping isometry $R$ ,

V_{t} ∣ ψ ⟩ = R (U (t) \otimes I_{D}) (W ∣ ψ ⟩ \otimes ∣ Ω_{bath} ⟩) .

All logical matrix units determine $N_{t}$ and $C_{t}$ .

The source is prepared before the diary arrives. A heralded preparation may be conditioned on a diary-independent success event. A deterministic attempted protocol must retain failure outputs and compare its complete channel.

The channel theorems apply to arbitrary finite $d$ . The specific two-mode diary construction above has $d = 2$ .

Displayed formulas in this passage

unlabeled · line 1353
$∣ Ω_{S} ⟩ = Z_{s} (β)^{- 1/2} e^{- β H_{S} /2} ∣ B_{s} ⟩, Z_{s} (β) = ⟨ B_{s} ∣ e^{- β H_{S}} ∣ B_{s} ⟩ .$
unlabeled · line 1361
$e^{- β M_{N}} \leq Z_{s} (β) \leq 1$
unlabeled · line 1371
$J_{D} ∣ 0 ⟩ = ∣ 00 ⟩_{D}, J_{D} ∣ 1 ⟩ = ∣ 11 ⟩_{D} .$
unlabeled · line 1377
$W ∣ ψ ⟩ = U_{sw} (J_{D} ∣ ψ ⟩ \otimes ∣ Ω_{S} ⟩) .$
unlabeled · line 1388
$r = span {p_{1}, \dots, p_{q}}, R = F_{s} (r),$
unlabeled · line 1392
$B = H_{S} \otimes H_{D} \otimes F_{s} (r^{⊥}) .$
unlabeled · line 1398
$V_{t} ∣ ψ ⟩ = R (U (t) \otimes I_{D}) (W ∣ ψ ⟩ \otimes ∣ Ω_{bath} ⟩) .$