Manuscript Hawk04-A@1 · Hawk04-A.tex · section · 526092ba

A coherent diary and its displaced carrier

A regulated interacting microscopic application › A coherent diary and its displaced carrier · lines 1133–1207

Hawk04-A@1#eq:filtered, Hawk04-A@1#eq:loading, Hawk04-A@1#eq:Vt

Choose a simultaneous spin eigenvector $∣ B_{s} ⟩$ of fixed source parity, and 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} ⟩ > 0.

This Euclidean-filtered preparation is motivated by Kourkoulou and Maldacena KourkoulouMaldacena. The exact normalization here follows from $e^{- β M_{N}} \leq Z_{s} (β) \leq 1$ .

The source preparation is completed before the diary arrives. If it is heralded, its success decision is independent of the diary. An attempted deterministic preparation protocol must retain failure outputs in its own complete channel.

Encode a qubit in two carrier modes:

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

Both basis vectors are even. Choose two source modes $T$ , fix a fermionic occupation ordering, and let $U_{sw}$ be the second quantization of their one-particle exchange with the carrier. Define

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

This is one message-independent, parity-preserving isometry.

The even diary introduces no logical-input-dependent permutation sign. The old source two-mode state is displaced into the carrier, including its correlations with the remaining source. The carrier marginal is diary independent, but it is generally neither vacuum nor uncorrelated with the source. It remains in the complete residual. This differs from the vacuum destination used in Hawk02-D CheungHawk02D.

Fix an orthonormal packet family $p_{1}, \dots, p_{q}$ and

r = span {p_{1}, \dots, p_{q}} .

The declared access partition is

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

After the fixed regrouping,

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

Its complete matrix units are

N_{t} (E_{ij}) = Tr_{B} ∣ V_{t} i ⟩ ⟨ V_{t} j ∣, C_{t} (E_{ij}) = Tr_{R} ∣ V_{t} i ⟩ ⟨ V_{t} j ∣ .

Displayed formulas in this passage

Hawk04-A@1#eq:filtered · line 1137
$∣ Ω_{S} ⟩ = Z_{s} (β)^{- 1/2} e^{- β H_{S} /2} ∣ B_{s} ⟩, Z_{s} (β) = ⟨ B_{s} ∣ e^{- β H_{S}} ∣ B_{s} ⟩ > 0.$
Hawk04-A@1#eq:loading · line 1166
$W ∣ ψ ⟩ = U_{sw} (J_{D} ∣ ψ ⟩ \otimes ∣ Ω_{S} ⟩) .$
Hawk04-A@1#eq:Vt · line 1194
$V_{t} ∣ ψ ⟩ = R (U (t) \otimes I_{D}) (W ∣ ψ ⟩ \otimes ∣ Ω_{bath} ⟩) .$
unlabeled · line 1156
$J_{D} ∣ 0 ⟩ = ∣ 00 ⟩_{D}, J_{D} ∣ 1 ⟩ = ∣ 11 ⟩_{D} .$
unlabeled · line 1184
$r = span {p_{1}, \dots, p_{q}} .$
unlabeled · line 1188
$R = F_{s} (r), B = H_{S} \otimes H_{D} \otimes F_{s} (r^{⊥}) .$
unlabeled · line 1202
$N_{t} (E_{ij}) = Tr_{B} ∣ V_{t} i ⟩ ⟨ V_{t} j ∣, C_{t} (E_{ij}) = Tr_{R} ∣ V_{t} i ⟩ ⟨ V_{t} j ∣ .$