Manuscript Hawk04-D@1 · Hawk04-D-v1.tex · section · 39b16f67

A model-specific finite-band regulator

A model-specific finite-band regulator · lines 977–1017

This section supplies one concrete realization of the approximation allowance. It reconstructs the finite-band regulator used in Hawk04-A/B CheungHawk04A,CheungHawk04B. It is independent of the thermal-pair calibration in the next section.

Let $H_{S}$ be a bounded self-adjoint operator on a finite source space. An optional quartic realization is

H_{S} = i < j < k < l \sum J_{ij k l} χ_{i} χ_{j} χ_{k} χ_{l} + B_{N} I, B_{N} = i < j < k < l \sum ∣ J_{ij k l} ∣,

where ${χ_{i}, χ_{j}} = 2 δ_{ij} I$ . Then $0 \leq H_{S} \leq 2 B_{N} I$ .

Fix $0 < μ < Λ < \infty$ , and set

h = L^{2} ([μ, Λ], d ω; C^{r}), (h f) (ω) = ω f (ω) .

On $F_{s} (h)$ , let $H_{B} = d Γ (h)$ . For fixed self-adjoint source operators $A_{a}$ , real $g_{a}$ , and $f_{a} \in h$ , define

H = H_{S} + H_{B} + a \sum g_{a} A_{a} \otimes Φ (f_{a}), Φ (f) = a (f) + a^{†} (f) .

Annihilation is conjugate-linear in $f$ . Put

G = a \sum ∣ g_{a} ∣ ∥ A_{a} ∥ ∥ f_{a} ∥, G_{ω} = a \sum ∣ g_{a} ∣ ∥ A_{a} ∥ h^{- 1/2} f_{a} .

The bath is initially vacuum, and all finite source/carrier preparation is given by one fixed code isometry. Choose a finite collected packet space $r = span {p_{1}, \dots, p_{q}}$ .

Displayed formulas in this passage

unlabeled · line 986
$H_{S} = i < j < k < l \sum J_{ij k l} χ_{i} χ_{j} χ_{k} χ_{l} + B_{N} I, B_{N} = i < j < k < l \sum ∣ J_{ij k l} ∣,$
unlabeled · line 996
$h = L^{2} ([μ, Λ], d ω; C^{r}), (h f) (ω) = ω f (ω) .$
unlabeled · line 1003
$H = H_{S} + H_{B} + a \sum g_{a} A_{a} \otimes Φ (f_{a}), Φ (f) = a (f) + a^{†} (f) .$
unlabeled · line 1009
$G = a \sum ∣ g_{a} ∣ ∥ A_{a} ∥ ∥ f_{a} ∥, G_{ω} = a \sum ∣ g_{a} ∣ ∥ A_{a} ∥ h^{- 1/2} f_{a} .$

A model-specific finite-band regulator

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