Manuscript Psy03-A@1 · Psy03-A.tex · section · b30eb6e9

State space and generator

A separately defined finite quantum stochastic network › State space and generator · lines 1277–1312

Psy03-A@1#eq:qgenerator

Let

H_{n} = span {∣ 1 ⟩, \dots, ∣ n ⟩}, P_{i} = ∣ i ⟩ ⟨ i ∣,

with a declared node basis. For real symmetric coupling data $J_{ij}$ , a symmetric binary mask $M_{ij}$ , and $M_{ii} = 0$ , define

H_{M} = i < j \sum M_{ij} J_{ij} (∣ i ⟩ ⟨ j ∣ + ∣ j ⟩ ⟨ i ∣) .

Its units are energy. For $i \neq = j$ , let $k_{ij} \geq 0$ be the transition rate from $j$ to $i$ , and let $γ_{i} \geq 0$ . Write

D_{L} (ρ) = L ρ L^{†} - \frac{1}{2} {L^{†} L, ρ},

and define

L (ρ) = - \frac{i}{ℏ} [H_{M}, ρ] + i \neq = j \sum k_{ij} D_{∣ i ⟩ ⟨ j ∣} (ρ) + i \sum γ_{i} D_{P_{i}} (ρ) .

This is a finite quantum stochastic-network model, following the general coherent–incoherent architecture of quantum stochastic walks Whitfield. The model is not the $512$ – $512$ – $256$ classical network.

Displayed formulas in this passage

Psy03-A@1#eq:qgenerator · line 1300
$L (ρ) = - \frac{i}{ℏ} [H_{M}, ρ] + i \neq = j \sum k_{ij} D_{∣ i ⟩ ⟨ j ∣} (ρ) + i \sum γ_{i} D_{P_{i}} (ρ) .$
unlabeled · line 1280
$H_{n} = span {∣ 1 ⟩, \dots, ∣ n ⟩}, P_{i} = ∣ i ⟩ ⟨ i ∣,$
unlabeled · line 1287
$H_{M} = i < j \sum M_{ij} J_{ij} (∣ i ⟩ ⟨ j ∣ + ∣ j ⟩ ⟨ i ∣) .$
unlabeled · line 1295
$D_{L} (ρ) = L ρ L^{†} - \frac{1}{2} {L^{†} L, ρ},$

State space and generator

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