Manuscript Psy03-B@1 · Psy03-B-v1.tex · section · 8ca3caf9

Sector dynamics

A quantum stochastic network with a scar register › Sector dynamics · lines 1831–1874

Psy03-B@1#eq:qgenerator

Reuse the finite node carrier of Psy03-A CheungPsy03A, not a different biological carrier:

H_{Q} = span {∣ 1 ⟩, \dots, ∣ n ⟩} .

Let $E$ be a finite permitted edge set,

S = 2^{E}, H_{S} = span {∣ s ⟩ : s \in S} .

The scar label $s$ records forbidden edges.

In sector $s$ , choose

H_{s} = {i, j} \in E ∖ s \sum J_{ij}^{(s)} (∣ i ⟩ ⟨ j ∣ + ∣ j ⟩ ⟨ i ∣),

with real energy-valued couplings. One permitted generator is

L_{s} (ρ) = - \frac{i}{ℏ} [H_{s}, ρ] + i \neq = j \sum k_{ij}^{(s)} D_{∣ i ⟩ ⟨ j ∣} (ρ) + i \sum γ_{i}^{(s)} D_{∣ i ⟩ ⟨ i ∣} (ρ),

where rates are nonnegative and rates associated with forbidden edges vanish. Here

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

Quantum stochastic walks provide this coherent–incoherent graph architecture Whitfield. Finite GKSL generators define unique CPTP sector propagators GKS,Lindblad. The graph is not the classical feed-forward architecture, and node labels do not themselves specify spatial tensor factors or a relativistic causal relation.

Displayed formulas in this passage

Psy03-B@1#eq:qgenerator · line 1852
$L_{s} (ρ) = - \frac{i}{ℏ} [H_{s}, ρ] + i \neq = j \sum k_{ij}^{(s)} D_{∣ i ⟩ ⟨ j ∣} (ρ) + i \sum γ_{i}^{(s)} D_{∣ i ⟩ ⟨ i ∣} (ρ),$
unlabeled · line 1835
$H_{Q} = span {∣ 1 ⟩, \dots, ∣ n ⟩} .$
unlabeled · line 1839
$S = 2^{E}, H_{S} = span {∣ s ⟩ : s \in S} .$
unlabeled · line 1846
$H_{s} = {i, j} \in E ∖ s \sum J_{ij}^{(s)} (∣ i ⟩ ⟨ j ∣ + ∣ j ⟩ ⟨ i ∣),$
unlabeled · line 1862
$D_{L} (ρ) = L ρ L^{†} - \frac{1}{2} {L^{†} L, ρ} .$