Weak rotational symmetry and decoupled operator sectors

PDF p. 17: Numerical method, first three sentences

Assume the Axiom of Choice, Countable Choice and Dependent Choice, and the finite-block emitter, bath, tensor, regularity and pointwise spectral hypotheses of Canonical angular-momentum downward jumps. Adopt the equal-rate rigid-rotor benchmark of Equal-rate rigid-rotor benchmark parameters, the diagonal lift of Phenomenological angular-momentum degeneracy lift, and the model choice that the lift does not change the three unperturbed jumps. The Hilbert space is the direct sum of the integer-spin blocks $J=0,\dots ,N$, with orthonormal magnetic basis $\mathrm{\mid }J,m⟩$ and $m=-J,\dots ,J$. Let $J_z\mathrm{\mid }J,m⟩=m\mathrm{\mid }J,m⟩$, $R_{\theta }=e^{-i\theta J_z}$, and $H={\sum }_{J,m}[{\omega }_{0}J(J+1)+\mathrm{\Delta }Jm]\mathrm{\mid }J,m⟩⟨J,m\mathrm{\mid }$, where $\mathrm{\Delta }$ is real. Let $L_{\delta }$, for $\delta =-1,0,1$, be the downward jumps of the cited result, with the benchmark coefficients.

For the finite generator $\mathcal{L}(X)=-i[H,X]+{\sum }_{\delta =-1}^1(L_{\delta }X{L}_{\delta }^{†}-\{L_{\delta }^{†}{L}_{\delta },X\}\mathrm{/}2)$, one has $R_{\theta }{L}_{\delta }{R}_{\theta }^{†}=e^{-i\theta \delta }{L}_{\delta }$ and $\mathcal{L}(R_{\theta }X{R}_{\theta }^{†})=R_{\theta }\mathcal{L}(X)R_{\theta }^{†}$ for every real $\theta$. For each integer $r$, the operator sector ${\mathcal{H}}_r=\mathrm{span}\{\mathrm{\mid }J,m⟩⟨J^{\mathrm{\prime }},m^{\mathrm{\prime }}\mathrm{\mid }:m-m^{\mathrm{\prime }}=r\}$ is invariant under $\mathcal{L}$. The matrix algebra is the direct sum of the nonzero sectors, and $e^{t\mathcal{L}}$ preserves each sector for $t\ge 0$; consequently their evolutions can be computed separately and in parallel. This is the finite-block form of the article's stated weak rotational symmetry and sector decomposition.

Equal-rate rigid-rotor benchmark parameters

Phenomenological angular-momentum degeneracy lift

Canonical angular-momentum downward jumps

Finite spin blocks, spherical tensors and Clebsch–Gordan phases

Finite GKSL evolution and the QEC channel interface

Rotation covariance and operator-weight sectors

The article states the sector decomposition using unrestricted $J$ notation. The deduction above uses completed finite-block suppliers and makes no assertion about domains or convergence for infinitely many blocks. The article's retained-jump approximation is an assumption here, not a consequence of adding the lift.