General bounded diagonal lift for fast-checking analysis

PDF p. 14: S4 Hamiltonian, jumps and timescale assumptions

Work in units $\mathrm{\hslash }=1$. Take the finite integer-spin benchmark with blocks $J=0,\dots ,N$, $N\ge 1$, and orthonormal magnetic basis $\mathrm{\mid }J,m⟩$ for $m=-J,\dots ,J$. Adopt the physical and spectral hypotheses, including the three choice assumptions, carried by Equal-rate rigid-rotor benchmark parameters. Let ${\omega }_0>0$, $\gamma >0$, and let $\mathrm{\Delta }(J,m)$ be real on these basis labels. Postulate the Hamiltonian $H={\sum }_{J=0}^N{\sum }_{m=-J}^J[{\omega }_{0}J(J+1)+\mathrm{\Delta }(J,m)]\mathrm{\mid }J,m⟩⟨J,m\mathrm{\mid }$.

Postulate that the same three benchmark downward jump matrices continue to describe the lifted model: $L_{\delta }=\sqrt{\gamma }{\sum }_{J=1}^N{\sum }_{m=-J}^J\sqrt{(2J+1)\mathrm{/}(2J-1)}{C}_{1,\delta ;J,m}^{J-1,m+\delta }\mathrm{\mid }J-1,m+\delta ⟩⟨J,m\mathrm{\mid }$ for $\delta =-1,0,1$, with out-of-range terms zero. Here $C_{1,\delta ;J,m}^{J-1,m+\delta }$ is the product-basis coefficient of the specified normalized coupled spin-$(J-1)$ basis in $V_1\otimes V_J$. The lifted-model use of these matrices is a separate phenomenological assumption: their derivation for unsplit levels alone does not establish it after lifting. Evolution is the finite GKSL channel generated by this $H$ and these jumps.

The main example takes $\mathrm{\Delta }(J,m)={\mathrm{\Delta }}_{0}Jm$ for real lifting strength ${\mathrm{\Delta }}_0$. For the short-checking regime, with interval $\tau \ge 0$, assume $\gamma \tau \ll 1$ and $D\tau \ll 1$, where $D=\max \mathrm{\mid }\mathrm{\Delta }(J,m)\mathrm{\mid }$ over the occupied labels relevant to the interval. The notation $x\ll 1$ specifies a small-parameter regime rather than a numerical error bound; if $D=0$, its condition is vacuous. For a code hosted in $J_0\ge 1$, the main lift has maximum $\mathrm{\mid }{\mathrm{\Delta }}_0\mathrm{\mid }{J}_0^2$ on the full host manifold, giving the sufficient condition $\mathrm{\mid }{\mathrm{\Delta }}_0\mathrm{\mid }{J}_0^2\tau \ll 1$ there. Downward reachable manifolds have no larger maximum.

The article imposes no separate ${\omega }_0\tau \ll 1$ condition in its single-manifold phase discussion: the rigid-rotor term is scalar on each fixed-$J$ manifold. This observation does not assert that a laboratory-frame first-order expansion of $e^{-iH\tau }$ is uniform for arbitrary ${\omega }_0\tau$, or that rotor phases are global across different $J$ manifolds.

Given: The finite model and the postulated Hamiltonian, jump matrices and checking regime above.

Goal: Explain the finite channel construction, the displayed host-manifold lift bound and the scope of the rotor-phase statement.

1.1 The real diagonal entries make $H$ Hermitian, and the three displayed jumps are finite matrices. Thus the finite construction in [F3] applies. The benchmark and coefficient conventions in [F1] and [F2] identify the matrices being postulated; they do not derive their continued physical validity for the lifted Hamiltonian. [F1, F2, F3, A1]

1.2 For $\mathrm{\mid }m\mathrm{\mid }\le J_0$, $\mathrm{\mid }{\mathrm{\Delta }}_0{J}_{0}m\mathrm{\mid }\le \mathrm{\mid }{\mathrm{\Delta }}_0\mathrm{\mid }{J}_0^2$, with equality at $m=\pm J_0$. The same maximum for any lower nonnegative integer $J$ is at most this value. On a fixed block the rotor term has the same eigenvalue ${\omega }_{0}J(J+1)$ for every $m$; its contribution cancels in the diagonal-entry difference governing conjugation of a matrix unit within that block by [F4]. Different blocks can have different scalar phases. These calculations establish the stated bounds and the limited phase claim. [F4, step 1.1, algebra] ∎