Exact and thresholded Kraus ranks for J at most two

In the stated fixed-jump model, the benchmark coefficients are exactly $\sqrt{\gamma }{b}_{J,m,q}$ from [F13], with $q=\delta$. The rotor energies are $E_J={\omega }_{0}J(J+1)$, and the lift adds $\mathrm{\Delta }Jm$. Complete-block invariance permits the $J\le 2$ restriction without adding an artificial top-boundary transition. Apply [F13] to these finite matrices to obtain six one-jump directions at $\mathrm{\Delta }=0$ and twelve at $\mathrm{\Delta }\mathrm{\ne }0$. Apply [F9] to the same matrices for the one no-jump direction, five two-jump directions and total Choi ranks, and [F5] for the asserted exact Kraus ranks. This is a conditional analytic deduction, not evidence for the physical accuracy of fixed jumps after lifting.

By [F4], [F7] and [F8], effective rank counts only eigenvalues above the stated strict threshold and is bounded above by exact rank. At $t=0$, the identity channel on dimension nine has a single nonzero Choi eigenvalue nine. The finite generator exponential is continuous at zero, so its Choi matrix is continuous there. To see the threshold conclusion without rank continuity, write $Q_t=Q_0+B_t$ with $\mathrm{\parallel }{B}_t\mathrm{\parallel }\to 0$. On the codimension-one subspace perpendicular to the identity vector, every Rayleigh quotient is at most $\mathrm{\parallel }{B}_t\mathrm{\parallel }$. There cannot be two eigenvalues above $\mathrm{\parallel }{B}_t\mathrm{\parallel }$, since their two-dimensional span intersects that subspace. The largest eigenvalue is at least $9-\mathrm{\parallel }{B}_t\mathrm{\parallel }$, by testing the normalized identity vector. This gives effective rank one for sufficiently small time at any fixed threshold in $(0,9)$. It also explains why the displayed early count four is a finite-window statement, not the exact endpoint.

Every jump lowers $J$; the total excited population eventually leaves both excited manifolds. In the equal-rate model $p_2(t)=e^{-\gamma t}{p}_2(0)$ and $p_1(t)=e^{-\gamma t}[p_1(0)+\gamma t{p}_2(0)]$, by the diagonal block trace equations. Both vanish. Positivity bounds ground–excited and excited–excited coherences by square roots of the respective populations, so each input state's output tends to the ground state; finite linearity extends this to the replacement channel $X\mapsto \mathrm{Tr}(X)\mathrm{\mid }0,0⟩⟨0,0\mathrm{\mid }$. Its nine matrix-unit Kraus operators are $\mathrm{\mid }0,0⟩⟨J,m\mathrm{\mid }$, with nine orthogonal vectorizations of norm one. Thus its Choi rank is nine, and for threshold ${10}^{-8}$ the effective rank is eventually nine by the same spectral perturbation reasoning as step 1.2 around eigenvalues zero and one. This is a limiting statement and does not change step 1.1 at any finite time.