C043 derivation / analysis

Construction of the reduced superoperator

Manifold $M=\{|dd\rangle,|uu\rangle,|\Psi^+\rangle,|\Psi^-\rangle\}$ with isometry $B=[\,|dd\rangle\,|uu\rangle\,|\Psi^+\rangle\,|\Psi^-\rangle\,]$ ($9\times4$, orthonormal columns). The reduced density-operator space is the $4\times4=16$-dim column-stacked space.

Paper's two approximations (Supp. S1): - Full transfer $\Phi=\pi/4$: numerically $U_A(\pi/4)$ maps $|dd\rangle\to|ee\rangle$ and leaves $|uu\rangle,|\Psi^+\rangle,|\Psi^-\rangle$ invariant (overlaps verified $=1$ in run.log). This is exactly the idealization the paper invokes. - Complete repump: the two-ion Kraus channel $B(\gamma)$ returns all $|ee\rangle$ population to the qubit manifold $\{\downarrow,\uparrow\}^{\otimes2}=M$. Measured population leakage out of $M$ after $B\!\circ\!A$ is $4\times10^{-16}$, i.e. $S_{BA}$ maps $M\to M$ as the paper states.

$S_{BA}(\gamma)$ is built column-by-column: for each manifold operator $|i\rangle\langle j|$ apply $U_A(\pi/4)$, then the nine two-ion Kraus maps, then re-express in $M$ via $B$. Drive C restricted to $M$ is unitary (unitarity error $7\times10^{-16}$), giving $S_C|M$, and $S(\gamma,\theta)=S_C|_M\,S(\gamma)$.

Results

1. Steady state: $\lambda_{ss}=+1$ is present at every $(\theta,\gamma)$ grid point (323/323), with eigenvector $|\Psi^-\rangle\langle\Psi^-|$.
2. Strict dominance (the second half of the claim) is CONFIRMED: $\lambda_{\max}$ strictly exceeds every other non-unit eigenvalue modulus at all 323 grid points (0 non-strict points; min gap $1.6\times10^{-5}$, median gap $0.068$), consistent with the paper.
3. Convergence rate match: at the paper's optimum $(\theta,\gamma)=(0.716\pi,0.221\pi)$ the reduced map gives $\lambda_{\max}=0.876964$, identical to the full 81-dim cycle ($|{\rm diff}|=4\times10^{-15}$) and $N_0=7.617$, matching the paper's $N_0\approx7.62$ (C007).
4. Eigenvalue count (the first half of the claim) does NOT reproduce. The faithful reduced map has ten non-zero eigenvalues at every grid point (histogram $\{10:323\}$), not six. At the optimum the non-zero spectrum is ${1,\,0.87696,\,0.638\,(\times2,\text{cmplx}),\,0.360\, (\times2),\,0.292\,(\times2),\,0.237,\,0.201}$.

Why the count differs

The paper's "six non-zero eigenvalues / degree-5 polynomial" must come from a further structural reduction beyond the manifold restriction stated in the text — e.g. restricting to the permutation-symmetric / physically-reachable population sub-block, or discarding the $|\Psi^-\rangle$-to-symmetric coherence modes that carry no population. The coupling graph of $S$ does split into population/symmetric blocks plus decoupled $|\Psi^\pm\rangle$ coherence triples, but no single $S$-invariant block of dimension exactly six containing both $\lambda_{ss}=1$ and $\lambda_{\max}$ could be isolated without an additional assumption the paper does not spell out. The dynamically relevant content — the unique $+1$ steady state, strict $\lambda_{\max}$ dominance, and the exact optimal convergence rate — is fully reproduced; only the bookkeeping count of non-zero eigenvalues differs (10 vs 6).

Verdict

partial, failure_reason: mismatch. The dominance statement and the quantitative convergence rate are verified; the specific "six non-zero eigenvalues" count is not reproduced by the faithful reduced construction (I obtain ten), so the structural sub-claim is reported honestly as a mismatch pending the paper's unstated additional reduction.