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Generation of a maximally entangled state using collective optical pumping

verification/C043/attempts/R002/claim_report.md

C043 - With full transfer (Phi=pi/4) and complete repump, the combined two-parameter superoperat...

Verdict: partial (mismatch) Location: Supp. Mat. S1 Type / expected artifact: math / math Claim: With full transfer (Phi=pi/4) and complete repump, the combined two-parameter superoperator S_{BA}(gamma) restricted to the ground manifold has six non-zero eigenvalues: lambda_ss=+1 and five roots of a degree-5 polynomial, with lambda_max strictly dominating the other non-unit eigenvalues over the whole parameter space. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial. Limitations: paper_text_only_reimplementation. Source location(s): source/supp_content.tex:25-27 (Supp. Mat. S1).

Conclusion

Built the REDUCED superoperator on the 4-state manifold M={dd,uu,Psi+,Psi-} under the paper's approximations: full transfer (numerically U_A(pi/4) maps dd->ee and leaves uu,Psi+,Psi- invariant) and complete repump (population leakage out of M after B.A = 4e-16), giving S(gamma,theta)=S_C|_M S_BA(gamma) on the 16-dim manifold density space. CONFIRMED parts: a unique lambda_ss=+1 (eigenvector |Psi-> 1e-9. The dominance statement and the optimal convergence rate are verified; the six-eigenvalue structural sub-claim is not reproduced -> partial / mismatch.

Verification details

Derivation excerpt: 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.

Executable rerun: run.py exited 0 in 0.696s; log verification/C043/attempts/R002/run.log.

Output excerpt:

manifold orthonormal: True
U_A(pi/4) action on manifold (overlaps onto {dd,uu,Psi+,Psi-,ee}):
   dd -> dd:0.00 uu:0.00 Psi+:0.00 Psi-:0.00 ee:1.00
   uu -> dd:0.00 uu:1.00 Psi+:0.00 Psi-:0.00 ee:0.00
   Psi+ -> dd:0.00 uu:0.00 Psi+:1.00 Psi-:0.00 ee:0.00
   Psi- -> dd:0.00 uu:0.00 Psi+:0.00 Psi-:1.00 ee:0.00
manifold leakage after B.A: 4.44e-16   drive-C unitarity err in M: 6.99e-16
non-zero eigenvalue-count histogram over grid: {10: 323}
steady-state +1 present at every grid point: True
min(lambda_max - next distinct |eig|) over grid: 0.000016  (>0 => lambda_max strictly dominates the other non-unit eigenvalues)

Supporting files