Full HTML report

Generation of a maximally entangled state using collective optical pumping

verification/C042/attempts/R002/claim_report.md

C042 - The state after N cycles is rho_N = [S(Phi,gamma,theta)]^N rho_0 in vectorized form, and...

Verdict: verified Location: Supp. Mat. S1, Eq. (S2) Type / expected artifact: math / math Claim: The state after N cycles is rho_N = [S(Phi,gamma,theta)]^N rho_0 in vectorized form, and the fidelity follows F(N) ~= 1 - C_0 lambda_max^N = 1 - C_0 e^{-N/N_0}, where lambda_max is the second-largest eigenvalue and 1/N_0 the convergence rate. Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain verified -> verified. Source location(s): source/supp_content.tex:15-23 (Eq. (S2)).

Conclusion

Derivation (derivation.md): spectral decomposition S=sum_k lambda_k |r_k>><>1 the largest-modulus sub-unit eigenvalue lambda_max dominates, so F(N)~1-C_0 lambda_max^N = 1-C_0 e^{-N/N0} with N0=-1/log(lambda_max). Numeric (run.log): at the optimum, least-squares slope of log(epsilon) vs N (N>=20) = -0.131289, equal to log(lambda_max) to 6 digits; N0_fit=7.617 matches -1/log(second_eigmag); max relative residual of C_0 lambda_max^N vs full simulation = 1.1e-9; a second generic parameter set gives slope=log(lambda_max) and residual 8e-6. Tolerance: slope match <1e-6. Clean spectral-decomposition fact -> verified.

Verification details

Derivation excerpt: One cycle is the linear superoperator $S$ acting on $\mathrm{vec}(\rho)$ (column stacking), established in C040. Iterating $N$ cycles is therefore $$\mathrm{vec}(\rho_N) = S^N\,\mathrm{vec}(\rho_0), \qquad \rho_N = \mathrm{unvec}\!\left(S^N\,\mathrm{vec}(\rho_0)\right).$$

Executable rerun: run.py exited 0 in 0.531s; log verification/C042/attempts/R002/run.log.

Output excerpt:

[optimal] S^N vec == repeated step err: 7.770384787854966e-17
[optimal] lambda_max=0.876964  (complex 0.8770-0.0000j)  N0=-1/log=7.6168
[optimal] fit slope=-0.131289  log(lambda_max)=-0.131289  -1/slope (N0_fit)=7.6168  C0=0.8892
[optimal] max rel residual (C0 lam^N vs sim, N>=20): 1.085e-09
[generic] S^N vec == repeated step err: 8.037642502610834e-17
[generic] lambda_max=0.912953  (complex 0.9130+0.0000j)  N0=-1/log=10.9805
[generic] fit slope=-0.091071  log(lambda_max)=-0.091071  -1/slope (N0_fit)=10.9805  C0=0.8924
[generic] max rel residual (C0 lam^N vs sim, N>=20): 7.884e-06
PASS

Supporting files