Full HTML report

Generation of a maximally entangled state using collective optical pumping

verification/C044/attempts/R002/claim_report.md

C044 - With alternating drives theta_1=pi (odd) and theta_2=pi/2 (even), the two-cycle superoper...

Verdict: verified Location: Supp. Mat. S1, Eq. (S3) Type / expected artifact: math / math Claim: With alternating drives theta_1=pi (odd) and theta_2=pi/2 (even), the two-cycle superoperator S(pi/4,gamma,theta_2)S(pi/4,gamma,theta_1) has only three non-zero eigenvalues: lambda_ss=+1 and lambda_pm = (1/4)(1 +/- 3 sqrt(1 - (2/9)[2+cos^4(gamma)] sin^2(2 gamma))). 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:43-46 (Eq. (S3)).

Conclusion

The two-cycle alternating superoperator S(pi/4,gamma,pi/2) S(pi/4,gamma,pi) has exactly three non-zero eigenvalues across a 47-point gamma grid (the next-largest discarded modulus is ~1.6e-8 numerical noise, well below TOL=1e-6): lambda_ss=+1, lambda_+ and lambda_-. The numeric lambda_+ and lambda_- match the closed form lambda_pm=(1/4)(1 +/- 3 sqrt(1-(2/9)[2+cos^4(gamma)] sin^2(2 gamma))) to <=1.8e-15 (lambda_+) and <=5.6e-16 (lambda_-) over the whole grid; a sympy spot-check at gamma=0.23pi gives formula 0.77675852 vs numeric 0.77675852. Tolerance 1e-8; satisfied. -> verified.

Verification details

Executable rerun: run.py exited 0 in 0.716s; log verification/C044/attempts/R002/run.log.

Output excerpt:

top-5 |eig| at 0.23pi: ['1.00e+00', '7.77e-01', '2.77e-01', '1.58e-08', '1.58e-08']
largest discarded |eig| (4th): 1.58e-08  (<< TOL=1e-06)
set of non-zero eigenvalue counts over gamma grid: [3]
max|numeric lambda_+ - formula| over grid: 1.776e-15
max|numeric lambda_- - formula| over grid: 5.551e-16
spot check (gamma/pi: numeric eigs | formula lam_pm):
  g=0.22pi  nonzero eigs=['1.00000', '0.77798', '-0.27798']  formula lam+=0.77798 lam-=-0.27798
  g=0.23pi  nonzero eigs=['1.00000', '0.77676', '-0.27676']  formula lam+=0.77676 lam-=-0.27676
  g=0.31pi  nonzero eigs=['1.00000', '0.82930', '-0.32930']  formula lam+=0.82930 lam-=-0.32930
symbolic two-cycle eigenvalues at gamma=0.23pi:

Supporting files