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Generation of a maximally entangled state using collective optical pumping

verification/C053/attempts/R002/claim_report.md

C053 - For both ions in the {down,e} subspace, U_i(epsilon) creates an error with probability p...

Verdict: verified Location: Supp. Mat. S2 Type / expected artifact: math / math Claim: For both ions in the {down,e} subspace, U_i(epsilon) creates an error with probability p ~= 2 (epsilon/2)^2 for p<<1, so the process is parametrized by epsilon = sqrt(2 p). 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:82 (Supp. Mat. S2).

Conclusion

P(exactly one ion flipped) for U_x(eps)|dd> = 2 sin^2(eps/2) cos^2(eps/2) = eps^2/2 - eps^4/6 + O(eps^6) (sympy); leading coeff 1/2 => p ~= 2(eps/2)^2 = eps^2/2, so eps=sqrt(2p). Numeric eps=0.05 -> 1.2490e-3 (paper hint 1.25e-3); p_one/(eps^2/2) -> 1 as eps->0; round-trip eps=sqrt(2p) recovers p. Clean asymptotic math fact -> verified.

Verification details

Executable rerun: run.py exited 0 in 0.756s; log verification/C053/attempts/R002/run.log.

Output excerpt:

P_one (exact) = 2 sin^2(e/2) cos^2(e/2)
series to O(eps^4): epsilon**2*(3 - epsilon**2)/6
leading coeff P_one/eps^2 -> 1/2 (= 1/2 => p ~= eps^2/2)
  eps        p_one(numeric)   eps^2/2        ratio
  0.0500    1.24895868e-03   1.25000000e-03   0.999167
  0.0200    1.99973335e-04   2.00000000e-04   0.999867
  0.0100    4.99983334e-05   5.00000000e-05   0.999967
  0.0050    1.24998958e-05   1.25000000e-05   0.999992
ground-truth check: eps=0.05 -> p_one=1.248959e-03 (paper hint 1.25e-3)
eps=sqrt(2p) with p=0.001: eps=0.044721 -> p_one=9.993335e-04 (target 1.000000e-03)

Supporting files