Full HTML report

Generation of a maximally entangled state using collective optical pumping

verification/C005/attempts/R002/claim_report.md

C005 - Drive (C) is described by the unitary U_C(theta) = exp(i (theta/2) sigma_x) tensor exp(i...

Verdict: verified Location: Dissipation scheme Type / expected artifact: math / math Claim: Drive (C) is described by the unitary U_C(theta) = exp(i (theta/2) sigma_x) tensor exp(i (theta/2) sigma_x), with sigma_x = |up><up|. 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/main.tex:107-108 (Dissipation scheme).

Conclusion

U_C(theta)=exp(i theta S_x) collective rotation. For theta in {0,0.2,0.5,0.716,1.0}pi: U_C|Psi->=|Psi-> (overlap 1.0, no phase) and triplet subspace invariant; U_C(0.5pi) cycles triplet population (|dd> changes). Matches ground truth (U_C leaves singlet invariant). -> verified.

Verification details

Derivation excerpt: Drive C is $U_C(\theta)=u(\theta)\otimes u(\theta)$ with the same single-ion rotation $u(\theta)=e^{i(\theta/2)\sigma_x}$, $\sigma_x=|{\uparrow}\rangle\langle{\downarrow}|+|{\downarrow}\rangle\langle{\uparrow}|$ on the qubit. Because both ions are rotated identically, $U_C$ is a collective rotation: $U_C=e^{i\theta S_x}$ with $S_x=\tfrac12(\sigma_x\otimes\mathbb1+\mathbb1\otimes\sigma_x)$.

Executable rerun: sympy_check.py exited 0 in 0.516s; log verification/C005/attempts/R002/sympy_check.log.

Output excerpt:

theta/pi  |<Psi-|U_C|Psi->|  U_C|Psi-> ?= |Psi->  triplet->triplet
  0.000   1.000000        True            True
  0.200   1.000000        True            True
  0.500   1.000000        True            True
  0.716   1.000000        True            True
  1.000   1.000000        True            True
U_C(0.5pi) cycles triplet population (|dd> changes): True
ALL_CHECKS_PASS = True

Supporting files