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Generation of a maximally entangled state using collective optical pumping

verification/C026/attempts/R002/claim_report.md

C026 - Correlated phase-flip errors (I_e Z_e + Z_e I_e) leave the fidelity unaffected because |P...

Verdict: verified Location: Theoretical error analysis Type / expected artifact: empirical / math Claim: Correlated phase-flip errors (I_e Z_e + Z_e I_e) leave the fidelity unaffected because |Psi^-> resides in a decoherence-free subspace. 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:145 (Theoretical error analysis).

Conclusion

Claim: correlated phase-flip (I_eZ_e+Z_eI_e) leaves fidelity unaffected because |Psi^-> is in a DFS. (1) Structural: U_z(eps)|Psi^->=e^{i phi}|Psi^-> for all eps; ||=1.0 and residual ~1.6e-16 for eps in {0.1..2.3}. |Psi^-> lives in the {down,up} sector, sigma_z,de gives a common collective phase to both terms => global phase only. (2) Dynamical: per-cycle U_z(sqrt(2p)) leaves steady-state F=1 to machine precision (max 1-F=1.1e-14) for p in {0..0.1}. Clean mathematical fact (collective-sigma_z invariance of the singlet), not a reproduced measurement. verified.

Verification details

Executable rerun: run.py exited 0 in 0.54s; log verification/C026/attempts/R002/run.log.

Output excerpt:

=== (1) Structural: U_z(eps)|Psi^-> = phase * |Psi^-> ? ===
  eps=0.10: |<Psi^-|U|Psi^->|=1.0000000000, ||U|Psi^-> - <..>|Psi^->|| = 1.573e-16
  eps=0.50: |<Psi^-|U|Psi^->|=1.0000000000, ||U|Psi^-> - <..>|Psi^->|| = 1.618e-16
  eps=1.00: |<Psi^-|U|Psi^->|=1.0000000000, ||U|Psi^-> - <..>|Psi^->|| = 1.755e-16
  eps=2.30: |<Psi^-|U|Psi^->|=1.0000000000, ||U|Psi^-> - <..>|Psi^->|| = 1.755e-16
=== (2) Dynamical: steady-state F with U_z(sqrt(2p)) per cycle ===
       p           F_ss          1-F
   0.000   1.0000000000    5.232e-12
   0.005   1.0000000000    5.412e-12
   0.010   1.0000000000    5.652e-12

Supporting files