C040 - One cycle of the protocol is the superoperator S(Phi,gamma,theta) = S_C(theta) S_B(gamma)...
Verdict: verified
Location: Supp. Mat. S1
Type / expected artifact: math / math
Claim: One cycle of the protocol is the superoperator S(Phi,gamma,theta) = S_C(theta) S_B(gamma) S_A(Phi), where S_A and S_C are the superoperator forms of U_A and U_C and S_B is the composition of two independent single-ion decay channels.
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:9-13 (Supp. Mat. S1).
Conclusion
Confirmed the column-stacking superoperator identities vec(U rho U^dag)=kron(U,U)vec(rho) (err 8e-17) and vec(sum_k K_k rho K_k^dag)=(sum_k kron(K_k,K_k))vec(rho) (err 0). S_B factorizes as two independent single-ion decay channels (err 1e-16). The ordered product S(Phi,gamma,theta)=S_C S_B S_A reproduces the by-hand physical cycle (apply U_A, then two-ion Kraus B, then U_C) to err <1e-16 at three parameter sets, while the reversed ordering disagrees (err ~0.3-0.5). Tolerance 1e-12; satisfied. Clean composition identity -> verified.
Verification details
Executable rerun: run.py exited 0 in 0.517s; log verification/C040/attempts/R002/run.log.
Output excerpt:
unitary vec identity err: 7.881846445770684e-17
kraus vec identity err: 0.0
S_B independent-channel factorization err: 1.35619434333351e-16
(Phi=0.785,g=0.974,th=3.142) cycle err=3.49e-17 wrong-order err=4.764e-01
(Phi=0.300,g=0.691,th=2.262) cycle err=9.70e-17 wrong-order err=2.871e-01
(Phi=0.785,g=0.694,th=2.249) cycle err=9.44e-17 wrong-order err=4.476e-01
PASS