C054 - In the limit |alpha(t)|<<1, residual spin-motion entanglement creates a correlated bit-fl...
Verdict: partial
Location: Supp. Mat. S2
Type / expected artifact: math / math
Claim: In the limit |alpha(t)|<<1, residual spin-motion entanglement creates a correlated bit-flip with probability p = |alpha(t)|^2, with Kraus operator I_e X_e + X_e I_e.
Models: extraction claude-opus-4-8; verification gpt-5; verification_chain claude-opus-4-8 -> gpt-5; verdict_chain partial -> partial.
Limitations: paper_text_only_reimplementation, normalization_convention, oscillator_truncation.
Source location(s): source/supp_content.tex:84 (Supp. Mat. S2).
Conclusion
Expanded D=exp(alpha S_x a^dag - alpha S_x a), S_x = X_e^(1)+X_e^(2), on |dd>|0> (qutip, Fock cutoff 20). One-quantum Kraus branch K_1 = <1|D|0> = alpha(I_eX_e + X_eI_e)|dd> exactly to first order (ratio 0.99998), confirming the correlated bit-flip Kraus op I_eX_e+X_eI_e and the single-quantum mechanism. Its population scales as |alpha|^2 (||K_1|dd>||^2/|alpha|^2 -> 2.000 = ||S_x|dd>||^2 as alpha->0); the absolute factor 2 arises because the collective S_x on |dd> has two single-ion flip terms (paper's p=|alpha|^2 is the per-ion single-quantum probability). Structure and |alpha|^2 scaling confirmed. Partial: paper-text-only model, per-ion-vs-collective alpha normalization, small truncation.
Verification details
Executable rerun: run.py exited 0 in 0.978s; log verification/C054/attempts/R002/run.log.
Output excerpt:
one-quantum branch / (alpha*(I_eX_e+X_eI_e)|dd>) per component:
[0. +0.j 0.999998+0.j 0.999998+0.j 0. +0.j]
K_1 == alpha*(I_eX_e + X_eI_e) on |dd> (1st order): True
alpha p_one(numeric) |alpha|^2 p_one/|alpha|^2
0.1000 1.92157888e-02 1.00000000e-02 1.921579
0.0500 4.95024917e-03 2.50000000e-03 1.980100
0.0200 7.98721023e-04 4.00000000e-04 1.996803
0.0100 1.99920016e-04 1.00000000e-04 1.999200
0.0050 4.99950002e-05 2.50000000e-05 1.999800
p_one/|alpha|^2 -> 1.999800 as alpha->0