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Generation of a maximally entangled state using collective optical pumping

verification/C061/attempts/R002/claim_report.md

C061 - Spontaneous emission from |e> with optical-qubit lifetime T_1 limits an MS gate of length...

Verdict: partial Location: Supp. Mat. S3 Type / expected artifact: math / math Claim: Spontaneous emission from |e> with optical-qubit lifetime T_1 limits an MS gate of length t to a Bell-state error ~= 0.5 t/T_1, and the amplified error gives a lower bound on the protocol steady-state infidelity of ~= 3 t/T_1. 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. Source location(s): source/supp_content.tex:117 (Supp. Mat. S3).

Conclusion

Paper: MS-gate Bell error ~0.5 t/T1; protocol steady-state infidelity lower bound ~3 t/T1. Derivation: expected decays over a gate = _avg * t/T1 with _avg ~ 0.5 (final |ee> has 1 excited ion, population rises 0->1), reproducing 0.5 t/T1 as an order-of-magnitude estimate (linear-area numeric 0.36, same order). the self-contained model reproduces the amplification factor exactly: steady-state singlet error = 3.18 p for per-cycle correlated bit-flip p (= main-text 3.2p). The ~3 t/T1 bound = amplification(3.18) x per-cycle decay(~t/T1); consistent in form but the factor 3 vs gate 0.5 requires the amplification model plus an O(1) per-cycle decay assumption, not a single clean identity. Both stated approximately. -> partial.

Verification details

Derivation excerpt: Spontaneous emission from $|e\rangle$ with optical-qubit lifetime $T_1$ limits an MS gate of length $t$ to a Bell-state error $\approx 0.5\, t/T_1$, and the amplified error gives a lower bound on the protocol steady-state infidelity of $\approx 3\, t/T_1$.

Executable rerun: sympy_check.py exited 0 in 0.585s; log verification/C061/attempts/R002/sympy_check.log.

Output excerpt:

Bell point n_excited (#ions in e) at end = 1.0000  (expect 1: half pop in |ee>)
time-averaged number of excited ions over the gate <n_e> = 0.3634
=> expected decays over gate = <n_e> * t/T1 = 0.363 t/T1
   each decay (from a Bell state) is ~ a Bell-state error, so MS error ~ 0.363 t/T1
   paper states ~0.5 t/T1; computed coefficient = 0.363
per-cycle correlated bit-flip p=1e-04: steady-state singlet error = 3.17713e-04  ratio err/p = 3.177
per-cycle correlated bit-flip p=1e-03: steady-state singlet error = 3.16697e-03  ratio err/p = 3.167
per-cycle correlated bit-flip p=5e-03: steady-state singlet error = 1.56130e-02  ratio err/p = 3.123
per-cycle correlated bit-flip p=1e-02: steady-state singlet error = 3.06888e-02  ratio err/p = 3.069
steady-state amplification factor (err/p) in small-p limit ~ 3.18

Supporting files