Theorem · T-SCH23-ASYMPTOTIC
Schiansky et al. 2023 · Supplementary Note 3B
P04 · Nat. Commun. 14, 3849 (published 2023-06-29; corrected 2023-08-21). Read after correction.
- Status
- VERIFIED FROM SOURCEField read in the primary source at the anchor given.VERIFIED_FROM_SOURCE; NOT_VERIFIED fields: coherent_attack_reduction
- Game
- D1
- Anchor
- Supplementary Note 3B, eqs.(16)-(18), SI PDF p.6-7; main text Fig.4a
- Source file
- research/THEOREM_REGISTRY_SEED.yaml
Statement summary
Analytic fit to the optimal cheating boundary: e = -(1/(4+2 sqrt2)) l + (1 - p_m/2)/(8 + 4 sqrt2) (16). Equivalently, the secure region is M(e,l) = (4+2 sqrt2) e + l <= (1/2)(1 - p_m/2) (18). N -> infinity is justified 'following the exponential de Finetti arguments' (qualitative).
Citations
Record
Every field of the source record, in file order.
- Proof obligationproof_obligation
- P-D1
- Gamegame
D1
- Source paper idsource_paper_id
- P04
- Source urlsource_url
https://doi.org/10.1038/s41467-023-39519-w
- Anchoranchor
Supplementary Note 3B, eqs.(16)-(18), SI PDF p.6-7; main text Fig.4a
- Statusstatus
VERIFIED_FROM_SOURCE
- Bound latexbound_latex
M(e,l)=(4+2\sqrt{2})\,e+l\le\frac{1}{2}\left(1-\frac{p_m}{2}\right)LaTeX source - Variablesvariables
e: fraction of declared outcomes that disagree with (b, B) l: fraction of token states declared lost p_m: multiphoton emission probability
- Recomputationrecomputation
RECOMPUTED_BY_LANE_A: e*(l=0.224) = 3.795% with p_m=6.76% (main-text value) and 3.822% with p_m=6.02% (SI Table 2 value). These match the main-text 3.79% and SI 3.82%. So the two documents use different p_m. Allowed mishaps M = 0.5(1-0.0602/2) = 0.48495 matches SI 0.4849.
- Caveatscaveats
- eq.(16) is described as 'an analytical expression that fits' the SDP optimum. It's a fit, not a derived theorem.
- Collective-to-coherent reduction (de Finetti) is argued, not quantified.
- Verificationverification
statement_summary: VERIFIED_FROM_SOURCE bound_latex: VERIFIED_FROM_SOURCE recomputation: RECOMPUTED_BY_LANE_A coherent_attack_reduction: NOT_VERIFIED
- Not verified reasonsnot_verified_reasons
coherent_attack_reduction: SI Note 3B gives only a qualitative de Finetti argument with no explicit finite-N correction term.