Theorem · T-SCH23-SDP-N1
Schiansky et al. 2023 · Supplementary Information
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.
- Game
- D1
- Anchor
- Supplementary Information, Supplementary Note 3A 'For N = 1', eqs.(9)-(15), SI PDF p.4-6; main text Methods 'Optimal cheating strategy'
- Source file
- research/THEOREM_REGISTRY_SEED.yaml
Statement summary
Two-merchant double-spend: from the token state rho_P produce two classical cryptograms kappa_0 (Z basis, M0) and kappa_1 (X basis, M1) that both pass. The SDP minimises the error e = Tr(E0 J(Lambda)) the adversary must cause while declaring at most l losses per cryptogram.
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 Information, Supplementary Note 3A 'For N = 1', eqs.(9)-(15), SI PDF p.4-6; main text Methods 'Optimal cheating strategy'
- Statusstatus
VERIFIED_FROM_SOURCE
- Sdp primalsdp_primal
min Tr(E0 J) s.t. Tr_{H0 (x) H1}(J) = 1_{HP}; Tr(E0 J) >= Tr(E1 J); Tr(L0 J) <= l; Tr(L1 J) <= l; J >= 0 (eq. 14)
- Sdp dual as printedsdp_dual_as_printed
max Tr[X1] + l (x2 + x3) s.t. 1_9 (x) X1 + L0 x2 + L1 x3 + (E1 - E0) x4 - E0 <= 0, X1 Hermitian 2x2, x2,x3,x4 real (eq. 15)
- Spacesspaces
H0, H1: 3-dim {|a0>,|a1>,|empty>}; HP: 7-dim {|v>,|q0>,|q1>,|m0>..|m3>} (vacuum, qubit, 4 orthogonal multiphoton flags)
- Statesstates
sigma_k = p0|v><v| + p1 |s_k><s_k| + pm |m_k><m_k| with s_k in {|+>,|+i>,|->,|-i>} (X and Y eigenstates in the SI); rho_P = (1/4) sum_k sigma_k (eq. 9)
- Operatorsoperators
E0 = (1/4) sum_{k=0}^{3} (1/2)|a_k^perp><a_k^perp| (x) 1 (x) conj(sigma_k); E1 = (1/4) sum_k 1 (x) (1/2)|a_k^perp><a_k^perp| (x) conj(sigma_k); L0 = (1/4) sum_k |empty><empty| (x) 1 (x) conj(sigma_k); L1 = (1/4) sum_k 1 (x) |empty><empty| (x) conj(sigma_k) (eqs. 12-13; the 1/2 appears in E0/E1 only, as typeset). P0 = Tr sum_k ((1/2)|a_k^perp><a_k^perp| (x) 1) Lambda((1/4) sigma_k) (eq. 10).
- Observations for lane Cobservations_for_lane_C
- The SI dual says 'X1 is a hermitian 2x2 matrix' but HP is 7-dimensional, so the dual variable for Tr_{H0H1} J = 1_HP should be 7x7. Either a typo or a different reduced space. Check before reusing it.
- SI eq.(6) defines the Choi operator with a 1/d prefactor, J = (1/d) sum Lambda(|i><j|) (x) |i><j|, and states trace preservation as Tr_{out} J = 1. With the 1/d normalisation that should read (1/d)1, so the normalisation convention is internally inconsistent as printed. MVW (P01) uses no 1/d. Lane C must fix one convention and document it.
- The SI writes the qubit states as X/Y eigenstates. The main text writes H/V and D/A. That's a relabeling of an unbiased basis pair, but lanes B/C must use one consistent convention (C7: Z/X).
- Verificationverification
statement_summary: VERIFIED_FROM_SOURCE sdp_primal: VERIFIED_FROM_SOURCE sdp_dual_as_printed: VERIFIED_FROM_SOURCE operators: VERIFIED_FROM_SOURCE