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Proof obligation · P-S2

KLPR2022 Lemmas 2,3,4 (Eqs. 2-6; proofs SI Note V), Theorem 1 (Eqs. 7-10; proof SI Note VII), Theorem 2 (Eq. 11; SI Not…

s_money_kent_2022_realistic · A. Kent, D. Lowndes, D. Pitalua-Garcia, J. Rarity. Practical quantum tokens without quantum memories and experimental tests. npj Quantum Information 8, 28 (2022). doi:10.1038/s41534-022-00524-4

Status
PUBLISHED REPRODUCEDLane D reproduced the published bound in code with an independent check.
Game
S2
Anchor
KLPR2022 Lemma 2, Eqs. (2)-(3); SI Eqs. (58)-(59); KLPR2022 Lemma 3, Eqs. (4)-(5); KLPR2022 Lemma 4, Eq. (6); 2^M points Eq. (11); KLPR2022 Theorem 1, Eqs. (7)-(10); SI Eqs. (99)-(102); 2^M points Theorem 2 Eq. (11), SI Theorem 3 Eqs. (194)-(198)
Source file
proofs/OBLIGATIONS.yaml

Source and theorem

KLPR2022 Lemmas 2,3,4 (Eqs. 2-6; proofs SI Note V), Theorem 1 (Eqs. 7-10; proof SI Note VII), Theorem 2 (Eq. 11; SI Note VIII, Theorem 3 Eqs. 194-198) for the practical schemes QT1/QT2 (Tables 2,3) with imperfections of Table 5 and assumptions of Table 6.

Citations

Record

Every field of the source record, in file order.

Protocolprotocol

s_money_kent_2022_realistic

Gamegame

S2

Sourcesource

KLPR2022

Assumptionsassumptions
  • Table 6 A: each qubit-labelled state is prepared from exactly two orthonormal qubit bases.
  • Table 6 B: t and u are generated with exactly product (independent per-position) distributions with biases beta_PS, beta_PB.
  • Table 6 C/D: the reported set Lambda gives Bob no information on Alice's bases W and bit z (follows from E,F with reporting strategy 1/2, Lemma 5).
  • Table 6 E: Bob cannot use undeclared degrees of freedom to affect/learn Alice's detectors (side channels excluded).
  • Table 6 F: Alice's detectors have equal efficiency; dark-count products equal; dark counts and detections independent.
  • Table 6 G: from Alice's view Bob's pulses are Fock-state mixtures (phase-randomised / low-coherence source).
  • Table 5 item 1: multiphoton probability P_noqub > 0 is known (upper bounded); Table 5 item 2: basis misalignment angle theta in (0, pi/4).
  • Lemmas 2,3 use the Chernoff bounds of SI Proposition 1, which require INDEPENDENT Bernoulli rounds (detections, errors).
  • Unforgeability: Bob honest, Alice arbitrary; robustness/correctness: both honest; privacy: Alice honest, Bob arbitrary (under C-F).
Required declarationsrequired_declarations
  • table6_A_two_bases
  • table6_B_product_distributions
  • table6_C_D_reporting_no_leak
  • table6_E_no_side_channels
  • table6_F_equal_detectors
  • table6_G_fock_mixtures
  • independent_rounds
Calibration familiescalibration_families
  • kent_2022
Finite n or asymptoticfinite_n_or_asymptotic

finite N (non-asymptotic Chernoff-type bounds)

Parametersparameters
Quantifier over adversariesquantifier_over_adversaries

eps_unf: for every strategy of dishonest Alice with unbounded quantum power (only indirect restriction via Table 6 G); eps_priv: for every strategy of Bob subject to Table 6 C-F.

Mathematical boundmathematical_bound

eps_rob = exp(-(P_det N/2)(1 - gamma_det/P_det)^2) [Eq. 3, domain Eq. 2] eps_cor = exp(-(P_det(1-2b_PB)N/4)(1 - 2 nu_cor/(P_det(1-2b_PB)))^2) + exp(-(E nu_cor N/3)(gamma_err/E - 1)^2) [Eq. 5, domain Eq. 4] eps_priv = beta_E [Eq. 6] eps_unf = exp(-(P_noqub N/3)(nu_unf/P_noqub - 1)^2) + exp(-N f) [Eq. 10, domain Eq. 7 and f > 0] f = (g_det - nu_unf)[lambda/2 (1 - delta/lambda)^2 - ln(1 + 2 b_PS)] - (1 - (g_det - nu_unf)) ln(1 + h) [Eq. 8] h = 2 b_PS sqrt(1/2 + 2 b_PB^2 + (1/2 - 2 b_PB^2) sin(2 theta)), delta = g_det g_err/(g_det - nu_unf) [Eq. 9] lambda = (1/2)(1 - sqrt(1 - [1 - O(theta)^2](1 - 4 b_PB^2))), O(theta) = (cos theta + sin theta)/sqrt 2 [Table 5 item 3, item 2] 2^M points (Eq. 11): rob^M = M rob, cor^M = M cor, priv^M = 2^-M[(1 + 2 priv)^M - 1], unf^M = C unf, C = L(L-1)/2

Numerical methodnumerical_method

natural-log space with log-sum-exp; constraints checked exactly as stated (strict inequalities)

Verification scriptverification_script

qmint/proofs/finite_size.py; tests/unit/proofs/test_finite_size.py; qmint/proofs/reproduce.py

Independent crosscheckindependent_crosscheck

qmint/proofs/checker.py (Decimal, own trig series, brute-force interval corners)

Caveatscaveats
  • Out-of-domain parameters (any constraint of Eqs. 2,4,7 violated, or f <= 0) return NO value, never an extrapolated exponential.
  • The paper's own experimental tuple (beta_PB=2.4e-3, beta_PS=3.6e-3, theta<=10deg, E=0.058) does NOT satisfy Theorem 1 (f <= 0): unforgeability is not certified by the paper for its experiment, as the paper itself states.
  • nu_cor, nu_unf, gamma_err are analysis/protocol choices; any admissible choice gives a valid bound.
  • Assumptions A,B,E,F,G are physical idealisations that cannot be verified from counts; they must be declared and are flagged as such.
Statusstatus

PUBLISHED_REPRODUCED

Reproduced examplesreproduced_examples
  • eps_rob = e^-1052 at N=4e7, P_det=0.019, gamma_det=0.018 (paper Methods p.12)
  • P_noqub = 1-(1+mu)e^-mu = 3.8e-3 at mu=0.09; P_det_theo = 2(e^{-mu eta/2} - e^{-mu eta}) = 0.01863 at eta=0.21 (Eq. 17)
  • beta_PB=2.4e-3, beta_PS=3.6e-3 and E=0.058 from the published counts (Methods p.13)
  • Fig. 5 beta_max: theta=10deg,E=0.058 -> paper 6e-6 (ours 6.9e-6); theta=5deg,E=0.03 -> paper 2.3e-4 (ours 2.25e-4); theta->0,E=0.01 -> paper ~6e-4 (ours 5.9e-4)
Boundsbounds

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