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Theorem · T-KLPR22-LEMMA1

Kent et al. 2022 · Lemma 1

P03 · npj Quantum Inf. 8, 28 (published 2022-03-11); equations transcribed from the article HTML TeX source

Status
VERIFIED FROM SOURCEField read in the primary source at the anchor given.
Game
S1
Anchor
Lemma 1, eq.(1), main text PDF p.5; protocols IQT1/IQT2 in Table 1 (PDF p.3); proof Supplementary Note IV, SI p.5-6 (eqs 47-48), using Lemma 6 (SI Note III)
Source file
research/THEOREM_REGISTRY_SEED.yaml

Statement verbatim

Lemma 1: The quantum token schemes IQT1 and IQT2 are eps_unf-unforgeable with eps_unf = (1/2 + 1/(2 sqrt 2))^N.

Citations

Record

Every field of the source record, in file order.

Proof obligationproof_obligation
P-S1
Gamegame

S1

Source paper idsource_paper_id
P03
Source urlsource_url

https://doi.org/10.1038/s41534-022-00524-4

Anchoranchor

Lemma 1, eq.(1), main text PDF p.5; protocols IQT1/IQT2 in Table 1 (PDF p.3); proof Supplementary Note IV, SI p.5-6 (eqs 47-48), using Lemma 6 (SI Note III)

Statusstatus

VERIFIED_FROM_SOURCE

Bound latexbound_latex
\epsilon_{\mathrm{unf}}=\left(\frac{1}{2}+\frac{1}{2\sqrt{2}}\right)^{N}
LaTeX source
Bound plainbound_plain

(0.5 + 1/(2*sqrt(2)))**N # base = cos^2(pi/8) = 0.853553390593...

Variablesvariables
N: number of BB84 qubits Bob sends Alice (no losses in the ideal scheme)
Definitionsdefinitions
eps_unf: 'eps_unf-unforgeable: probability that Bob accepts Alice''s tokens as valid at any two or more
  different presentation points is <= eps_unf, if Bob follows the scheme honestly (main text p.3).'
Game definitiongame_definition

Ideal two-presentation-point scheme (M=1). IQT1 (Table 1): B sends N BB84 states |phi_{t_k u_k}>. A measures each in random basis y_k, keeps outcomes x, sends d = y (z=0) or the bitwise complement of y (z=1) to B. In stage II, A chooses b, sends c = b xor z. B_i computes d~_i, and B_b validates x if x_b = t_b on Delta_b = {k : d~_{b,k} = u_k}. IQT2: A measures all qubits in one basis z. Dishonest Alice wins if she gets valid tokens at BOTH Q0 and Q1 (SI p.5: a_0 = t_0 and b_1 = t_1). Proof: Lemma 6 with gamma_err=0, beta_PB=0, theta=0 (O = 1/sqrt2), so lambda = 1/2 - 1/(2 sqrt2) and max ||D_{a,b}|| <= (1-lambda)^N.

Assumptionsassumptions
  • no losses, no errors, exact BB84 states, uniform bases/states (beta_PB = beta_PS = 0)
  • secure and authenticated classical channels among agents of the same party; secure labs
  • Bob honest (unforgeability is against dishonest Alice)
Quantifier over adversariesquantifier_over_adversaries

general cheating strategy: joint projective measurement on the N qubits plus an arbitrary finite-dimensional ancilla (SI p.5)

Tightnesstightness

upper bound (no tightness claim found in the main text)

Not valid fornot_valid_for
  • W1 / W1_CV Wiesner games (different game; MVW W1_CV base is 3/4+sqrt2/8 = 0.9268, not 0.8536)
  • the realistic schemes QT1/QT2 with loss/noise/bias (use T-KLPR22-THM1)
  • 2^M presentation points without the Theorem 2 pair-count factor
Verificationverification
statement_verbatim: VERIFIED_FROM_SOURCE
bound_latex: VERIFIED_FROM_SOURCE
definitions: VERIFIED_FROM_SOURCE
game_definition: VERIFIED_FROM_SOURCE

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