Kent et al. 2022 · Theorem 1

Citations

• https://doi.org/10.1038/s41534-022-00524-4(opens in a new tab)
• https://www.nature.com/articles/s41534-022-00524-4.pdf(opens in a new tab)
• https://media.springernature.com/original/springer-static/esm/art%3A10.1038%2Fs41534-022-00524-4/MediaObjects/41534_2022_524_MOESM1_ESM.pdf(opens in a new tab)

Record

Every field of the source record, in file order.

Proof obligationproof_obligation

P-S2

Gamegame

S2

Source paper idsource_paper_id

P03

Source urlsource_url

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

Anchoranchor

Theorem 1, eqs.(7)-(10), main text PDF p.5-6; lambda and O(theta) defined in Table 5 items 2-3 (online table 5); assumptions Table 6 A-G; proof Supplementary Note VII

Statusstatus

VERIFIED_FROM_SOURCE

Preconditions latexpreconditions_latex

• –0<\gamma_{\mathrm{err}}<\lambda(\theta,\beta_{\mathrm{PB}})
• –0<P_{\mathrm{noqub}}<\nu_{\mathrm{unf}}<\min\left\{2P_{\mathrm{noqub}},\ \gamma_{\mathrm{det}}\left(1-\frac{\gamma_{\mathrm{err}}}{\lambda(\theta,\beta_{\mathrm{PB}})}\right)\right\}
• –0<\beta_{\mathrm{PS}}<\frac{1}{2}\left[e^{\frac{\lambda(\theta,\beta_{\mathrm{PB}})}{2}\left(1-\frac{\delta}{\lambda(\theta,\beta_{\mathrm{PB}})}\right)^{2}}-1\right]
• –f(\gamma_{\mathrm{err}},\beta_{\mathrm{PS}},\beta_{\mathrm{PB}},\theta,\nu_{\mathrm{unf}},\gamma_{\mathrm{det}})>0

Auxiliary latexauxiliary_latex

f: f=(\gamma_{\mathrm{det}}-\nu_{\mathrm{unf}})\left[\frac{\lambda(\theta,\beta_{\mathrm{PB}})}{2}\left(1-\frac{\delta}{\lambda(\theta,\beta_{\mathrm{PB}})}\right)^{2}-\ln(1+2\beta_{\mathrm{PS}})\right]-\left(1-(\gamma_{\mathrm{det}}-\nu_{\mathrm{unf}})\right)\ln\left[1+h(\beta_{\mathrm{PS}},\beta_{\mathrm{PB}},\theta)\right]
h: h(\beta_{\mathrm{PS}},\beta_{\mathrm{PB}},\theta)=2\beta_{\mathrm{PS}}\sqrt{\frac{1}{2}+2\beta_{\mathrm{PB}}^{2}+\left(\frac{1}{2}-2\beta_{\mathrm{PB}}^{2}\right)\sin(2\theta)}
delta: \delta=\frac{\gamma_{\mathrm{det}}\gamma_{\mathrm{err}}}{\gamma_{\mathrm{det}}-\nu_{\mathrm{unf}}}
lambda: \lambda(\theta,\beta_{\mathrm{PB}})=\frac{1}{2}\left(1-\sqrt{1-[1-(O(\theta))^{2}](1-4\beta_{\mathrm{PB}}^{2})}\right)
O: O(\theta)=\frac{1}{\sqrt2}(\cos\theta+\sin\theta)

Bound latexbound_latex

\epsilon_{\mathrm{unf}}=e^{-\frac{P_{\mathrm{noqub}}N}{3}\left(\frac{\nu_{\mathrm{unf}}}{P_{\mathrm{noqub}}}-1\right)^{2}}+e^{-Nf(\gamma_{\mathrm{err}},\beta_{\mathrm{PS}},\beta_{\mathrm{PB}},\theta,\nu_{\mathrm{unf}},\gamma_{\mathrm{det}})}

Bound plainbound_plain

exp(-(P_noqub*N/3)*(nu_unf/P_noqub - 1)**2) + exp(-N*f)

Variablesvariables

Show structured value21 lines

Assumptions table6assumptions_table6

A: Bob prepares each qubit state from exactly two orthonormal qubit bases
B: t and u generated with exactly product distributions (no correlations)
C: Lambda gives Bob no information about W and z
D: 'QT1: conditioned on k in Lambda, Alice''s basis is uniform (P_MB = 1/2)'
E: Bob cannot use un-agreed degrees of freedom to affect/learn Alice's measurement statistics (no side
  channels)
F: 'photonic: Alice''s detectors have equal efficiency eta, dark counts with (1-d0)(1-d1) = (1-d+)(1-d-),
  independent'
G: 'photonic: from Alice''s perspective Bob''s pulses are Fock-state mixtures (phase randomised)'

Allowed imperfections table5allowed_imperfections_table5

items 1-8: multi-photon/non-qubit states (P_noqub); basis uncertainty theta; biases beta_PS, beta_PB; losses (Lambda, gamma_det); arbitrary pair of measurement bases D0, D1; measurement errors (gamma_err); bias beta_E; threshold detectors with dark counts

Quantifier over adversariesquantifier_over_adversaries

dishonest Alice, arbitrary quantum strategy; only possible restriction is indirect, through assumption G (Discussion, PDF p.7)

Out of domain ruleout_of_domain_rule

If (7) fails or f <= 0, the theorem gives no bound. The QMint status must be UNCERTIFIED and must not be extrapolated.

Published parameter tuplespublished_parameter_tuples

Show structured value30 lines

Lane a sanity checklane_a_sanity_check

RECOMPUTED_BY_LANE_A: with mu=0.09 (P_noqub=0.0038150) and N=4e7, the first term of (10) reaches 5e-10 at nu_unf ~ 3.89e-3, which matches the reported 3.9e-3 'minimum value' after rounding. At theta=5 deg, beta=2.3e-4 and gamma_err just above E=0.03, f is about 1.8e-5 > 0, which agrees with the Fig.5 region. This is a consistency check of the transcription, not a reproduction of Fig.5.

Verificationverification

statement_summary: VERIFIED_FROM_SOURCE
preconditions_latex: VERIFIED_FROM_SOURCE
auxiliary_latex: VERIFIED_FROM_SOURCE
bound_latex: VERIFIED_FROM_SOURCE
assumptions_table6: VERIFIED_FROM_SOURCE
published_parameter_tuples: VERIFIED_FROM_SOURCE
lane_a_sanity_check: RECOMPUTED_BY_LANE_A
supplementary_proof_correctness: NOT_VERIFIED

Not verified reasonsnot_verified_reasons

supplementary_proof_correctness: Lane A transcribed the statements and didn't audit the SI Note VII proof
  line by line. Lane D/M need to do that.

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