| import OAI.InformationTheory.SecretKey.Extensions |
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| namespace OAI |
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| noncomputable section |
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| open Matrix MeasureTheory ProbabilityTheory Filter |
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| open scoped ComplexOrder MatrixOrder Matrix.Norms.L2Operator ENNReal |
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| namespace ZeroKey |
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| universe u |
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| theorem class_uniformProtocolGap [Nonempty (Fin a)] [Nonempty (Fin b)] |
| (R : Matrix (Fin a × Fin b) (Fin a × Fin b) ℂ) (hR : Density R) (hc : InClass R) : |
| UniformProtocolGap. R hR := by |
| intro k O M _ _ _ _ _ _ d A B _ _ _ _ _ _ _ _ TA TB CA CB _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ p e P hP sigma hs |
| exact p.uniform_gap_purification (positiveCopies R k) (hc.positiveCopies k) P hP sigma hs |
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| theorem class_uniformCompletedGap [Nonempty (Fin a)] [Nonempty (Fin b)] |
| (R : Matrix (Fin a × Fin b) (Fin a × Fin b) ℂ) (hR : Density R) (hc : InClass R) : |
| UniformCompletedGap. R hR := by |
| intro k ell hell p |
| exact p.uniform_error (hc.positiveCopies k) hell |
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| namespace Explicit |
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| /-- The explicit density witness removes a proof dependency from the challenge statement. |
| This is equivalent to the original five-clause formulation. -/ |
| theorem mainClaim_iff_original : MainClaim. ↔ |
| (Density rhoFin ∧ ¬ Separable rhoFin ∧ |
| distillableSecretKey. rhoFin rhoFin_density = 0 ∧ |
| UniformProtocolGap. rhoFin rhoFin_density ∧ |
| UniformCompletedGap. rhoFin rhoFin_density) := by |
| constructor |
| · rintro ⟨hR, hsep, hkey, hprotocol, hcompleted⟩ |
| exact ⟨hR, hsep, hkey, hprotocol, hcompleted⟩ |
| · rintro ⟨hR, hsep, hkey, hprotocol, hcompleted⟩ |
| exact ⟨hR, hsep, hkey, hprotocol, hcompleted⟩ |
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| theorem main : MainClaim. := by |
| exact ⟨rhoFin_density, rhoFin_not_separable, |
| class_distillableSecretKey_zero rhoFin rhoFin_density rhoFin_inClass, |
| class_uniformProtocolGap rhoFin rhoFin_density rhoFin_inClass, |
| class_uniformCompletedGap rhoFin rhoFin_density rhoFin_inClass⟩ |
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| end Explicit |
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| end ZeroKey |
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| end |
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| end OAI |
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