| import OAI.InformationTheory.SecretKey.Main |
| import OAI.InformationTheory.SecretKey.ChoiTransferProof |
| import OAI.Analysis.Quantum.DimensionTen.Channel |
| import OAI.InformationTheory.SecretKey.DimensionTenModel |
|
|
| namespace OAI |
|
|
| /-! Key separation for the dimension-ten state in the finite terminal-readout model. -/ |
|
|
| noncomputable section |
| open Matrix |
| open scoped ComplexOrder MatrixOrder |
|
|
| namespace ZeroKey.TenDimensional |
|
|
| universe u |
|
|
| lemma rho_density : Density rho := DimensionTen.rho_density |
|
|
| lemma trace_ne_zero : trace DimensionTen.compositeChoi ≠ 0 := |
| DimensionTen.compositeChoi_trace_ne |
|
|
| lemma rho_not_separable : ¬ Separable rho := DimensionTen.rho_entangled |
|
|
| lemma rho_productfree : |
| ∀ u v : Fin 10 → ℂ, ∀ w : Fin 10 × Fin 10 → ℂ, |
| rho *ᵥ w = DimensionTen.productVector u v → u = 0 ∨ v = 0 := |
| DimensionTen.rho_productfree |
|
|
| def TransferInput : Prop := |
| ChoiTransfer.PublishedTransfer DimensionTen.phiOneLinear DimensionTen.phiTwoLinear |
|
|
| def PPTInput : Prop := |
| ChoiTransfer.PublishedPPT DimensionTen.phiOneLinear DimensionTen.phiTwoLinear |
|
|
| lemma rho_inClass_conditional (hppt : PPTInput) (htransfer : TransferInput) : |
| InClass rho := by |
| have h := ChoiTransfer.normalized_inClass_conditional |
| DimensionTen.phiOneLinear DimensionTen.phiTwoLinear hppt htransfer |
| rw [DimensionTen.linear_composite_choi] at h |
| exact h DimensionTen.compositeChoi_posSemidef DimensionTen.compositeChoi_ne |
|
|
| theorem main_state_conditional (hppt : PPTInput) (htransfer : TransferInput) : |
| MainClaim.{u} := by |
| have hc := rho_inClass_conditional hppt htransfer |
| exact ⟨rho_density, rho_not_separable, rho_productfree, |
| class_distillableSecretKey_zero rho rho_density hc, |
| class_uniformProtocolGap rho rho_density hc, |
| class_uniformCompletedGap rho rho_density hc⟩ |
|
|
| end ZeroKey.TenDimensional |
|
|
| noncomputable section |
|
|
| namespace ZeroKey.TenDimensional |
|
|
| universe u |
|
|
| theorem pptInput : PPTInput := |
| ChoiTransfer.publishedPPT DimensionTen.phiOneLinear_ppt DimensionTen.phiTwoLinear_ppt |
|
|
| theorem transferInput : TransferInput := |
| ChoiTransfer.publishedTransfer DimensionTen.phiOneLinear DimensionTen.phiTwoLinear |
| DimensionTen.phiOneLinear_cp |
|
|
| theorem rho_inClass : InClass rho := |
| rho_inClass_conditional pptInput transferInput |
|
|
| theorem main_state : MainClaim.{u} := |
| main_state_conditional pptInput transferInput |
|
|
| end ZeroKey.TenDimensional |
|
|
| end |
| end |
|
|
| end OAI |
|
|
