| import Mathlib |
| import OAI.Analysis.TracialSplitting.PreparedBlocks |
| import OAI.Analysis.JiangSu.DimensionDrop.IteratedSpectrum |
| import OAI.Analysis.JiangSu.DimensionDrop.SortingNetwork |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
|
|
| namespace JiangSuConstruction |
| open scoped CStarAlgebra ComplexOrder Kronecker |
|
|
| theorem continuous_matching_control_two_scales (p n : ℕ) {δ r : ℝ} (hr : 0<r) |
| (x y : Fin n → Interval) |
| (hxy : ∀ t : ℝ,(Finset.univ.filter fun i => (x i:ℝ)≤t).card≤ |
| (Finset.univ.filter fun i => (y i:ℝ)≤t+δ).card) |
| (hyx : ∀ t : ℝ,(Finset.univ.filter fun i => (y i:ℝ)≤t).card≤ |
| (Finset.univ.filter fun i => (x i:ℝ)≤t+δ).card) |
| (f : Interval → Mat p) {η ηs : ℝ} (hη : 0≤η) (hηs : 0≤ηs) |
| (hf : ∀ s t,dist s t≤δ → ‖f s-f t‖≤η) |
| (hfs : ∀ s t,dist s t≤r → ‖f s-f t‖≤ηs) : |
| ‖unitaryConjugation (continuousMatchingUnitary p n r (x,y)) |
| (blockDiagonalHom (f ∘ x))-blockDiagonalHom (f ∘ y)‖≤ |
| 6*((sortingNetwork n).length:ℝ)*ηs+η := by |
| let ux := networkUnitary p n r (sortingNetwork n) x |
| let uy := networkUnitary p n r (sortingNetwork n) y |
| let X := blockDiagonalHom (f ∘ x) |
| let Y := blockDiagonalHom (f ∘ y) |
| let SX := blockDiagonalHom (f ∘ (x ∘ Tuple.sort x)) |
| let SY := blockDiagonalHom (f ∘ (y ∘ Tuple.sort y)) |
| have hx : ‖unitaryConjugation ux X-SX‖≤3*((sortingNetwork n).length:ℝ)*ηs := |
| continuous_sort_unitary_control p n hr f hηs hfs x |
| have hy : ‖unitaryConjugation uy Y-SY‖≤3*((sortingNetwork n).length:ℝ)*ηs := |
| continuous_sort_unitary_control p n hr f hηs hfs y |
| have hs : ‖SX-SY‖≤η := by |
| change ‖blockDiagonalHom _-blockDiagonalHom _‖≤_ |
| rw [←map_sub] |
| apply (NonUnitalStarAlgHom.norm_apply_le |
| (blockDiagonalHom : (Fin n → Mat p) →⋆ₐ[ℂ] TensorMat p n) _).trans |
| apply (pi_norm_le_iff_of_nonneg hη).mpr |
| intro i |
| exact hf _ _ (interval_sorted_displacement x y δ hxy hyx i) |
| have hcancel : unitaryConjugation uy⁻¹ (unitaryConjugation uy Y)=Y := by |
| rw [←unitaryConjugation_mul,inv_mul_cancel,unitaryConjugation_one] |
| have hb : ‖unitaryConjugation ux X-unitaryConjugation uy Y‖≤ |
| 6*((sortingNetwork n).length:ℝ)*ηs+η := by |
| have h₁ := norm_sub_le_norm_sub_add_norm_sub (unitaryConjugation ux X) SX SY |
| have h₂ := norm_sub_le_norm_sub_add_norm_sub (unitaryConjugation ux X) SY |
| (unitaryConjugation uy Y) |
| rw [norm_sub_rev SY] at h₂ |
| linarith |
| change ‖unitaryConjugation (uy⁻¹*ux) X-Y‖≤_ |
| rw [unitaryConjugation_mul,←hcancel,←map_sub] |
| exact (NonUnitalStarAlgHom.norm_apply_le (unitaryConjugation uy⁻¹) _).trans hb |
|
|
|
|
| lemma filter_card_equiv {I J : Type*} [Fintype I] [Fintype J] |
| (e : I ≃ J) (P : J → Prop) [DecidablePred P] : |
| (Finset.univ.filter fun i : I => P (e i)).card= |
| (Finset.univ.filter P).card := by |
| classical |
| rw [←Fintype.card_subtype,←Fintype.card_subtype] |
| exact Fintype.card_congr (e.subtypeEquiv (fun _ => Iff.rfl)) |
|
|
| abbrev iteratedSize (n k : ℕ) := Fintype.card (IteratedIndex n k) |
| def iteratedTuple (n k : ℕ) (s : Interval) : Fin (iteratedSize n k) → Interval := |
| fun j => iteratedFamily n k ((Fintype.equivFin (IteratedIndex n k)).symm j) s |
|
|
| lemma iteratedTuple_continuous (n k : ℕ) : Continuous (iteratedTuple n k) := by |
| apply continuous_pi |
| intro j |
| exact (iteratedFamily n k _).continuous |
|
|
| lemma iteratedTuple_shifted_counts (n : ℕ) {δ : ℝ} (hδ : 0<δ) : |
| ∃ k : ℕ,∀ l,k≤l → ∀ s t : Interval,∀ v : ℝ, |
| (Finset.univ.filter fun i => (iteratedTuple n l s i:ℝ)≤v).card≤ |
| (Finset.univ.filter fun i => (iteratedTuple n l t i:ℝ)≤v+δ).card := by |
| classical |
| obtain ⟨k,hk⟩ := iterated_spectrum_shifted_counts n hδ |
| refine ⟨k,?_⟩ |
| intro l hl s t v |
| change (Finset.univ.filter fun i => (iteratedFamily n l |
| ((Fintype.equivFin (IteratedIndex n l)).symm i) s:ℝ)≤v).card≤_ |
| rw [filter_card_equiv (Fintype.equivFin (IteratedIndex n l)).symm |
| (fun i => (iteratedFamily n l i s:ℝ)≤v)] |
| change _≤(Finset.univ.filter fun i => (iteratedFamily n l |
| ((Fintype.equivFin (IteratedIndex n l)).symm i) t:ℝ)≤v+δ).card |
| rw [filter_card_equiv (Fintype.equivFin (IteratedIndex n l)).symm |
| (fun i => (iteratedFamily n l i t:ℝ)≤v+δ)] |
| exact hk l hl s t v |
|
|
| lemma finite_continuous_modulus {p : ℕ} (F : Finset C(Interval,Mat p)) |
| {η : ℝ} (hη : 0<η) : |
| ∃ δ : ℝ,0<δ ∧ ∀ f∈F,∀ s t : Interval,dist s t≤δ → ‖f s-f t‖≤η := by |
| classical |
| let g : Interval → (F → Mat p) := fun s f => f.val s |
| have hc : Continuous g := by apply continuous_pi; intro f; exact f.val.continuous |
| have hu := CompactSpace.uniformContinuous_of_continuous hc |
| obtain ⟨δ,hδ,hd⟩ := Metric.uniformContinuous_iff.mp hu (η/2) (by positivity) |
| refine ⟨δ/2,by positivity,?_⟩ |
| intro f hf s t hst |
| have hh : dist (g s) (g t)<η/2 := hd (by linarith) |
| have hi := dist_le_pi_dist (g s) (g t) (⟨f,hf⟩:F) |
| have hh' : dist (g s (⟨f,hf⟩:F)) (g t (⟨f,hf⟩:F))≤η := |
| hi.trans (le_trans hh.le (by linarith)) |
| simpa only [g,dist_eq_norm] using hh' |
|
|
| theorem actual_iterated_continuous_matching (n p : ℕ) |
| (F : Finset C(Interval,Mat p)) {ε : ℝ} (hε : 0<ε) : |
| ∃ k : ℕ,∀ l,k≤l → ∃ U : C(Interval×Interval,unitary (TensorMat p (iteratedSize n l))), |
| ∀ f∈F,∀ s t : Interval, |
| ‖unitaryConjugation (U (s,t)) (blockDiagonalHom (f ∘ iteratedTuple n l s))- |
| blockDiagonalHom (f ∘ iteratedTuple n l t)‖<ε := by |
| obtain ⟨δ,hδ,hF⟩ := finite_continuous_modulus F (show 0<ε/3 by positivity) |
| obtain ⟨k,hk⟩ := iteratedTuple_shifted_counts n hδ |
| refine ⟨k,?_⟩ |
| intro l hl |
| let L : ℝ := ((sortingNetwork (iteratedSize n l)).length:ℝ) |
| have hL : 0≤L := Nat.cast_nonneg _ |
| let ηs := ε/(3*(6*L+1)) |
| have hηs : 0<ηs := by dsimp [ηs]; positivity |
| obtain ⟨r,hr,hFs⟩ := finite_continuous_modulus F hηs |
| let U : C(Interval×Interval,unitary (TensorMat p (iteratedSize n l))) := |
| (continuousMatchingUnitary p (iteratedSize n l) r).comp |
| ⟨fun z => (iteratedTuple n l z.1,iteratedTuple n l z.2), |
| ((iteratedTuple_continuous n l).comp continuous_fst).prodMk |
| ((iteratedTuple_continuous n l).comp continuous_snd)⟩ |
| refine ⟨U,?_⟩ |
| intro f hf s t |
| have hh := continuous_matching_control_two_scales p (iteratedSize n l) hr |
| (iteratedTuple n l s) (iteratedTuple n l t) (hk l hl s t) (hk l hl t s) |
| f (show 0≤ε/3 by positivity) hηs.le (hF f hf) (hFs f hf) |
| change ‖unitaryConjugation (continuousMatchingUnitary p (iteratedSize n l) r |
| (iteratedTuple n l s,iteratedTuple n l t)) _-_‖<ε |
| apply hh.trans_lt |
| change 6*L*ηs+ε/3<ε |
| have heq : ηs*(3*(6*L+1))=ε := by |
| dsimp [ηs] |
| exact div_mul_cancel₀ _ (by positivity) |
| nlinarith |
|
|
| end JiangSuConstruction |
|
|
| end |
|
|
| end OAI |
|
|
