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f1f7953 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 | 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
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