| import OAI.Analysis.CharacterCriterion.MatrixSimplex |
|
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| namespace OAI |
|
|
| noncomputable section |
| open scoped TensorProduct |
| open scoped InnerProductSpace |
| open scoped BigOperators |
| open Set Module |
| open Topology |
| open NormedSpace |
| open scoped ComplexOrder |
| open scoped ComplexOrder |
| open scoped InnerProductSpace |
| open scoped InnerProductSpace |
| open NormedSpace |
| open scoped TensorProduct InnerProductSpace |
| open UniformSpace |
| variable [KirchbergRordam.GenericRW.Size] |
| local notation "rankNum" => KirchbergRordam.GenericRW.Size.n |
| variable [KirchbergRordam.GenericRW.PauliData] |
|
|
| namespace KirchbergRordam.GenericRW.CubeField |
| open scoped BigOperators Topology |
| open Filter MatrixMix MatrixSimplex CubeWeights |
| variable {A : Type*} [CStarAlgebra A] |
| variable {n : ℕ} {φ : Fin (n+1) → Mat (Fin Size.n) →⋆ₐ[ℂ] A} |
| variable (F : Family φ) |
|
|
| lemma norm_matrix_unit (ψ : Mat (Fin Size.n) →⋆ₐ[ℂ] A) (i j : Fin Size.n) : ‖ψ (e i j)‖≤1 := by |
| have hp : IsStarProjection (ψ (e j j)) := ⟨by change ψ _*ψ _=ψ _; rw [← map_mul,e_mul,ite_eq_left rfl],by change star (ψ _) = ψ _; rw [← map_star,e_star]⟩ |
| have hn := hp.norm_le _ |
| have he : ‖ψ (e i j)‖*‖ψ (e i j)‖=‖ψ (e j j)‖ := by |
| rw [← CStarRing.norm_star_mul_self,← map_star,← map_mul,e_star,e_mul,ite_eq_left rfl] |
| nlinarith [norm_nonneg (ψ (e i j))] |
|
|
| def dampedFun (f : C(Cube n,ℝ)) (i j : Fin Size.n) (t : Cube n) : A := |
| if ht : t≠0 then f t • F.hom (normalized ⟨t,ht⟩) (e i j) else 0 |
| lemma damped_zero (f : C(Cube n,ℝ)) (i j : Fin Size.n) : dampedFun F f i j 0=0 := by simp [dampedFun] |
| lemma damped_norm (f : C(Cube n,ℝ)) (i j : Fin Size.n) (t : Cube n) : ‖dampedFun F f i j t‖≤‖f t‖ := by |
| unfold dampedFun |
| split_ifs |
| · rw [norm_smul] |
| exact mul_le_of_le_one_right (norm_nonneg _) (norm_matrix_unit _ _ _) |
| · simp only [norm_zero]; exact norm_nonneg (f t) |
| lemma damped_continuous (f : C(Cube n,ℝ)) (hf : f 0=0) (i j : Fin Size.n) : |
| Continuous (dampedFun F f i j) := by |
| have ho : ContinuousOn (dampedFun F f i j) {t | t≠0} := by |
| rw [continuousOn_iff_continuous_domRestrict] |
| change Continuous (fun t : Punctured n => dampedFun F f i j t.1) |
| have hh := ((f.continuous.comp continuous_subtype_val).smul |
| ((F.continuous (e i j)).comp normalized_continuous)) |
| exact hh.congr (fun t => by simp [dampedFun,t.2]) |
| rw [continuous_iff_continuousAt] |
| intro t |
| by_cases ht : t=0 |
| · subst t |
| change Tendsto (dampedFun F f i j) _ (𝓝 (dampedFun F f i j 0)) |
| rw [damped_zero] |
| apply squeeze_zero_norm (damped_norm F f i j) |
| simpa [ContinuousAt,hf] using f.continuous.norm.continuousAt (x:=0) |
| · exact ho.continuousAt (isOpen_ne.mem_nhds ht) |
|
|
| def damped (f : C(Cube n,ℝ)) (hf : f 0=0) (i j : Fin Size.n) : C(Cube n,A) := |
| ⟨dampedFun F f i j,damped_continuous F f hf i j⟩ |
|
|
| def scalarH (δ : ℝ) : C(Cube n,ℝ) := ⟨H δ,H_continuous δ⟩ |
| def scalarSqrt (δ : ℝ) : C(Cube n,ℝ) := ⟨fun t => Real.sqrt (H δ t),(H_continuous δ).sqrt⟩ |
| def x (δ : ℝ) (i j : Fin Size.n) : C(Cube n,A) := damped F (scalarH δ) (H_zero δ) i j |
| def v (δ : ℝ) : C(Cube n,A) := damped F (scalarSqrt δ) (by simp [scalarSqrt,H_zero]) 1 0 |
|
|
| def h (δ : ℝ) : C(Cube n,A) := ⟨fun t => H δ t • (1:A),(H_continuous δ).smul continuous_const⟩ |
| lemma x_apply (δ : ℝ) (i j : Fin Size.n) (t : Punctured n) : |
| x F δ i j t.1=H δ t.1 • F.hom (normalized t) (e i j) := by simp [x,damped,dampedFun,t.2,scalarH] |
| lemma x_zero (δ : ℝ) (i j : Fin Size.n) : x F δ i j 0=0 := damped_zero F _ _ _ |
| lemma v_apply (δ : ℝ) (t : Punctured n) : |
| v F δ t.1=Real.sqrt (H δ t.1) • F.hom (normalized t) (e 1 0) := by simp [v,damped,dampedFun,t.2,scalarSqrt] |
| lemma v_zero (δ : ℝ) : v F δ 0=0 := damped_zero F _ _ _ |
|
|
| lemma x_star (δ : ℝ) (i j : Fin Size.n) : star (x F δ i j)=x F δ j i := by |
| ext t |
| by_cases ht : t=0 |
| · subst t; simp [x_zero] |
| · change star (x F δ i j t)=_ |
| rw [x_apply F δ i j ⟨t,ht⟩,x_apply F δ j i ⟨t,ht⟩,star_smul,star_trivial,← map_star,e_star] |
| lemma x_mul (δ : ℝ) (i j k l : Fin Size.n) : |
| x F δ i j*x F δ k l=if j=k then h δ*x F δ i l else 0 := by |
| ext t |
| by_cases ht : t=0 |
| · subst t; by_cases hjk : j=k <;> simp [hjk,x_zero,h,H_zero] |
| · change x F δ i j t*x F δ k l t=_ |
| rw [x_apply F δ i j ⟨t,ht⟩,x_apply F δ k l ⟨t,ht⟩,smul_mul_smul_comm,← map_mul,e_mul] |
| by_cases hjk : j=k |
| · rw [ite_eq_left hjk,ite_eq_left hjk] |
| change _=H δ t • (1:A)*x F δ i l t |
| rw [x_apply F δ i l ⟨t,ht⟩,smul_mul_smul_comm,one_mul] |
| · simp [hjk] |
| lemma x_diagonal (δ : ℝ) : ∑ i : Fin Size.n,x F δ i i=h δ := by |
| ext t |
| by_cases ht : t=0 |
| · subst t; simp [x_zero,h,H_zero] |
| · simp only [ContinuousMap.sum_apply] |
| simp_rw [x_apply F δ _ _ ⟨t,ht⟩] |
| rw [← Finset.smul_sum,← map_sum,e_sum,map_one] |
| rfl |
| lemma v_star_mul {δ : ℝ} (hδ : 0<δ) : star (v F δ)*v F δ=x F δ 0 0 := by |
| ext t |
| by_cases ht : t=0 |
| · subst t; simp [v_zero,x_zero] |
| · change star (v F δ t)*v F δ t=_ |
| rw [v_apply F δ ⟨t,ht⟩,x_apply F δ 0 0 ⟨t,ht⟩,star_smul,star_trivial,← map_star, |
| e_star,smul_mul_smul_comm,Real.mul_self_sqrt (H_nonneg hδ t),← map_mul,e_mul,ite_eq_left rfl] |
| lemma v_mul_star {δ : ℝ} (hδ : 0<δ) : v F δ*star (v F δ)=x F δ 1 1 := by |
| ext t |
| by_cases ht : t=0 |
| · subst t; simp [v_zero,x_zero] |
| · change v F δ t*star (v F δ t)=_ |
| rw [v_apply F δ ⟨t,ht⟩,x_apply F δ 1 1 ⟨t,ht⟩,star_smul,star_trivial,← map_star, |
| e_star,smul_mul_smul_comm,Real.mul_self_sqrt (H_nonneg hδ t),← map_mul,e_mul,ite_eq_left rfl] |
|
|
| lemma x_support (δ : ℝ) (t : Cube n) (S : StarSubalgebra ℂ A) |
| (hs : ∀ k,0<(t k:ℝ) → ∀ a,φ k a∈S) (i j : Fin Size.n) : x F δ i j t∈S := by |
| by_cases ht : t=0 |
| · subst t; rw [x_zero]; exact S.zero_mem |
| · rw [x_apply F δ i j ⟨t,ht⟩] |
| exact S.smul_mem (F.support _ S (fun k hk => hs k (normalized_support ⟨t,ht⟩ k hk)) _) _ |
|
|
| lemma x_localized {δ : ℝ} (hδ : 0<δ) (t : Punctured n) (j : Fin (n+1)) |
| (hj : δ<(t.1 j:ℝ)) (hi : ∀ i,i≠j → (t.1 i:ℝ)<δ/2) (a b : Fin Size.n) : |
| x F δ a b t.1=φ j (e a b) := by |
| rw [x_apply,localized_normalized t hδ j hj hi,F.vertex,H_of_large hδ t.1 j hj.le,one_smul] |
| end KirchbergRordam.GenericRW.CubeField |
|
|
| namespace KirchbergRordam.GenericRW.CubeField |
| open scoped BigOperators CStarAlgebra |
| open MatrixMix MatrixSimplex CubeWeights |
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
| variable {n : ℕ} {φ : Fin (n+1) → Mat (Fin Size.n) →⋆ₐ[ℂ] A} (F : Family φ) |
| lemma h_nonneg {δ : ℝ} (hδ : 0<δ) : 0≤(h δ : C(Cube n,A)) := by |
| intro t |
| exact smul_nonneg (H_nonneg hδ t) zero_le_one |
| omit [PartialOrder A] [StarOrderedRing A] in |
| lemma h_norm {δ : ℝ} (hδ : 0<δ) : ‖(h δ : C(Cube n,A))‖≤1 := by |
| apply (ContinuousMap.norm_le _ (by norm_num)).mpr |
| intro t |
| change ‖H δ t • (1:A)‖≤1 |
| rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg (H_nonneg hδ t)] |
| have hn : ‖(1:A)‖≤1 := (IsStarProjection.one (R:=A)).norm_le _ |
| exact (mul_le_of_le_one_right (H_nonneg hδ t) hn).trans (H_le_one δ t) |
|
|
| def weighted {δ : ℝ} (hδ : 0<δ) : OrderZero.WeightedUnits (Fin Size.n) C(Cube n,A) where |
| h := h δ |
| x := x F δ |
| nonneg := h_nonneg hδ |
| norm_le := h_norm hδ |
| star_x := x_star F δ |
| mul_x := x_mul F δ |
| diagonal := x_diagonal F δ |
|
|
| lemma first_nonneg {δ : ℝ} (hδ : 0<δ) : 0≤x F δ 0 0 := by |
| rw [← v_star_mul F hδ] |
| intro t |
| exact star_mul_self_nonneg (v F δ t) |
| lemma second_nonneg {δ : ℝ} (hδ : 0<δ) : 0≤x F δ 1 1 := by |
| rw [← v_mul_star F hδ] |
| intro t |
| exact mul_star_self_nonneg (v F δ t) |
| omit [PartialOrder A] [StarOrderedRing A] in |
| lemma pair_orthogonal (δ : ℝ) : x F δ 0 0*x F δ 1 1=0 := by |
| have hn : 1 < Size.n := by have := PauliData.two_le; omega |
| have h01 : (0:Fin Size.n) ≠ 1 := by |
| intro he |
| have hv := congrArg Fin.val he |
| simp only [Fin.val_zero,Fin.val_one',Nat.mod_eq_of_lt hn] at hv |
| omega |
| rw [x_mul,ite_eq_right h01] |
| omit [PartialOrder A] [StarOrderedRing A] in |
| lemma entry_norm {δ : ℝ} (hδ : 0<δ) (i j : Fin Size.n) : ‖x F δ i j‖≤1 := by |
| apply (ContinuousMap.norm_le _ (by norm_num)).mpr |
| intro t |
| exact (damped_norm F (scalarH δ) i j t).trans (by |
| change ‖H δ t‖≤1 |
| rw [Real.norm_eq_abs,abs_of_nonneg (H_nonneg hδ t)] |
| exact H_le_one δ t) |
| end KirchbergRordam.GenericRW.CubeField |
| |
| namespace KirchbergRordam.GenericRW.ConeCube |
| open scoped CStarAlgebra ComplexOrder BigOperators |
| open ConeModel MatrixSimplex |
| local instance (n : ℕ) : PartialOrder (Tensor n) := CStarAlgebra.spectralOrder _ |
| local instance (n : ℕ) : StarOrderedRing (Tensor n) := CStarAlgebra.spectralOrderedRing _ |
| |
| def family (n : ℕ) : Family (matLeg n) := (exists_family n (matLeg n) (matLeg_commute n)).some |
| |
| lemma x_face_commute (n : ℕ) (δ : ℝ) (i l : Fin Size.n) (t : Cube n) (j : Fin (n+1)) |
| (hj : t j=0) (b : M2) : Commute (matLeg n j b) (CubeField.x (family n) δ i l t) := by |
| let S := StarSubalgebra.centralizer ℂ (Set.range (matLeg n j)) |
| have hm : CubeField.x (family n) δ i l t∈S := CubeField.x_support _ _ _ S (by |
| intro k hk a |
| have hkj : k≠j := by intro h; subst k; rw [hj] at hk; exact (lt_irrefl (0:ℝ)) hk |
| rw [StarSubalgebra.mem_centralizer_iff] |
| rintro _ ⟨c,rfl⟩ |
| constructor |
| · exact (matLeg_commute n j k hkj.symm c a).eq |
| · rw [← map_star] |
| exact (matLeg_commute n j k hkj.symm (star c) a).eq) i l |
| exact (((StarSubalgebra.mem_centralizer_iff ℂ).mp hm) _ ⟨b,rfl⟩).1 |
| |
| lemma x_liftable (n : ℕ) (δ : ℝ) (i j : Fin Size.n) : |
| ∃! b : Tensor n,repr n b=CubeField.x (family n) δ i j ∧ |
| ‖b‖=‖CubeField.x (family n) δ i j‖ := by |
| apply exists_unique_lift |
| intro a t ha |
| obtain ⟨k,hk,hne⟩ := p_zero n a t ha |
| exact ccoeff_zero n a k hne _ (x_face_commute n δ i j t k hk) |
| |
| def x (n : ℕ) (δ : ℝ) (i j : Fin Size.n) : Tensor n := (x_liftable n δ i j).exists.choose |
| lemma repr_x (n : ℕ) (δ : ℝ) (i j : Fin Size.n) : |
| repr n (x n δ i j)=CubeField.x (family n) δ i j := ((x_liftable n δ i j).exists.choose_spec).1 |
| |
| def hScalar (n : ℕ) (δ : ℝ) : C(Cube n,ℂ) := |
| ⟨fun t => ((CubeWeights.H δ t:ℝ):ℂ),Complex.continuous_ofReal.comp (CubeWeights.H_continuous δ)⟩ |
| def h (n : ℕ) (δ : ℝ) : Tensor n := κ n (hScalar n δ) |
| lemma repr_h (n : ℕ) (δ : ℝ) : repr n (h n δ)=CubeField.h δ := by |
| apply ContinuousMap.ext |
| intro t |
| rw [h,repr_κ] |
| change algebraMap ℂ (Fiber n) (((CubeWeights.H δ t):ℝ):ℂ)=CubeWeights.H δ t • (1:Fiber n) |
| rw [Algebra.algebraMap_eq_smul_one] |
| exact Complex.coe_smul _ _ |
| lemma h_nonneg (n : ℕ) {δ : ℝ} (hδ : 0<δ) : 0≤h n δ := by |
| apply map_nonneg (κ n) |
| intro t |
| exact Complex.zero_le_real.mpr (CubeWeights.H_nonneg hδ t) |
| lemma h_norm (n : ℕ) {δ : ℝ} (hδ : 0<δ) : ‖h n δ‖≤1 := by |
| have hn : ‖h n δ‖=‖repr n (h n δ)‖ := by |
| exact (NonUnitalStarAlgHom.norm_map (repr n) (faithful n) _).symm |
| rw [hn,repr_h] |
| exact CubeField.h_norm hδ |
| |
| |
| def weighted (n : ℕ) {δ : ℝ} (hδ : 0<δ) : OrderZero.WeightedUnits (Fin Size.n) (Tensor n) where |
| h := h n δ |
| x := x n δ |
| nonneg := h_nonneg n hδ |
| norm_le := h_norm n hδ |
| star_x i j := by apply faithful n; rw [map_star,repr_x,repr_x]; exact CubeField.x_star _ _ _ _ |
| mul_x i j k l := by |
| have product_eq := CubeField.x_mul (family n) δ i j k l |
| apply faithful n |
| rw [map_mul,repr_x,repr_x] |
| refine product_eq.trans ?_ |
| split_ifs <;> simp [map_mul,repr_h,repr_x] |
| diagonal := by |
| apply faithful n |
| rw [map_sum,repr_h] |
| simp_rw [repr_x] |
| exact CubeField.x_diagonal _ _ |
| end KirchbergRordam.GenericRW.ConeCube |
| end |
| |
| end OAI |
| |
