| import Mathlib |
| import OAI.Probability.BinarySweep.TensorBounds.SignedBlock |
| import OAI.Probability.BinarySweep.Representations.IsotypicDensity |
| import OAI.Probability.BinarySweep.TensorBounds.PiTensorMatrices |
| import OAI.Probability.BinarySweep.YoungTheory.KoszulBlocks |
| |
| namespace OAI |
|
|
| noncomputable section |
|
|
| section |
|
|
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps.Signed |
| open Matrix Density |
|
|
| variable {I A : Type*} [Fintype I] [LinearOrder I] |
| [Fintype A] [DecidableEq A] {n : I → ℕ} {N : ℕ} |
|
|
| omit [Fintype I] in |
| @[simp] lemma blockPerm_inv (e : (Σₗ i, Fin (n i)) ≃o Fin N) |
| (g : ∀i, Equiv.Perm (Fin (n i))) : (blockPerm e g)⁻¹=blockPerm e (fun i => (g i)⁻¹) := by |
| apply Equiv.ext |
| intro x |
| obtain ⟨⟨i,a⟩,rfl⟩ := (toLex.trans e.toEquiv).surjective x |
| apply (blockPerm e g).injective |
| change blockPerm e g ((blockPerm e g).symm (e (toLex ⟨i,a⟩))) = _ |
| rw [Equiv.apply_symm_apply] |
| simp |
|
|
| lemma signC_blocks (e : (Σₗ i, Fin (n i)) ≃o Fin N) |
| (g : ∀i, Equiv.Perm (Fin (n i))) (β : Fin N → Bool) : |
| signC β (blockPerm e g) = ∏i, signC (fun a => β (e (toLex ⟨i,a⟩))) (g i) := by |
| simp only [signC,koszulSign_blocks,Units.coe_prod,Int.cast_prod] |
|
|
| def blockWords (e : (Σₗ i, Fin (n i)) ≃o Fin N) : |
| (Fin N → A) ≃ (∀i, Fin (n i) → A) := |
| ((toLex.trans e.toEquiv).symm.arrowCongr (Equiv.refl A)).trans (Equiv.piCurry (fun _ _ => A)) |
|
|
| omit [Fintype I] [Fintype A] [DecidableEq A] in |
| @[simp] lemma blockWords_apply (e : (Σₗ i, Fin (n i)) ≃o Fin N) |
| (x : Fin N → A) (i : I) (a : Fin (n i)) : blockWords e x i a=x (e (toLex ⟨i,a⟩)) := rfl |
|
|
| omit [Fintype I] [Fintype A] [DecidableEq A] in |
| lemma blockWords_comp (e : (Σₗ i, Fin (n i)) ≃o Fin N) |
| (g : ∀i, Equiv.Perm (Fin (n i))) (x : Fin N → A) : |
| blockWords e (x ∘ blockPerm e g) = fun i => (blockWords e x i) ∘ g i := by |
| funext i a |
| simp |
|
|
| theorem actionMatrix_blocks (e : (Σₗ i, Fin (n i)) ≃o Fin N) |
| (g : ∀i, Equiv.Perm (Fin (n i))) (p : A → Bool) : |
| LinearMap.toMatrix' (act p (blockPerm e g)) = |
| (piTensorMatrices (fun i => LinearMap.toMatrix' (act p (g i)))).submatrix |
| (blockWords e) (blockWords e) := by |
| ext x y |
| change LinearMap.toMatrix' (act p (blockPerm e g)) x y = |
| ∏i, LinearMap.toMatrix' (act p (g i)) (blockWords e x i) (blockWords e y i) |
| simp only [actionMatrix_apply] |
| have he : x ∘ blockPerm e g=y ↔ ∀i, (blockWords e x i) ∘ g i=blockWords e y i := by |
| rw [← (blockWords e).injective.eq_iff,blockWords_comp] |
| exact funext_iff |
| by_cases h : x ∘ blockPerm e g=y |
| · rw [ite_eq_left h,blockPerm_inv,signC_blocks] |
| apply Finset.prod_congr rfl |
| intro i _ |
| rw [ite_eq_left (he.mp h i)] |
| rfl |
| · rw [ite_eq_right h] |
| have hh := h |
| rw [he] at hh |
| push Not at hh |
| obtain ⟨i,hi⟩ := hh |
| symm |
| exact Finset.prod_eq_zero (Finset.mem_univ i) (ite_eq_right hi) |
|
|
| end BinaryCoordinateSweeps.Signed |
| |
| end |
|
|
| open scoped BigOperators Classical ComplexOrder MonoidAlgebra |
| open Matrix |
|
|
| namespace BinaryCoordinateSweeps.Signed |
| open Density Irrep Representation |
|
|
| variable {A : Type*} [Fintype A] [DecidableEq A] [Nonempty A] |
|
|
| def vacuumDensity : Matrix A A ℂ := Matrix.diagonal (fun a => if a=Classical.choice (inferInstance : Nonempty A) then 1 else 0) |
|
|
| omit [Fintype A] in |
| lemma vacuumDensity_psd : (vacuumDensity (A:=A)).PosSemidef := by |
| apply Matrix.posSemidef_diagonal_iff.mpr |
| intro a |
| split < |
|
|
| lemma vacuumDensity_trace : (vacuumDensity (A:=A)).trace=1 := by |
| simp [vacuumDensity,Matrix.trace,Matrix.diag] |
| |
| omit [Fintype A] in |
| lemma vacuumDensity_even (p : A → Bool) (a b : A) (h : p a ≠ p b) : |
| vacuumDensity a b=0 := by |
| have hh : a≠b := fun he => h (congrArg p he) |
| simp [vacuumDensity,hh] |
|
|
| abbrev LocalDensityIndex (p : A → Bool) (n : ℕ) := |
| WordType (MatchedPair p) n × (MatchedPair p → ZMod (n+1)) |
|
|
| def localDensity (p : A → Bool) (n : ℕ) (u : LocalDensityIndex p n) : Matrix A A ℂ := |
| if n=0 then vacuumDensity else reducedPhaseDensity n (matchedEmbedding p) u.1 u.2 |
|
|
| lemma localDensity_psd (p : A → Bool) (n : ℕ) (u : LocalDensityIndex p n) : |
| (localDensity p n u).PosSemidef := by |
| unfold localDensity |
| split |
| · exact vacuumDensity_psd |
| · exact reducedPhaseDensity_psd _ _ _ _ |
|
|
| lemma localDensity_trace (p : A → Bool) (n : ℕ) (u : LocalDensityIndex p n) : |
| (localDensity p n u).trace=1 := by |
| unfold localDensity |
| split |
| · exact vacuumDensity_trace |
| · exact reducedPhaseDensity_trace n (by omega) _ (matchedEmbedding_injective p) _ _ |
|
|
| lemma localDensity_even (p : A → Bool) (n : ℕ) (u : LocalDensityIndex p n) |
| (a b : A) (h : p a ≠ p b) : localDensity p n u a b=0 := by |
| unfold localDensity |
| split |
| · exact vacuumDensity_even p a b h |
| · exact reducedPhaseDensity_even n p p (matchedEmbedding p) (fun k => k.property) _ _ a b h |
|
|
| omit [Nonempty A] in |
| lemma localDensityIndex_card (p : A → Bool) (n : ℕ) : |
| Fintype.card (LocalDensityIndex p n) ≤ (n+1)^(2*(Fintype.card A)^2) := by |
| have hc : Fintype.card (MatchedPair p) ≤ (Fintype.card A)^2 := by |
| simpa only [Fintype.card_prod,pow_two] using Fintype.card_le_of_injective |
| (matchedEmbedding p) (matchedEmbedding_injective p) |
| calc |
| _ = Fintype.card (WordType (MatchedPair p) n) * (n+1)^Fintype.card (MatchedPair p) := by |
| rw [Fintype.card_prod,Fintype.card_fun,ZMod.card] |
| _ ≤ (n+1)^Fintype.card (MatchedPair p) * (n+1)^Fintype.card (MatchedPair p) := |
| Nat.mul_le_mul_right _ (by simpa only [Fintype.card_coe] using card_types_le (A:=MatchedPair p) n) |
| _ = (n+1)^(2*Fintype.card (MatchedPair p)) := by rw [two_mul,pow_add] |
| _ ≤ _ := Nat.pow_le_pow_right (by omega) (Nat.mul_le_mul_left 2 hc) |
|
|
| lemma localDensityIndex_nonempty (p : A → Bool) (n : ℕ) : Nonempty (LocalDensityIndex p n) := by |
| let a : A := Classical.choice inferInstance |
| exact ⟨wordType n (fun _ => ⟨(a,a),rfl⟩),fun _ => 0⟩ |
|
|
| lemma projection_complement_psd {I : Type*} [Fintype I] [DecidableEq I] |
| (P : Matrix I I ℂ) (hP : P.IsHermitian) (hPP : P*P=P) : (1-P).PosSemidef := by |
| have he : (1-P).conjTranspose*(1-P)=1-P := by |
| simp only [Matrix.conjTranspose_sub,Matrix.conjTranspose_one,hP.eq,mul_sub,sub_mul, |
| one_mul,mul_one,hPP] |
| abel |
| rw [← he] |
| exact Matrix.posSemidef_conjTranspose_mul_self _ |
|
|
| variable {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] |
|
|
| theorem local_type_density_domination (p : A → Bool) (n : ℕ) |
| (ρ : Representation ℂ (Equiv.Perm (Fin n)) V) [ρ.IsIrreducible] : |
| (((Module.finrank ℂ V : ℂ)*(n+1 : ℂ)^(2*(Fintype.card A)^2)) • |
| (∑u : LocalDensityIndex p n, matrixTensorPower n (localDensity p n u)) - |
| isotypicProjection p ρ).PosSemidef := by |
| by_cases hn : n=0 |
| · subst n |
| have : Nonempty (LocalDensityIndex p 0) := localDensityIndex_nonempty p 0 |
| have hc : Fintype.card (LocalDensityIndex p 0)=1 := le_antisymm |
| (by simp) (Fintype.card_pos_iff.mpr inferInstance) |
| have he (u : LocalDensityIndex p 0) : matrixTensorPower 0 (localDensity p 0 u)=1 := by |
| ext x y |
| simp [matrixTensorPower,Matrix.one_apply,Subsingleton.elim x y] |
| simp only [he,Finset.sum_const,Finset.card_univ,hc,one_nsmul,Nat.cast_zero,zero_add,one_pow,mul_one] |
| have : Nontrivial ρ.asModule := IsSimpleModule.nontrivial ℂ[Equiv.Perm (Fin 0)] ρ.asModule |
| have hd : (1:ℂ) ≤ (Module.finrank ℂ V : ℂ) := by |
| have hh : 0 < Module.finrank ℂ V := by |
| rw [← ρ.asModuleEquiv.finrank_eq] |
| exact Module.finrank_pos |
| exact_mod_cast hh |
| have h := (Matrix.PosSemidef.one : (1 : Matrix (Fin 0 → A) (Fin 0 → A) ℂ).PosSemidef).smul |
| (sub_nonneg.mpr hd) |
| have hp := projection_complement_psd (isotypicProjection p ρ) |
| (isotypicProjection_psd p ρ).isHermitian (projectionMatrix_idempotent _) |
| convert h.add hp using 1 |
| simp only [sub_smul,one_smul] |
| abel |
| · simpa only [localDensity,ite_eq_right hn,Fintype.sum_prod_type] using |
| isotypic_density_domination n (by omega) p ρ |
|
|
| end BinaryCoordinateSweeps.Signed |
| |
| end |
|
|
| end OAI |
|
|
