import Mathlib import OAI.MathematicalPhysics.PEPSFilters.LocalOperators import OAI.MathematicalPhysics.PEPSSubvolume.NestedLower import OAI.MathematicalPhysics.PEPSSubvolume.RectangleContours import OAI.MathematicalPhysics.PEPSSubvolume.HarmonicWeights namespace OAI /-! Uniform geometric-buffer entropy lower bounds. -/ noncomputable section open scoped BigOperators ComplexOrder open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator open scoped BigOperators open scoped Topology open Filter open scoped MatrixOrder open scoped BigOperators Matrix.Norms.L2Operator open scoped ComplexOrder BigOperators Matrix.Norms.L2Operator open Matrix open Filter Topology open PolynomialPEPS.PinnedEntropy namespace PolynomialPEPS.Subvolume.GeometricLower open scoped BigOperators open PolynomialPEPS.Subvolume.RectangleContours PolynomialPEPS.Subvolume.HarmonicWeights open PolynomialPEPS.Subvolume.ContourEnergy PolynomialPEPS.Subvolume.NestedLower variable {L q : ℕ} def blocks (q : ℕ) (J Δ : ℝ) : ℕ := 96+Nat.ceil (147456*(q:ℝ)^2*J/Δ) def bufferConstant (q : ℕ) (J Δ : ℝ) : ℝ := (2:ℝ)^(blocks q J Δ)*Real.log 2+1 lemma blocks_large (q : ℕ) (J Δ : ℝ) : 96 ≤ blocks q J Δ ∧ 147456*(q:ℝ)^2*J/Δ ≤ (blocks q J Δ:ℝ) := by constructor · exact Nat.le_add_right _ _ · exact (Nat.le_ceil _).trans (by simp only [blocks,Nat.cast_add,Nat.cast_ofNat]; linarith) lemma rectangle_boundary_weight (lo₁ hi₁ lo₂ hi₂ : ℤ) (r j : ℕ) (hr : 0 < r) (h₁ : hi₁-lo₁+1 ≤ r) (h₂ : hi₂-lo₂+1 ≤ r) : (boundaryCard (rectangle (L:=L) lo₁ hi₁ lo₂ hi₂ (j+1)):ℝ) ≤ 12*((r:ℝ)+j) := by have h := boundaryCard_le (L:=L) lo₁ hi₁ lo₂ hi₂ (j+1) r h₁ h₂ have h' : (boundaryCard (rectangle (L:=L) lo₁ hi₁ lo₂ hi₂ (j+1)):ℝ) ≤ 4*((r:ℝ)+2*((j:ℝ)+1)) := by exact_mod_cast h have hr' : (1:ℝ) ≤ r := by exact_mod_cast hr linarith [Nat.cast_nonneg (α:=ℝ) j] lemma rectangle_unique_split (lo₁ hi₁ lo₂ hi₂ : ℤ) (e : Edge L) (j k : ℕ) (hj : Splits (EdgeSites e) (rectangle lo₁ hi₁ lo₂ hi₂ (j+1))) (hk : Splits (EdgeSites e) (rectangle lo₁ hi₁ lo₂ hi₂ (k+1))) : j=k := by have h := unique_crossing lo₁ hi₁ lo₂ hi₂ e (j+1) (k+1) ((edge_splits_iff e _).mp hj) ((edge_splits_iff e _).mp hk) omega lemma buffer_energy (J Δ : ℝ) (hJ : 0 ≤ J) (hΔ : 0 < Δ) (lo₁ hi₁ lo₂ hi₂ : ℤ) (r : ℕ) (hr : 0 < r) (h₁ : hi₁-lo₁+1 ≤ r) (h₂ : hi₂-lo₂+1 ≤ r) : let n := 2^(blocks q J Δ)*r ((q:ℝ)^2*J/Δ)*(∑ j ∈ Finset.range n, (weight r n j)^2*(boundaryCard (rectangle (L:=L) lo₁ hi₁ lo₂ hi₂ (j+1)):ℝ)) ≤ 1/256 := by let s := blocks q J Δ have hs := blocks_large q J Δ have hs0 : 0 < (s:ℝ) := by have hs' : (96:ℝ) ≤ s := by exact_mod_cast hs.1 linarith have hw := (buffer_weights r s hr hs.1 (fun j => (boundaryCard (rectangle (L:=L) lo₁ hi₁ lo₂ hi₂ (j+1)):ℝ)) (fun j _ => rectangle_boundary_weight lo₁ hi₁ lo₂ hi₂ r j hr h₁ h₂)).2.2 apply (mul_le_mul_of_nonneg_left hw (by positivity)).trans apply (le_div_iff₀ (by norm_num : (0:ℝ) < 256)).mpr change (q:ℝ)^2*J/Δ*(576/(s:ℝ))*256 ≤ 1 calc _ = (147456*(q:ℝ)^2*J/Δ)/(s:ℝ) := by ring _ ≤ 1 := by apply (div_le_iff₀ hs0).mpr simpa only [one_mul] using hs.2 theorem rectangle_optimizer_entropy_lower (hq : 0 < q) (J Δ E₀ : ℝ) (hJ : 0 ≤ J) (hΔ : 0 < Δ) (hv : Vertex L → Operator L q) (he : Edge L → Operator L q) (Ω : State L q) (hΩ : ‖Ω‖=1) (hH : IsGridHamiltonian J hv he) (hg : asMap (Hamiltonian hv he) Ω=(E₀:ℂ) • Ω) (hgap : FullSystemGap (Hamiltonian hv he) Ω E₀ Δ) (lo₁ hi₁ lo₂ hi₂ : ℤ) (r : ℕ) (hr : 0 < r) (h₁ : hi₁-lo₁+1 ≤ r) (h₂ : hi₂-lo₂+1 ≤ r) : let n := 2^(blocks q J Δ)*r let X := fun j => rectangle (L:=L) lo₁ hi₁ lo₂ hi₂ (j+1) ∃ F : FilterFamily q n (fun j => X j.val), IsFilterOptimizer Ω (fun j => weight r n j.val) F ∧ (∑ j ∈ Finset.range n, weight r n j)=4 ∧ (∑ j ∈ Finset.range n, weight r n j*vonNeumannEntropy Ω (X j))- bufferConstant q J Δ*(r:ℝ) ≤ -Real.log (‖filteredVector F Ω‖^2) := by let s := blocks q J Δ let n := 2^s*r let X := fun j => rectangle (L:=L) lo₁ hi₁ lo₂ hi₂ (j+1) have hs := blocks_large q J Δ have hn : 0 < n := Nat.mul_pos (pow_pos (by omega) _) hr have hw := buffer_weights r s hr hs.1 (fun j => (boundaryCard (X j):ℝ)) (fun j _ => rectangle_boundary_weight lo₁ hi₁ lo₂ hi₂ r j hr h₁ h₂) obtain ⟨F,hF⟩ := OptimizerExistence.exists_optimizer hq (X := fun j : Fin n => X j.val) (fun j : Fin n => weight r n j.val) (fun j => (hw.1 j).1) Ω refine ⟨F,hF,hw.2.1,?_⟩ have hb := buffer_energy (q:=q) (L:=L) J Δ hJ hΔ lo₁ hi₁ lo₂ hi₂ r hr h₁ h₂ change ((q:ℝ)^2*J/Δ)*(∑ j ∈ Finset.range n, (weight r n j)^2*(boundaryCard (X j):ℝ)) ≤ 1/256 at hb have hterm (j : ℕ) (hj : j < n) : ((q:ℝ)^2*J/Δ)*((weight r n j)^2*(boundaryCard (X j):ℝ)) ≤ 1/256 := by apply le_trans _ hb apply mul_le_mul_of_nonneg_left _ (by positivity) exact Finset.single_le_sum (f := fun k => (weight r n k)^2*(boundaryCard (X k):ℝ)) (fun k _ => by positivity) (Finset.mem_range.mpr hj) have hm := optimizer_entropy_lower hq J Δ E₀ hJ hΔ hv he Ω hΩ hH hg hgap X (fun j k hjk => monotone_rectangle lo₁ hi₁ lo₂ hi₂ (Nat.add_le_add_right hjk 1)) (weight r n) n (fun j _ => hw.1 j) (fun e j k _ _ => rectangle_unique_split lo₁ hi₁ lo₂ hi₂ e j k) (fun j hj => by have ht := hterm j hj simp only [div_eq_mul_inv] at ht ⊢ nlinarith only [ht]) (by simp only [div_eq_mul_inv] at hb ⊢ nlinarith only [hb]) F hF have herr : (∑ j ∈ Finset.range n, (256*(q:ℝ)^2*J*(boundaryCard (X j):ℝ)/Δ)*(weight r n j)^2) ≤ 1 := by have heq : (∑ j ∈ Finset.range n, (256*(q:ℝ)^2*J*(boundaryCard (X j):ℝ)/Δ)*(weight r n j)^2) = 256*((q:ℝ)^2*J/Δ)*(∑ j ∈ Finset.range n, (weight r n j)^2*(boundaryCard (X j):ℝ)) := by rw [Finset.mul_sum] apply Finset.sum_congr rfl intro j _ ring rw [heq] simp only [div_eq_mul_inv] at hb ⊢ nlinarith only [hb] have hnr : (n:ℝ)*Real.log 2+1 ≤ bufferConstant q J Δ*(r:ℝ) := by have hr' : (1:ℝ) ≤ r := by exact_mod_cast hr simp only [n,s,bufferConstant,Nat.cast_mul,Nat.cast_pow,Nat.cast_ofNat] nlinarith only [hr'] exact le_trans (by linarith only [herr,hnr]) hm end PolynomialPEPS.Subvolume.GeometricLower end end OAI