| import Mathlib |
| import OAI.RepresentationTheory.SignedTensor.Blocks |
| import OAI.RepresentationTheory.Unitary.Spectral |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Matrix.Norms.L2Operator ComplexOrder |
| attribute [local instance] Classical.propDecidable |
| noncomputable section |
| open scoped BigOperators ComplexConjugate Matrix.Norms.L2Operator |
| attribute [local instance] Classical.propDecidable |
| noncomputable section |
| open scoped BigOperators ComplexOrder MatrixOrder |
| attribute [local instance] Classical.propDecidable |
| noncomputable section |
| open scoped BigOperators ENNReal |
| open MeasureTheory |
| noncomputable section |
| open scoped BigOperators Matrix.Norms.L2Operator ComplexOrder |
| noncomputable section |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
| noncomputable section |
| open scoped BigOperators Matrix.Norms.L2Operator ComplexOrder |
| attribute [local instance] Classical.propDecidable |
| noncomputable section |
| attribute [local instance] Classical.propDecidable |
| namespace CoordinateSweeps.TensorBoard |
| variable {R C J : Type*} [Fintype R] [DecidableEq R] [Nonempty R] |
| [Fintype C] [DecidableEq C] [Fintype J] [DecidableEq J] |
| |
| omit [Nonempty R] in |
| lemma board_trace_le_reindexed {l : ℕ} (F : Finset (R × C)) (e : Fin l ≃ F) |
| (r : R → Matrix J J ℂ) (v : C → Matrix J J ℂ) |
| (hr : ∀ i, (r i).PosSemidef) (hv : ∀ j, (v j).PosSemidef) |
| (ht : ∀ j, Matrix.trace (v j)=1) (A : Matrix J J ℂ) (hA : A.IsHermitian) |
| (hw : ((Fintype.card R : ℂ) • (1 : Matrix J J ℂ)-∑ i, A*r i*A).PosSemidef) : |
| (Matrix.trace (tensor (fun _ : Fin l => A)*tensor (fun x => r (e x).val.1)* |
| tensor (fun _ : Fin l => A)*tensor (fun x => v (e x).val.2))).re ≤ |
| Real.exp (Fintype.card R*Fintype.card C-F.card) := by |
| have hh := board_trace_le F r v hr hv ht A hA hw |
| rw [← tensor_mul,tensor_trace] at hh |
| rw [← tensor_mul,← tensor_mul,← tensor_mul,tensor_trace] |
| have he := Equiv.prod_comp e (fun x : F => Matrix.trace (A*r x.val.1*A*v x.val.2)) |
| simpa only [he] using hh |
|
|
| end CoordinateSweeps.TensorBoard |
|
|
| namespace CoordinateSweeps.UnitaryIrrep |
| variable {Γ n : Type*} [Group Γ] [Fintype Γ] [Fintype n] [DecidableEq n] |
| lemma projector_commute_lift (ρ : UnitaryIrrep Γ) (τ : Γ →* Matrix n n ℂ) |
| (x : MonoidAlgebra ℂ Γ) : |
| ρ.projector τ*MonoidAlgebra.lift ℂ _ _ τ x=MonoidAlgebra.lift ℂ _ _ τ x*ρ.projector τ := by |
| induction x using MonoidAlgebra.induction_on with |
| | of g => simpa using (ρ.projector_commute τ g).symm |
| | add x y hx hy => simp only [map_add,add_mul,mul_add,hx,hy] |
| | smul c x hx => simp only [map_smul,smul_mul_assoc,mul_smul_comm,hx] |
| end CoordinateSweeps.UnitaryIrrep |
| |
| namespace CoordinateSweeps.SignedTensor |
| |
| lemma densityPower_commute_projector {p l : ℕ} (r : Matrix (Letter p) (Letter p) ℂ) |
| (hr : IsEven r) (ρ : UnitaryIrrep (Equiv.Perm (Fin l))) : |
| densityPower l r*ρ.projector (action p l)=ρ.projector (action p l)*densityPower l r := by |
| simp only [UnitaryIrrep.projector,Matrix.mul_sum,Matrix.sum_mul,Matrix.mul_smul,Matrix.smul_mul] |
| apply Finset.sum_congr rfl |
| intro σ hσ |
| exact congrArg (fun A => ρ.coefficient σ • A) (densityPower_commute_of_even r hr σ).symm |
| |
| lemma even_average_trace {R : Type*} [Fintype R] [Nonempty R] {p : ℕ} |
| (r : R → EvenDensity p) (ε : ℝ) : |
| Matrix.trace ((Fintype.card R : ℂ)⁻¹ • (∑ i, (r i).val)+ |
| (ε : ℂ) • (1 : Matrix (Letter p) (Letter p) ℂ)) = 1+ε*(2*p) := by |
| rw [Matrix.trace_add,Matrix.trace_smul,Matrix.trace_sum,Matrix.trace_smul,Matrix.trace_one] |
| simp only [(r _).property.2.1,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one, |
| smul_eq_mul,Fintype.card_fin] |
| have hc : (Fintype.card R : ℂ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero |
| rw [inv_mul_cancel₀ hc] |
| push_cast |
| rfl |
| |
| end CoordinateSweeps.SignedTensor |
| |
| namespace CoordinateSweeps.SignedTensor |
| open TensorBoard Overlap |
| variable {R C I J : Type*} [Fintype R] [DecidableEq R] [Nonempty R] |
| [Fintype C] [DecidableEq C] [Fintype I] [Fintype J] |
| |
| abbrev groupOperator {p l : ℕ} (x : MonoidAlgebra ℂ (Equiv.Perm (Fin l))) : |
| Matrix (Word p l) (Word p l) ℂ := MonoidAlgebra.lift ℂ _ _ (action p l) x |
| |
| def rowProduct {p l : ℕ} {F : Finset (R × C)} (e : Fin l ≃ F) (r : R → EvenDensity p) : |
| Matrix (Word p l) (Word p l) ℂ := TensorBoard.tensor (fun x => (r (e x).val.1).val) |
| |
| def columnProduct {p l : ℕ} {F : Finset (R × C)} (e : Fin l ≃ F) (r : C → EvenDensity p) : |
| Matrix (Word p l) (Word p l) ℂ := TensorBoard.tensor (fun x => (r (e x).val.2).val) |
| |
| /- Literal deleted-board overlap with actual signed tensor action. The two |
| mixture dominations will be obtained by tensoring the local density theorem; |
| the whitening and the full-group dimension loss are constructed here. -/ |
| theorem board_overlap_regularized {p l : ℕ} (hp : 0 < p) |
| (F : Finset (R × C)) (e : Fin l ≃ F) |
| (ρ : UnitaryIrrep (Equiv.Perm (Fin l))) |
| (PR : Matrix (Word p l) (Word p l) ℂ) (xC : MonoidAlgebra ℂ (Equiv.Perm (Fin l))) |
| (hPR : PR.IsHermitian) (hPR2 : PR*PR=PR) |
| (hPC : (groupOperator (p := p) xC).IsHermitian) |
| (hPC2 : groupOperator (p := p) xC*groupOperator (p := p) xC=groupOperator (p := p) xC) |
| (u : I → ℝ) (v : J → ℝ) (hu : ∀ i, 0 ≤ u i) (hv : ∀ j, 0 ≤ v j) |
| (hsu : ∑ i, u i=1) (hsv : ∑ j, v j=1) |
| (r : I → R → EvenDensity p) (c : J → C → EvenDensity p) |
| (cr cc : ℝ) (hcr : 0 ≤ cr) (hcc : 0 ≤ cc) |
| (hrow : ((cr : ℂ) • (∑ i, (u i : ℂ) • rowProduct e (r i))-PR).PosSemidef) |
| (hcol : ((cc : ℂ) • (∑ j, (v j : ℂ) • columnProduct e (c j))-groupOperator xC).PosSemidef) |
| {ε : ℝ} (hε : 0 < ε) : |
| (ρ.dimension : ℝ)*‖groupOperator (p := p) xC*ρ.projector (action p l)*PR‖^2 ≤ |
| cr*cc*(1+ε*(2*p : ℕ))^l*Real.exp (Fintype.card R*Fintype.card C-F.card) := by |
| let : NeZero (2*p) := ⟨by omega⟩ |
| let τ := action p l |
| let PC := groupOperator (p := p) xC |
| let B (i : I) : Matrix (Letter p) (Letter p) ℂ := |
| (Fintype.card R : ℂ)⁻¹ • (∑ k, (r i k).val)+(ε : ℂ) • 1 |
| choose A hA hAe hABA hAw using fun i => exists_even_whitening (r i) hε |
| have hb (i : I) : (B i).PosSemidef := |
| (average_regularization_posDef (fun k => (r i k).val) (fun k => (r i k).property.1) hε).posSemidef |
| have he (i : I) : IsEven (B i) := isEven_add |
| (isEven_smul _ (isEven_sum _ (fun k => (r i k).property.2.2))) (isEven_smul _ isEven_one) |
| have htau (g : Equiv.Perm (Fin l)) : (τ g).conjTranspose=τ g⁻¹ := (matrix_inv g).symm |
| have hpq : ρ.projector τ*PC=PC*ρ.projector τ := ρ.projector_commute_lift τ xC |
| have htt : 0 ≤ (1+ε*(2*p : ℕ))^l := pow_nonneg (by positivity) _ |
| have hwhite (i : I) : |
| (((1+ε*(2*p : ℕ))^l : ℝ) : ℂ) • (densityPower l (A i)*PC*densityPower l (A i)) - |
| (ρ.dimension : ℂ) • (PC*ρ.projector τ) |>.PosSemidef := by |
| have hWA : (densityPower l (A i)).IsHermitian := (densityPower_posSemidef l (A i) (hA i)).isHermitian |
| have hWABA : densityPower l (A i)*densityPower l (B i)*densityPower l (A i)=1 := by |
| rw [← densityPower_mul,← densityPower_mul,hABA i,densityPower_one] |
| have hh := Overlap.whitened_projector_domination ρ τ htau |
| (densityPower l (B i)) (densityPower l (A i)) PC (densityPower_posSemidef l (B i) (hb i)) |
| (fun g => densityPower_commute_of_even (B i) (he i) g) hWA hWABA |
| (densityPower_commute_projector (A i) (hAe i) ρ) |
| (densityPower_commute_lift (A i) (hAe i) xC).symm hPC hPC2 hpq |
| have ht : Matrix.trace (densityPower l (B i))=(((1+ε*(2*p : ℕ))^l : ℝ) : ℂ) := by |
| rw [densityPower_trace,even_average_trace] |
| push_cast |
| rfl |
| rwa [ht] at hh |
| apply Overlap.finite_mixture_overlap PR PC (ρ.projector τ) hPR hPR2 hPC hPC2 |
| (ρ.projector_hermitian τ htau) (ρ.projector_idempotent τ) hpq u v hu hv hsu hsv |
| (fun i => rowProduct e (r i)) (fun j => columnProduct e (c j)) (fun i => densityPower l (A i)) |
| (fun i => TensorBoard.tensor_posSemidef _ (fun x => (r i (e x).val.1).property.1)) |
| (fun i => (densityPower_posSemidef l (A i) (hA i)).isHermitian) |
| cr cc ρ.dimension ((1+ε*(2*p : ℕ))^l) _ hcr hcc (Nat.cast_nonneg _) htt hrow hcol hwhite |
| intro i j |
| exact board_trace_le_reindexed F e (fun k => (r i k).val) (fun k => (c j k).val) |
| (fun k => (r i k).property.1) (fun k => (c j k).property.1) (fun k => (c j k).property.2.1) |
| (A i) (hA i).isHermitian (hAw i) |
|
|
| end CoordinateSweeps.SignedTensor |
|
|
| namespace CoordinateSweeps.SignedTensor |
| open TensorBoard Overlap |
| variable {R C I J : Type*} [Fintype R] [DecidableEq R] [Nonempty R] |
| [Fintype C] [DecidableEq C] [Fintype I] [Fintype J] |
| |
| theorem board_overlap {p l : ℕ} (hp : 0 < p) |
| (F : Finset (R × C)) (e : Fin l ≃ F) |
| (ρ : UnitaryIrrep (Equiv.Perm (Fin l))) |
| (PR : Matrix (Word p l) (Word p l) ℂ) (xC : MonoidAlgebra ℂ (Equiv.Perm (Fin l))) |
| (hPR : PR.IsHermitian) (hPR2 : PR*PR=PR) |
| (hPC : (groupOperator (p := p) xC).IsHermitian) |
| (hPC2 : groupOperator (p := p) xC*groupOperator (p := p) xC=groupOperator (p := p) xC) |
| (u : I → ℝ) (v : J → ℝ) (hu : ∀ i, 0 ≤ u i) (hv : ∀ j, 0 ≤ v j) |
| (hsu : ∑ i, u i=1) (hsv : ∑ j, v j=1) |
| (r : I → R → EvenDensity p) (c : J → C → EvenDensity p) |
| (cr cc : ℝ) (hcr : 0 ≤ cr) (hcc : 0 ≤ cc) |
| (hrow : ((cr : ℂ) • (∑ i, (u i : ℂ) • rowProduct e (r i))-PR).PosSemidef) |
| (hcol : ((cc : ℂ) • (∑ j, (v j : ℂ) • columnProduct e (c j))-groupOperator xC).PosSemidef) |
| : |
| (ρ.dimension : ℝ)*‖groupOperator (p := p) xC*ρ.projector (action p l)*PR‖^2 ≤ |
| cr*cc*Real.exp (Fintype.card R*Fintype.card C-F.card) := by |
| let f (ε : ℝ) := cr*cc*(1+ε*(2*p : ℕ))^l*Real.exp (Fintype.card R*Fintype.card C-F.card) |
| have hf : Continuous f := by fun_prop |
| have hh : ∀ᶠ ε : ℝ in nhdsWithin 0 (Set.Ioi 0), |
| (ρ.dimension : ℝ)*‖groupOperator (p := p) xC*ρ.projector (action p l)*PR‖^2 ≤ f ε := by |
| filter_upwards [self_mem_nhdsWithin] with ε hε |
| exact board_overlap_regularized hp F e ρ PR xC hPR hPR2 hPC hPC2 |
| u v hu hv hsu hsv r c cr cc hcr hcc hrow hcol hε |
| have hh' := ge_of_tendsto (hf.continuousAt.tendsto.mono_left |
| (nhdsWithin_le_nhds : nhdsWithin (0 : ℝ) (Set.Ioi 0) ≤ nhds 0)) hh |
| simpa [f] using hh' |
|
|
| end CoordinateSweeps.SignedTensor |
|
|
| namespace CoordinateSweeps.SignedTensor |
| open TensorBoard |
| variable {p r c : ℕ} [NeZero r] |
|
|
| /- Source 06:eq19 for literal row and column subgroup projectors on an |
| arbitrary board with holes. The mixtures and signed regroupings are constructed. -/ |
| theorem actual_board_overlap (F : Finset (Fin r × Fin c)) |
| (e : Fin (Fintype.card F) ≃ F) (s : ℕ) (hp : 0 < p) (hps : p ≤ s) |
| (hnR : ∀ i, FiberBlocks.size (fun x : F => x.val.1) i ≤ s) |
| (hnC : ∀ j, FiberBlocks.size (fun x : F => x.val.2) j ≤ s) |
| (ρR : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (FiberBlocks.size (fun x : F => x.val.1) i)))) |
| (ρC : ∀ j, UnitaryIrrep (Equiv.Perm (Fin (FiberBlocks.size (fun x : F => x.val.2) j)))) |
| (ρ : UnitaryIrrep (Equiv.Perm (Fin (Fintype.card F)))) : |
| (ρ.dimension : ℝ)*‖fiberProjector (p := p) (fun x : F => x.val.2) e ρC* |
| ρ.projector (action p (Fintype.card F))*fiberProjector (fun x : F => x.val.1) e ρR‖^2 ≤ |
| ((∏ i, ((ρR i).dimension : ℝ))*(s+1 : ℕ)^(r*(41*p^2)))* |
| ((∏ j, ((ρC j).dimension : ℝ))*(s+1 : ℕ)^(c*(41*p^2)))* |
| Real.exp (r*c-F.card) := by |
| obtain ⟨I,hI,nI,a,ha⟩ := finite_fiber_density_domination (fun x : F => x.val.1) e s hp hps hnR ρR |
| obtain ⟨J,hJ,nJ,b,hb⟩ := finite_fiber_density_domination (fun x : F => x.val.2) e s hp hps hnC ρC |
| let := hI |
| let := hJ |
| let := nI |
| let := nJ |
| have hI0 : (Fintype.card I : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero |
| have hJ0 : (Fintype.card J : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero |
| have hu : ∑ _ : I, (Fintype.card I : ℝ)⁻¹=1 := by |
| rw [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_inv_cancel₀ hI0] |
| have hv : ∑ _ : J, (Fintype.card J : ℝ)⁻¹=1 := by |
| rw [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_inv_cancel₀ hJ0] |
| have hr := fiberProjector_posSemidef (p := p) (fun x : F => x.val.1) e ρR |
| have hc := fiberProjector_posSemidef (p := p) (fun x : F => x.val.2) e ρC |
| have hr2 := fiberProjector_idempotent (p := p) (fun x : F => x.val.1) e ρR |
| have hc2 := fiberProjector_idempotent (p := p) (fun x : F => x.val.2) e ρC |
| let xC := fiberProjectorAlgebra (fun x : F => x.val.2) e ρC |
| have hx : groupOperator (p := p) xC=fiberProjector (p := p) (fun x : F => x.val.2) e ρC := |
| fiberProjectorAlgebra_lift _ _ _ |
| have hrow : (((∏ i, ((ρR i).dimension : ℝ))*(s+1 : ℕ)^(r*(41*p^2)) : ℝ) : ℂ) • |
| (∑ i : I, (((Fintype.card I : ℝ)⁻¹ : ℝ) : ℂ) • rowProduct e (a i))- |
| fiberProjector (p := p) (fun x : F => x.val.1) e ρR |>.PosSemidef := by |
| simpa only [Complex.ofReal_mul,Complex.ofReal_prod,Complex.ofReal_natCast,Complex.ofReal_pow, |
| Complex.ofReal_inv,← Finset.smul_sum,rowProduct] using ha |
| have hcol : (((∏ j, ((ρC j).dimension : ℝ))*(s+1 : ℕ)^(c*(41*p^2)) : ℝ) : ℂ) • |
| (∑ j : J, (((Fintype.card J : ℝ)⁻¹ : ℝ) : ℂ) • columnProduct e (b j))- |
| groupOperator (p := p) xC |>.PosSemidef := by |
| rw [hx] |
| simpa only [Complex.ofReal_mul,Complex.ofReal_prod,Complex.ofReal_natCast,Complex.ofReal_pow, |
| Complex.ofReal_inv,← Finset.smul_sum,columnProduct] using hb |
| have hh := board_overlap hp F e ρ _ xC hr.isHermitian hr2 (hx ▸ hc.isHermitian) |
| (by rw [hx] |
| (fun _ : J => (Fintype.card J : ℝ)⁻¹) (fun _ => by positivity) (fun _ => by positivity) |
| hu hv a b ((∏ i, ((ρR i).dimension : ℝ))*(s+1 : ℕ)^(r*(41*p^2))) |
| ((∏ j, ((ρC j).dimension : ℝ))*(s+1 : ℕ)^(c*(41*p^2))) (by positivity) (by positivity) hrow hcol |
| simpa only [hx,Fintype.card_fin] using hh |
|
|
| end CoordinateSweeps.SignedTensor |
|
|
| namespace CoordinateSweeps.TensorBoard |
| variable {R C J : Type*} [Fintype R] [DecidableEq R] [Nonempty R] |
| [Fintype C] [DecidableEq C] [Fintype J] [DecidableEq J] |
| omit [Nonempty R] in |
| lemma board_trace_le_labels {l : ℕ} (e : Fin l → R × C) (he : Function.Injective e) |
| (r : R → Matrix J J ℂ) (v : C → Matrix J J ℂ) |
| (hr : ∀ i, (r i).PosSemidef) (hv : ∀ j, (v j).PosSemidef) |
| (ht : ∀ j, Matrix.trace (v j)=1) (A : Matrix J J ℂ) (hA : A.IsHermitian) |
| (hw : ((Fintype.card R : ℂ) • (1 : Matrix J J ℂ)-∑ i, A*r i*A).PosSemidef) : |
| (Matrix.trace (tensor (fun _ : Fin l => A)*tensor (fun x => r (e x).1)* |
| tensor (fun _ : Fin l => A)*tensor (fun x => v (e x).2))).re ≤ |
| Real.exp (Fintype.card R*Fintype.card C-l) := by |
| let F : Finset (R × C) := Finset.univ.image e |
| let E : Fin l ≃ F := Equiv.ofBijective (fun x => ⟨e x,by simp [F]⟩) (by |
| constructor |
| · intro x y h |
| · intro y |
| obtain ⟨x,_,hx⟩ := Finset.mem_image.mp y.property |
| exact ⟨x,Subtype.ext hx⟩) |
| have hc : F.card=l := by |
| dsimp [F] |
| rw [Finset.card_image_of_injective _ he,Finset.card_univ,Fintype.card_fin] |
| have hh := board_trace_le_reindexed F E r v hr hv ht A hA hw |
| have hE (x : Fin l) : (E x).val=e x := rfl |
| simpa only [hE,hc] using hh |
| end CoordinateSweeps.TensorBoard |
|
|
| namespace CoordinateSweeps.SignedTensor |
| open TensorBoard |
| variable {R C I J : Type*} [Fintype R] [DecidableEq R] [Nonempty R] |
| [Fintype C] [DecidableEq C] [Fintype I] [Fintype J] |
| |
| def labelRowProduct {p l : ℕ} (e : Fin l → R × C) (r : R → EvenDensity p) : |
| Matrix (Word p l) (Word p l) ℂ := TensorBoard.tensor (fun x => (r (e x).1).val) |
|
|
| def labelColumnProduct {p l : ℕ} (e : Fin l → R × C) (r : C → EvenDensity p) : |
| Matrix (Word p l) (Word p l) ℂ := TensorBoard.tensor (fun x => (r (e x).2).val) |
|
|
| /- Literal deleted-board overlap with actual signed tensor action. The two |
| mixture dominations will be obtained by tensoring the local density theorem; |
| the whitening and the full-group dimension loss are constructed here. -/ |
| theorem label_board_overlap_regularized {p l : ℕ} (hp : 0 < p) |
| (e : Fin l → R × C) (hinj : Function.Injective e) |
| (ρ : UnitaryIrrep (Equiv.Perm (Fin l))) |
| (PR : Matrix (Word p l) (Word p l) ℂ) (xC : MonoidAlgebra ℂ (Equiv.Perm (Fin l))) |
| (hPR : PR.IsHermitian) (hPR2 : PR*PR=PR) |
| (hPC : (groupOperator (p := p) xC).IsHermitian) |
| (hPC2 : groupOperator (p := p) xC*groupOperator (p := p) xC=groupOperator (p := p) xC) |
| (u : I → ℝ) (v : J → ℝ) (hu : ∀ i, 0 ≤ u i) (hv : ∀ j, 0 ≤ v j) |
| (hsu : ∑ i, u i=1) (hsv : ∑ j, v j=1) |
| (r : I → R → EvenDensity p) (c : J → C → EvenDensity p) |
| (cr cc : ℝ) (hcr : 0 ≤ cr) (hcc : 0 ≤ cc) |
| (hrow : ((cr : ℂ) • (∑ i, (u i : ℂ) • labelRowProduct e (r i))-PR).PosSemidef) |
| (hcol : ((cc : ℂ) • (∑ j, (v j : ℂ) • labelColumnProduct e (c j))-groupOperator xC).PosSemidef) |
| {ε : ℝ} (hε : 0 < ε) : |
| (ρ.dimension : ℝ)*‖groupOperator (p := p) xC*ρ.projector (action p l)*PR‖^2 ≤ |
| cr*cc*(1+ε*(2*p : ℕ))^l*Real.exp (Fintype.card R*Fintype.card C-l) := by |
| let : NeZero (2*p) := ⟨by omega⟩ |
| let τ := action p l |
| let PC := groupOperator (p := p) xC |
| let B (i : I) : Matrix (Letter p) (Letter p) ℂ := |
| (Fintype.card R : ℂ)⁻¹ • (∑ k, (r i k).val)+(ε : ℂ) • 1 |
| choose A hA hAe hABA hAw using fun i => exists_even_whitening (r i) hε |
| have hb (i : I) : (B i).PosSemidef := |
| (average_regularization_posDef (fun k => (r i k).val) (fun k => (r i k).property.1) hε).posSemidef |
| have he (i : I) : IsEven (B i) := isEven_add |
| (isEven_smul _ (isEven_sum _ (fun k => (r i k).property.2.2))) (isEven_smul _ isEven_one) |
| have htau (g : Equiv.Perm (Fin l)) : (τ g).conjTranspose=τ g⁻¹ := (matrix_inv g).symm |
| have hpq : ρ.projector τ*PC=PC*ρ.projector τ := ρ.projector_commute_lift τ xC |
| have htt : 0 ≤ (1+ε*(2*p : ℕ))^l := pow_nonneg (by positivity) _ |
| have hwhite (i : I) : |
| (((1+ε*(2*p : ℕ))^l : ℝ) : ℂ) • (densityPower l (A i)*PC*densityPower l (A i)) - |
| (ρ.dimension : ℂ) • (PC*ρ.projector τ) |>.PosSemidef := by |
| have hWA : (densityPower l (A i)).IsHermitian := (densityPower_posSemidef l (A i) (hA i)).isHermitian |
| have hWABA : densityPower l (A i)*densityPower l (B i)*densityPower l (A i)=1 := by |
| rw [← densityPower_mul,← densityPower_mul,hABA i,densityPower_one] |
| have hh := Overlap.whitened_projector_domination ρ τ htau |
| (densityPower l (B i)) (densityPower l (A i)) PC (densityPower_posSemidef l (B i) (hb i)) |
| (fun g => densityPower_commute_of_even (B i) (he i) g) hWA hWABA |
| (densityPower_commute_projector (A i) (hAe i) ρ) |
| (densityPower_commute_lift (A i) (hAe i) xC).symm hPC hPC2 hpq |
| have ht : Matrix.trace (densityPower l (B i))=(((1+ε*(2*p : ℕ))^l : ℝ) : ℂ) := by |
| rw [densityPower_trace,even_average_trace] |
| push_cast |
| rfl |
| rwa [ht] at hh |
| apply Overlap.finite_mixture_overlap PR PC (ρ.projector τ) hPR hPR2 hPC hPC2 |
| (ρ.projector_hermitian τ htau) (ρ.projector_idempotent τ) hpq u v hu hv hsu hsv |
| (fun i => labelRowProduct e (r i)) (fun j => labelColumnProduct e (c j)) (fun i => densityPower l (A i)) |
| (fun i => TensorBoard.tensor_posSemidef _ (fun x => (r i (e x).1).property.1)) |
| (fun i => (densityPower_posSemidef l (A i) (hA i)).isHermitian) |
| cr cc ρ.dimension ((1+ε*(2*p : ℕ))^l) _ hcr hcc (Nat.cast_nonneg _) htt hrow hcol hwhite |
| intro i j |
| exact board_trace_le_labels e hinj (fun k => (r i k).val) (fun k => (c j k).val) |
| (fun k => (r i k).property.1) (fun k => (c j k).property.1) (fun k => (c j k).property.2.1) |
| (A i) (hA i).isHermitian (hAw i) |
| |
| end CoordinateSweeps.SignedTensor |
| |
| namespace CoordinateSweeps.SignedTensor |
| open TensorBoard Overlap |
| variable {R C I J : Type*} [Fintype R] [DecidableEq R] [Nonempty R] |
| [Fintype C] [DecidableEq C] [Fintype I] [Fintype J] |
| |
| theorem label_board_overlap {p l : ℕ} (hp : 0 < p) |
| (e : Fin l → R × C) (hinj : Function.Injective e) |
| (ρ : UnitaryIrrep (Equiv.Perm (Fin l))) |
| (PR : Matrix (Word p l) (Word p l) ℂ) (xC : MonoidAlgebra ℂ (Equiv.Perm (Fin l))) |
| (hPR : PR.IsHermitian) (hPR2 : PR*PR=PR) |
| (hPC : (groupOperator (p := p) xC).IsHermitian) |
| (hPC2 : groupOperator (p := p) xC*groupOperator (p := p) xC=groupOperator (p := p) xC) |
| (u : I → ℝ) (v : J → ℝ) (hu : ∀ i, 0 ≤ u i) (hv : ∀ j, 0 ≤ v j) |
| (hsu : ∑ i, u i=1) (hsv : ∑ j, v j=1) |
| (r : I → R → EvenDensity p) (c : J → C → EvenDensity p) |
| (cr cc : ℝ) (hcr : 0 ≤ cr) (hcc : 0 ≤ cc) |
| (hrow : ((cr : ℂ) • (∑ i, (u i : ℂ) • labelRowProduct e (r i))-PR).PosSemidef) |
| (hcol : ((cc : ℂ) • (∑ j, (v j : ℂ) • labelColumnProduct e (c j))-groupOperator xC).PosSemidef) |
| : |
| (ρ.dimension : ℝ)*‖groupOperator (p := p) xC*ρ.projector (action p l)*PR‖^2 ≤ |
| cr*cc*Real.exp (Fintype.card R*Fintype.card C-l) := by |
| let f (ε : ℝ) := cr*cc*(1+ε*(2*p : ℕ))^l*Real.exp (Fintype.card R*Fintype.card C-l) |
| have hf : Continuous f := by fun_prop |
| have hh : ∀ᶠ ε : ℝ in nhdsWithin 0 (Set.Ioi 0), |
| (ρ.dimension : ℝ)*‖groupOperator (p := p) xC*ρ.projector (action p l)*PR‖^2 ≤ f ε := by |
| filter_upwards [self_mem_nhdsWithin] with ε hε |
| exact label_board_overlap_regularized hp e hinj ρ PR xC hPR hPR2 hPC hPC2 |
| u v hu hv hsu hsv r c cr cc hcr hcc hrow hcol hε |
| have hh' := ge_of_tendsto (hf.continuousAt.tendsto.mono_left |
| (nhdsWithin_le_nhds : nhdsWithin (0 : ℝ) (Set.Ioi 0) ≤ nhds 0)) hh |
| simpa [f] using hh' |
| |
| end CoordinateSweeps.SignedTensor |
| |
| namespace CoordinateSweeps.SignedTensor |
| open TensorBoard |
| variable {p r c : ℕ} [NeZero r] |
| {L : Type*} [Fintype L] [DecidableEq L] |
| |
| /- Source 06:eq19 for literal row and column subgroup projectors on an |
| arbitrary board with holes. The mixtures and signed regroupings are constructed. -/ |
| omit [DecidableEq L] in |
| theorem actual_label_overlap (f : L → Fin r) (g : L → Fin c) |
| (hinj : Function.Injective (fun x => (f x,g x))) |
| (e : Fin (Fintype.card L) ≃ L) (s : ℕ) (hp : 0 < p) (hps : p ≤ s) |
| (hnR : ∀ i, FiberBlocks.size f i ≤ s) |
| (hnC : ∀ j, FiberBlocks.size g j ≤ s) |
| (ρR : ∀ i, UnitaryIrrep (Equiv.Perm (Fin (FiberBlocks.size f i)))) |
| (ρC : ∀ j, UnitaryIrrep (Equiv.Perm (Fin (FiberBlocks.size g j)))) |
| (ρ : UnitaryIrrep (Equiv.Perm (Fin (Fintype.card L)))) : |
| (ρ.dimension : ℝ)*‖fiberProjector (p := p) g e ρC* |
| ρ.projector (action p (Fintype.card L))*fiberProjector f e ρR‖^2 ≤ |
| ((∏ i, ((ρR i).dimension : ℝ))*(s+1 : ℕ)^(r*(41*p^2)))* |
| ((∏ j, ((ρC j).dimension : ℝ))*(s+1 : ℕ)^(c*(41*p^2)))* |
| Real.exp (r*c-Fintype.card L) := by |
| obtain ⟨I,hI,nI,a,ha⟩ := finite_fiber_density_domination f e s hp hps hnR ρR |
| obtain ⟨J,hJ,nJ,b,hb⟩ := finite_fiber_density_domination g e s hp hps hnC ρC |
| let := hI |
| let := hJ |
| let := nI |
| let := nJ |
| have hI0 : (Fintype.card I : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero |
| have hJ0 : (Fintype.card J : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero |
| have hu : ∑ _ : I, (Fintype.card I : ℝ)⁻¹=1 := by |
| rw [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_inv_cancel₀ hI0] |
| have hv : ∑ _ : J, (Fintype.card J : ℝ)⁻¹=1 := by |
| rw [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_inv_cancel₀ hJ0] |
| have hr := fiberProjector_posSemidef (p := p) f e ρR |
| have hc := fiberProjector_posSemidef (p := p) g e ρC |
| have hr2 := fiberProjector_idempotent (p := p) f e ρR |
| have hc2 := fiberProjector_idempotent (p := p) g e ρC |
| let xC := fiberProjectorAlgebra g e ρC |
| have hx : groupOperator (p := p) xC=fiberProjector (p := p) g e ρC := |
| fiberProjectorAlgebra_lift _ _ _ |
| have hrow : (((∏ i, ((ρR i).dimension : ℝ))*(s+1 : ℕ)^(r*(41*p^2)) : ℝ) : ℂ) • |
| (∑ i : I, (((Fintype.card I : ℝ)⁻¹ : ℝ) : ℂ) • labelRowProduct (fun x => (f (e x),g (e x))) (a i))- |
| fiberProjector (p := p) f e ρR |>.PosSemidef := by |
| simpa only [Complex.ofReal_mul,Complex.ofReal_prod,Complex.ofReal_natCast,Complex.ofReal_pow, |
| Complex.ofReal_inv,← Finset.smul_sum,labelRowProduct] using ha |
| have hcol : (((∏ j, ((ρC j).dimension : ℝ))*(s+1 : ℕ)^(c*(41*p^2)) : ℝ) : ℂ) • |
| (∑ j : J, (((Fintype.card J : ℝ)⁻¹ : ℝ) : ℂ) • labelColumnProduct (fun x => (f (e x),g (e x))) (b j))- |
| groupOperator (p := p) xC |>.PosSemidef := by |
| rw [hx] |
| simpa only [Complex.ofReal_mul,Complex.ofReal_prod,Complex.ofReal_natCast,Complex.ofReal_pow, |
| Complex.ofReal_inv,← Finset.smul_sum,labelColumnProduct] using hb |
| have hh := label_board_overlap hp (fun x => (f (e x),g (e x))) (hinj.comp e.injective) ρ _ xC hr.isHermitian hr2 (hx ▸ hc.isHermitian) |
| (by rw [hx]; exact hc2) (fun _ : I => (Fintype.card I : ℝ)⁻¹) |
| (fun _ : J => (Fintype.card J : ℝ)⁻¹) (fun _ => by positivity) (fun _ => by positivity) |
| hu hv a b ((∏ i, ((ρR i).dimension : ℝ))*(s+1 : ℕ)^(r*(41*p^2))) |
| ((∏ j, ((ρC j).dimension : ℝ))*(s+1 : ℕ)^(c*(41*p^2))) (by positivity) (by positivity) hrow hcol |
| simpa only [hx,Fintype.card_fin] using hh |
| |
| end CoordinateSweeps.SignedTensor |
| |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| open scoped Matrix.Norms.L2Operator |
| |
| end OAI |
| |
