| import Mathlib |
| import OAI.Probability.BinarySweep.GridBounds.GroupedMoments |
| import OAI.Probability.BinarySweep.MatrixBounds.TraceDensityComparison |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| section |
|
|
| open scoped BigOperators Classical ComplexOrder MatrixOrder Matrix.Norms.L2Operator |
| open Matrix |
|
|
| namespace BinaryCoordinateSweeps.Signed |
| open Density Irrep Representation |
|
|
| variable {A I J U W V : Type*} [Fintype A] [DecidableEq A] [Nonempty A] |
| [Fintype I] [DecidableEq I] [Nonempty I] [Fintype J] [DecidableEq J] |
| [Fintype U] [Fintype W] |
| [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] |
|
|
| omit [FiniteDimensional ℂ V] in |
| theorem regularized_board_overlap {n : ℕ} (e : Fin n ↪ I × J) (p : A → Bool) |
| (ρ : Representation ℂ (Equiv.Perm (Fin n)) V) [ρ.IsIrreducible] |
| (R Q : Matrix (Fin n → A) (Fin n → A) ℂ) |
| (hR : R.IsHermitian) (hQ : Q.IsHermitian) (hRR : R*R=R) (hQQ : Q*Q=Q) |
| (hPQ : isotypicProjection p ρ * Q = Q * isotypicProjection p ρ) |
| (hCQ : ∀ M : Matrix A A ℂ, M.PosSemidef → |
| (∀ a b, p a ≠ p b → M a b = 0) → matrixTensorPower n M * Q = Q * matrixTensorPower n M) |
| (r : U → I → Matrix A A ℂ) (v : W → J → Matrix A A ℂ) |
| (hr : ∀ u i, (r u i).PosSemidef) (hv : ∀ w j, (v w j).PosSemidef) |
| (htr : ∀ u i, (r u i).trace=1) (htv : ∀ w j, (v w j).trace=1) |
| (her : ∀ u i a b, p a ≠ p b → r u i a b=0) |
| (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) |
| (hdr : ((a:ℂ) • ∑u, tensorMatrices (fun t : Fin n => r u (e t).1)-R).PosSemidef) |
| (hdv : ((b:ℂ) • ∑w, tensorMatrices (fun t : Fin n => v w (e t).2)-Q).PosSemidef) |
| {ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) : |
| (Module.finrank ℂ V : ℝ) * (1-ε)^n * ‖Q * isotypicProjection p ρ * R‖^2 ≤ |
| a*b*Fintype.card U*Fintype.card W* |
| Real.exp (Fintype.card I * Fintype.card J - (n : ℝ)) := by |
| let P := isotypicProjection p ρ |
| let d : ℝ := Module.finrank ℂ V |
| let L : ℝ := (1-ε)^n |
| let E := Real.exp (Fintype.card I * Fintype.card J - (n : ℝ)) |
| let T (u : U) := tensorMatrices (fun t : Fin n => r u (e t).1) |
| let Z (w : W) := tensorMatrices (fun t : Fin n => v w (e t).2) |
| have hL : 0 ≤ L := pow_nonneg (sub_nonneg.mpr hε1) n |
| have hd : 0 ≤ d := Nat.cast_nonneg _ |
| have hP : P.PosSemidef := isotypicProjection_psd p ρ |
| have hPP : P*P=P := projectionMatrix_idempotent _ |
| have hn := projection_overlap_norm_trace R Q P hR hQ hP.isHermitian hRR hQQ hPP hPQ |
| have hQP := commuting_projection_product_psd P Q hP hQ hQQ hPQ |
| have hrow := trace_right_density_domination R (Q*P) a T hQP hdr |
| rw [← mul_assoc R Q P] at hrow |
| have hlocal (u : U) : L * (d * (T u*Q*P).trace.re) ≤ b*(Fintype.card W*E) := by |
| let M := regularizedMean (r u) ε |
| have hM : M.PosDef := regularizedMean_posDef (r u) (hr u) hε hε1 |
| have heM := regularizedMean_even (r u) p (her u) ε |
| obtain ⟨B,C,hB,hC,hBB,hBC,hCB,heB,heC⟩ := exists_even_whitening p M hM heM |
| let H := matrixTensorPower n C |
| have hH : H.PosSemidef := tensorMatrices_psd _ (fun _ => hC) |
| have hcap := signed_tensor_whitening p ρ M B C hB hC hBB hBC |
| (regularizedMean_trace (r u) (htr u) ε) heM heC |
| change (H.conjTranspose*H-(d:ℂ)•P).PosSemidef at hcap |
| rw [hH.isHermitian.eq] at hcap |
| have hcomp := compress_whitening_cap P Q H (d:ℂ) hQ hQQ hPQ (hCQ C hC heC) hcap |
| have htu : (T u).PosSemidef := tensorMatrices_psd _ (fun t => hr u (e t).1) |
| have ht := trace_density_whitening (T u) Q P H d b Z htu hH.isHermitian hcomp hdv |
| have hCMC : C * regularizedMean (r u) ε * C = 1 := by |
| change C * M * C = 1 |
| rw [← hBB,← mul_assoc C B B,hCB,one_mul,hBC] |
| have hz (w : W) : L * (H*T u*H*Z w).trace.re ≤ E := by |
| have hh := whitened_board_budget e (r u) (v w) C (hr u) (hv w) (htv w) |
| hε.le hε1 hC.isHermitian hCMC |
| change L * (tensorMatrices (fun _ : Fin n => C) * T u * |
| tensorMatrices (fun _ : Fin n => C) * Z w).trace.re ≤ E |
| simpa only [T,Z,L,E,tensorMatrices_mul] using hh |
| calc |
| L*(d*(T u*Q*P).trace.re) ≤ L*(b*∑w, (H*T u*H*Z w).trace.re) := |
| mul_le_mul_of_nonneg_left ht hL |
| _ = b*∑w, L*(H*T u*H*Z w).trace.re := by simp only [Finset.mul_sum]; congr 1; ring_nf |
| _ ≤ b*∑w : W, E := mul_le_mul_of_nonneg_left |
| (Finset.sum_le_sum (fun w _ => hz w)) hb |
| _ = b*(Fintype.card W*E) := by simp |
| change d*L*‖Q*P*R‖^2 ≤ a*b*Fintype.card U*Fintype.card W*E |
| calc |
| _ ≤ d*L*(a*∑u, (T u*(Q*P)).trace.re) := |
| mul_le_mul_of_nonneg_left (hn.trans hrow) (mul_nonneg hd hL) |
| _ = a*∑u, L*(d*(T u*Q*P).trace.re) := by |
| simp only [← mul_assoc,Finset.mul_sum] |
| ring_nf |
| _ ≤ a*∑u : U, b*(Fintype.card W*E) := |
| mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun u _ => hlocal u)) ha |
| _ = _ := by simp; ring |
|
|
| omit [FiniteDimensional ℂ V] in |
| theorem board_overlap {n : ℕ} (e : Fin n ↪ I × J) (p : A → Bool) |
| (ρ : Representation ℂ (Equiv.Perm (Fin n)) V) [ρ.IsIrreducible] |
| (R Q : Matrix (Fin n → A) (Fin n → A) ℂ) |
| (hR : R.IsHermitian) (hQ : Q.IsHermitian) (hRR : R*R=R) (hQQ : Q*Q=Q) |
| (hPQ : isotypicProjection p ρ * Q = Q * isotypicProjection p ρ) |
| (hCQ : ∀ M : Matrix A A ℂ, M.PosSemidef → |
| (∀ a b, p a ≠ p b → M a b = 0) → matrixTensorPower n M * Q = Q * matrixTensorPower n M) |
| (r : U → I → Matrix A A ℂ) (v : W → J → Matrix A A ℂ) |
| (hr : ∀ u i, (r u i).PosSemidef) (hv : ∀ w j, (v w j).PosSemidef) |
| (htr : ∀ u i, (r u i).trace=1) (htv : ∀ w j, (v w j).trace=1) |
| (her : ∀ u i a b, p a ≠ p b → r u i a b=0) |
| (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) |
| (hdr : ((a:ℂ) • ∑u, tensorMatrices (fun t : Fin n => r u (e t).1)-R).PosSemidef) |
| (hdv : ((b:ℂ) • ∑w, tensorMatrices (fun t : Fin n => v w (e t).2)-Q).PosSemidef) |
| : (Module.finrank ℂ V : ℝ) * ‖Q * isotypicProjection p ρ * R‖^2 ≤ |
| a*b*Fintype.card U*Fintype.card W* |
| Real.exp (Fintype.card I * Fintype.card J - (n : ℝ)) := by |
| have hε1 : ∀ᶠ ε : ℝ in nhdsWithin 0 (Set.Ioi 0), ε < 1 := |
| (eventually_lt_nhds (by norm_num : (0:ℝ)<1)).filter_mono nhdsWithin_le_nhds |
| have hbound : ∀ᶠ ε : ℝ in nhdsWithin 0 (Set.Ioi 0), |
| (Module.finrank ℂ V : ℝ) * (1-ε)^n * ‖Q*isotypicProjection p ρ*R‖^2 ≤ |
| a*b*Fintype.card U*Fintype.card W* |
| Real.exp (Fintype.card I * Fintype.card J - (n : ℝ)) := by |
| filter_upwards [self_mem_nhdsWithin,hε1] with ε hε hε1 |
| exact regularized_board_overlap e p ρ R Q hR hQ hRR hQQ hPQ hCQ r v hr hv |
| htr htv her a b ha hb hdr hdv hε hε1.le |
| apply le_of_tendsto _ hbound |
| have ht : Filter.Tendsto (fun ε : ℝ => (Module.finrank ℂ V:ℝ)*(1-ε)^n* |
| ‖Q*isotypicProjection p ρ*R‖^2) (nhdsWithin 0 (Set.Ioi 0)) |
| (nhds ((Module.finrank ℂ V:ℝ)*(1-(0:ℝ))^n*‖Q*isotypicProjection p ρ*R‖^2)) := by |
| exact (show Continuous (fun ε : ℝ => (Module.finrank ℂ V:ℝ)*(1-ε)^n* |
| ‖Q*isotypicProjection p ρ*R‖^2) by fun_prop).continuousWithinAt.tendsto |
| simpa using ht |
|
|
| end BinaryCoordinateSweeps.Signed |
|
|
| end |
|
|
| open scoped BigOperators Classical ComplexOrder MatrixOrder Matrix.Norms.L2Operator |
| open Matrix |
|
|
| namespace BinaryCoordinateSweeps.Signed |
| open Density Irrep Representation |
|
|
| variable {A I J : Type*} [Fintype A] [DecidableEq A] [Nonempty A] |
| [Fintype I] [LinearOrder I] [Nonempty I] [Fintype J] [LinearOrder J] |
| {nR : I → ℕ} {nC : J → ℕ} {N : ℕ} |
| {VR : I → Type*} [∀i, NormedAddCommGroup (VR i)] |
| [∀i, InnerProductSpace ℂ (VR i)] [∀i, FiniteDimensional ℂ (VR i)] |
| {VC : J → Type*} [∀j, NormedAddCommGroup (VC j)] |
| [∀j, InnerProductSpace ℂ (VC j)] [∀j, FiniteDimensional ℂ (VC j)] |
| {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] |
|
|
| omit [FiniteDimensional ℂ V] in |
| theorem actual_board_overlap (board : Fin N ↪ I × J) (p : A → Bool) |
| (eR : (Σₗ i, Fin (nR i)) ≃o Fin N) (eC : (Σₗ j, Fin (nC j)) ≃o Fin N) |
| (gR gC : Equiv.Perm (Fin N)) |
| (hR : ∀t, groupOf eR gR t=(board t).1) (hC : ∀t, groupOf eC gC t=(board t).2) |
| (ρ : Representation ℂ (Equiv.Perm (Fin N)) V) [ρ.IsIrreducible] |
| (α : ∀i, Representation ℂ (Equiv.Perm (Fin (nR i))) (VR i)) [∀i, (α i).IsIrreducible] |
| (β : ∀j, Representation ℂ (Equiv.Perm (Fin (nC j))) (VC j)) [∀j, (β j).IsIrreducible] : |
| (Module.finrank ℂ V:ℝ)* |
| ‖groupedProjection eC gC p β*isotypicProjection p ρ*groupedProjection eR gR p α‖^2 ≤ |
| (∏i, (Module.finrank ℂ (VR i):ℝ)*(nR i+1:ℝ)^(2*(Fintype.card A)^2))* |
| (∏j, (Module.finrank ℂ (VC j):ℝ)*(nC j+1:ℝ)^(2*(Fintype.card A)^2))* |
| Fintype.card (∀i, LocalDensityIndex p (nR i))*Fintype.card (∀j, LocalDensityIndex p (nC j))* |
| Real.exp (Fintype.card I*Fintype.card J-(N:ℝ)) := by |
| apply board_overlap board p ρ _ _ |
| (groupedProjection_psd eR gR p α).isHermitian |
| (groupedProjection_psd eC gC p β).isHermitian |
| (groupedProjection_idempotent eR gR p α) |
| (groupedProjection_idempotent eC gC p β) |
| (groupedProjection_commutes eC gC p β _ (isotypicProjection_invariant p ρ)) |
| (fun M _ he => groupedProjection_commutes eC gC p β _ (even_tensor_invariant p M he)) |
| (fun u i => localDensity p (nR i) (u i)) (fun w j => localDensity p (nC j) (w j)) |
| (fun u i => localDensity_psd p _ _) (fun w j => localDensity_psd p _ _) |
| (fun u i => localDensity_trace p _ _) (fun w j => localDensity_trace p _ _) |
| (fun u i => localDensity_even p _ _) |
| · positivity |
| · positivity |
| · have h := grouped_density_domination eR gR p α |
| rw [show groupOf eR gR = (fun t => (board t).1) from funext hR] at h |
| simpa only [hR,Complex.ofReal_prod,Complex.ofReal_mul,Complex.ofReal_natCast, |
| Complex.ofReal_pow,Complex.ofReal_add,Complex.ofReal_one] using h |
| · have h := grouped_density_domination eC gC p β |
| rw [show groupOf eC gC = (fun t => (board t).2) from funext hC] at h |
| simpa only [hC,Complex.ofReal_prod,Complex.ofReal_mul,Complex.ofReal_natCast, |
| Complex.ofReal_pow,Complex.ofReal_add,Complex.ofReal_one] using h |
|
|
| end BinaryCoordinateSweeps.Signed |
|
|
| end |
|
|
| end OAI |
|
|
