| import Mathlib | |
| import OAI.MathematicalPhysics.PEPSFilters.LocalOperators | |
| import OAI.MathematicalPhysics.PEPSSubvolume.SpectralPowers | |
| namespace OAI | |
| /-! Zero-eigenspace perturbations and superquadratic normalization. -/ | |
| noncomputable section | |
| open scoped BigOperators ComplexOrder | |
| open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator | |
| open PolynomialPEPS.PinnedEntropy | |
| namespace PolynomialPEPS.Subvolume.KernelPerturbation | |
| open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator | |
| open PolynomialPEPS.Subvolume.SpectralCurve | |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] | |
| def kernel (U : unitary (Matrix ι ι ℂ)) (e : ι → ℝ) : Matrix ι ι ℂ := | |
| spectralHom U (fun i => if e i = 0 then 1 else 0) | |
| def magnitude (e : ι → ℝ) (t : ℝ) (i : ι) : ℝ := | |
| if e i = 0 then |t| else e i | |
| def phaseValue (e : ι → ℝ) (ζ : unitary ℂ) (t : ℝ) (i : ι) : ℂ := | |
| if e i = 0 then (if t < 0 then -(ζ : ℂ) else (ζ : ℂ)) else 1 | |
| omit [Fintype ι] [DecidableEq ι] in | |
| theorem phaseValue_unitary (e : ι → ℝ) (ζ : unitary ℂ) (t : ℝ) : | |
| (fun i => phaseValue e ζ t i) ∈ unitary (ι → ℂ) := by | |
| constructor | |
| · ext i | |
| change star (phaseValue e ζ t i) * phaseValue e ζ t i = 1 | |
| by_cases hi : e i = 0 | |
| · by_cases ht : t < 0 | |
| · simp only [phaseValue, ite_eq_left hi, ite_eq_left ht, star_neg, neg_mul_neg] | |
| exact ζ.property.1 | |
| · simp only [phaseValue, ite_eq_left hi, ite_eq_right ht] | |
| exact ζ.property.1 | |
| · simp [phaseValue,hi] | |
| · ext i | |
| change phaseValue e ζ t i * star (phaseValue e ζ t i) = 1 | |
| by_cases hi : e i = 0 | |
| · by_cases ht : t < 0 | |
| · simp only [phaseValue, ite_eq_left hi, ite_eq_left ht, star_neg, neg_mul_neg] | |
| exact ζ.property.2 | |
| · simp only [phaseValue, ite_eq_left hi, ite_eq_right ht] | |
| exact ζ.property.2 | |
| · simp [phaseValue,hi] | |
| def perturbUnitary (U : unitary (Matrix ι ι ℂ)) (e : ι → ℝ) | |
| (ζ : unitary ℂ) (t : ℝ) : unitary (Matrix ι ι ℂ) := | |
| ⟨spectralHom U (phaseValue e ζ t), | |
| Unitary.map_mem (spectralHom U) (phaseValue_unitary e ζ t)⟩ | |
| omit [Fintype ι] [DecidableEq ι] in | |
| theorem magnitude_nonneg (e : ι → ℝ) (he : ∀ i, 0 ≤ e i) (t : ℝ) (i : ι) : | |
| 0 ≤ magnitude e t i := by | |
| unfold magnitude | |
| split | |
| · exact abs_nonneg t | |
| · exact he i | |
| theorem magnitude_posSemidef (U : unitary (Matrix ι ι ℂ)) (e : ι → ℝ) | |
| (he : ∀ i, 0 ≤ e i) (t : ℝ) : | |
| (spectralHom U (fun i => (magnitude e t i : ℂ))).PosSemidef := by | |
| apply Matrix.PosSemidef.mul_mul_conjTranspose_same | |
| apply Matrix.PosSemidef.diagonal | |
| intro i | |
| exact Complex.nonneg_iff.mpr ⟨magnitude_nonneg e he t i,rfl⟩ | |
| theorem perturb_factorization (U : unitary (Matrix ι ι ℂ)) (e : ι → ℝ) | |
| (ζ : unitary ℂ) (t : ℝ) : | |
| (perturbUnitary U e ζ t : Matrix ι ι ℂ) * | |
| spectralHom U (fun i => (magnitude e t i : ℂ)) = | |
| spectralHom U (fun i => (e i : ℂ)) + ((t : ℂ)*(ζ : ℂ)) • kernel U e := by | |
| change spectralHom U _ * spectralHom U _ = _ | |
| unfold kernel | |
| rw [← map_mul, ← map_smul, ← map_add] | |
| congr 1 | |
| ext i | |
| simp only [Pi.mul_apply, Pi.add_apply, Pi.smul_apply, smul_eq_mul, | |
| phaseValue, magnitude] | |
| by_cases hi : e i = 0 | |
| · simp only [hi, ite_eq_left, Complex.ofReal_zero, zero_add, mul_one] | |
| by_cases ht : t < 0 | |
| · rw [ite_eq_left ht, abs_of_neg ht, Complex.ofReal_neg] | |
| ring | |
| · rw [ite_eq_right ht, abs_of_nonneg (le_of_not_gt ht)] | |
| ring | |
| · simp [hi] | |
| omit [DecidableEq ι] in | |
| theorem magnitude_power_sum (e : ι → ℝ) (p : ℝ) (hp : 0 < p) | |
| (hs : ∑ i, (e i)^p = 1) (t : ℝ) : | |
| ∑ i, (magnitude e t i)^p = 1 + | |
| (∑ i : ι, if e i = 0 then (1:ℝ) else 0) * |t|^p := by | |
| have heach (i : ι) : (magnitude e t i)^p = | |
| (e i)^p + (if e i = 0 then (1:ℝ) else 0)*|t|^p := by | |
| unfold magnitude | |
| by_cases hi : e i = 0 | |
| · simp [hi, Real.zero_rpow (ne_of_gt hp)] | |
| · simp [hi] | |
| simp_rw [heach] | |
| rw [Finset.sum_add_distrib, ← Finset.sum_mul, hs] | |
| end PolynomialPEPS.Subvolume.KernelPerturbation | |
| open scoped BigOperators | |
| namespace PolynomialPEPS.Subvolume.KernelNormalization | |
| variable {ι : Type*} [Fintype ι] | |
| def scale (S p : ℝ) : ℝ := S^(-(1/p)) | |
| theorem scale_pos (S p : ℝ) (hS : 0 < S) : 0 < scale S p := | |
| Real.rpow_pos_of_pos hS _ | |
| theorem scale_power (S p : ℝ) (hS : 0 < S) (hp : p ≠ 0) : | |
| (scale S p)^p * S = 1 := by | |
| rw [scale, ← Real.rpow_mul hS.le] | |
| have he : -(1/p)*p = -1 := by field_simp | |
| rw [he, Real.rpow_neg_one, inv_mul_cancel₀ (ne_of_gt hS)] | |
| theorem power_sum (d : ι → ℝ) (hd : ∀ i, 0 ≤ d i) | |
| (S p : ℝ) (hS : 0 < S) (hp : p ≠ 0) (hs : ∑ i, (d i)^p = S) : | |
| ∑ i, (scale S p*d i)^p = 1 := by | |
| simp_rw [Real.mul_rpow (scale_pos S p hS).le (hd _)] | |
| rw [← Finset.mul_sum, hs, scale_power S p hS hp] | |
| theorem scale_inv_sq_le (S p : ℝ) (hS : 1 ≤ S) (hp : 2 < p) : | |
| (scale S p)⁻¹^2 ≤ S := by | |
| have hSp : 0 < S := lt_of_lt_of_le zero_lt_one hS | |
| rw [scale, ← Real.rpow_neg hSp.le] | |
| simp only [neg_neg] | |
| rw [← Real.rpow_natCast _ 2, ← Real.rpow_mul hSp.le] | |
| conv_rhs => rw [← Real.rpow_one S] | |
| apply Real.rpow_le_rpow_of_exponent_le hS | |
| have hp' : 0 < p := by linarith | |
| have he : (1/p)*2 = 2/p := by ring | |
| norm_num only [Nat.cast_ofNat] | |
| rw [he] | |
| exact (div_le_one hp').mpr (le_of_lt hp) | |
| theorem norm_sq_bound (N M S p : ℝ) (hN : 0 ≤ N) (hM : 0 ≤ M) | |
| (hS : 1 ≤ S) (hp : 2 < p) (hb : scale S p * M ≤ N) : | |
| M^2 ≤ N^2*S := by | |
| have hspos := scale_pos S p (lt_of_lt_of_le zero_lt_one hS) | |
| have hm : M ≤ N*(scale S p)⁻¹ := by | |
| apply (le_div_iff₀ hspos).mpr | |
| simpa [mul_comm, div_eq_mul_inv] using hb | |
| have hm' : M^2 ≤ (N*(scale S p)⁻¹)^2 := by | |
| exact (sq_le_sq₀ hM (mul_nonneg hN (inv_nonneg.mpr hspos.le))).mpr hm | |
| rw [mul_pow] at hm' | |
| exact hm'.trans (mul_le_mul_of_nonneg_left (scale_inv_sq_le S p hS hp) (sq_nonneg N)) | |
| end PolynomialPEPS.Subvolume.KernelNormalization | |
| namespace PolynomialPEPS.Subvolume.KernelPerturbation | |
| open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator | |
| open PolynomialPEPS.Subvolume.SpectralCurve | |
| open PolynomialPEPS.Subvolume.KernelNormalization | |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] | |
| def cost (e : ι → ℝ) (p t : ℝ) : ℝ := | |
| 1 + (∑ i : ι, if e i = 0 then (1:ℝ) else 0) * |t|^p | |
| omit [DecidableEq ι] in | |
| theorem one_le_cost (e : ι → ℝ) (p t : ℝ) : 1 ≤ cost e p t := by | |
| unfold cost | |
| have hk : 0 ≤ ∑ i : ι, if e i = 0 then (1:ℝ) else 0 := | |
| Finset.sum_nonneg (fun i _ => by split <;> norm_num) | |
| exact le_add_of_nonneg_right (mul_nonneg hk (Real.rpow_nonneg (abs_nonneg _) _)) | |
| def normalized (U : unitary (Matrix ι ι ℂ)) (e : ι → ℝ) (p t : ℝ) : Matrix ι ι ℂ := | |
| spectralHom U (fun i => ((scale (cost e p t) p * magnitude e t i : ℝ) : ℂ)) | |
| theorem normalized_posSemidef (U : unitary (Matrix ι ι ℂ)) | |
| (e : ι → ℝ) (he : ∀ i, 0 ≤ e i) (p t : ℝ) : | |
| (normalized U e p t).PosSemidef := by | |
| apply Matrix.PosSemidef.mul_mul_conjTranspose_same | |
| apply Matrix.PosSemidef.diagonal | |
| intro i | |
| refine Complex.nonneg_iff.mpr ⟨?_,rfl⟩ | |
| exact mul_nonneg (scale_pos _ _ (lt_of_lt_of_le zero_lt_one (one_le_cost e p t))).le | |
| (magnitude_nonneg e he t i) | |
| theorem normalized_capacity (U : unitary (Matrix ι ι ℂ)) | |
| (e : ι → ℝ) (he : ∀ i, 0 ≤ e i) (p : ℝ) (hp : 0 < p) | |
| (hs : ∑ i, (e i)^p = 1) (t : ℝ) : | |
| ∑ i, ((normalized_posSemidef U e he p t).isHermitian.eigenvalues i)^p = 1 := by | |
| have hsum := sum_eigenvalues_spectralHom U | |
| (fun i => scale (cost e p t) p * magnitude e t i) | |
| (normalized_posSemidef U e he p t).isHermitian (fun x => x^p) | |
| apply hsum.trans | |
| exact power_sum (magnitude e t) (magnitude_nonneg e he t) (cost e p t) p | |
| (lt_of_lt_of_le zero_lt_one (one_le_cost e p t)) (ne_of_gt hp) | |
| (magnitude_power_sum e p hp hs t) | |
| theorem normalized_eq_smul (U : unitary (Matrix ι ι ℂ)) | |
| (e : ι → ℝ) (p t : ℝ) : | |
| normalized U e p t = (scale (cost e p t) p : ℂ) • | |
| spectralHom U (fun i => (magnitude e t i : ℂ)) := by | |
| unfold normalized | |
| rw [← map_smul] | |
| apply congrArg (spectralHom U) | |
| funext i | |
| exact Complex.ofReal_mul _ _ | |
| theorem normalized_factorization (U : unitary (Matrix ι ι ℂ)) | |
| (e : ι → ℝ) (ζ : unitary ℂ) (p t : ℝ) : | |
| (perturbUnitary U e ζ t : Matrix ι ι ℂ) * normalized U e p t = | |
| (scale (cost e p t) p : ℂ) • | |
| (spectralHom U (fun i => (e i : ℂ)) + ((t : ℂ)*(ζ : ℂ)) • kernel U e) := by | |
| rw [normalized_eq_smul, mul_smul_comm, perturb_factorization] | |
| end PolynomialPEPS.Subvolume.KernelPerturbation | |
| end | |
| end OAI | |