| import OAI.RepresentationTheory.SpinAngle.Core |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators ComplexConjugate InnerProductSpace |
| open scoped BigOperators ComplexConjugate InnerProductSpace Matrix |
| open scoped Matrix ComplexOrder |
| open scoped TensorProduct InnerProductSpace BigOperators |
| open scoped BigOperators ComplexConjugate |
| open scoped InnerProductSpace |
| open scoped TensorProduct InnerProductSpace ComplexOrder |
| open scoped BigOperators |
| open scoped InnerProductSpace ComplexOrder BigOperators |
| open scoped InnerProductSpace BigOperators ComplexOrder |
| open scoped BigOperators |
| open scoped BigOperators |
| open MeasureTheory TopologicalSpace Set |
| open scoped BigOperators |
| open scoped BigOperators |
| open Set |
| open scoped BigOperators Matrix.Norms.L2Operator MatrixOrder ComplexOrder |
| open scoped BigOperators |
| open scoped BigOperators Matrix.Norms.L2Operator MatrixOrder ComplexOrder |
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| noncomputable section |
| open scoped BigOperators |
| namespace SpinAngle.Analytic |
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| variable {A : Type*} [contextInstance35_21 : CStarAlgebra A] [contextInstance35_38 : PartialOrder A] [contextInstance35_55 : StarOrderedRing A] |
| [contextInstance36_2 : NonnegSpectrumClass ℝ A] |
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| lemma norm_mul_projection_sq |
| {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
| [NonnegSpectrumClass ℝ A] (a p : A) (hp : IsStarProjection p) : |
| ‖a * p‖ ^ 2 = ‖a * p * star a‖ := by |
| rw [sq, ← CStarRing.norm_self_mul_star, star_mul, hp.isSelfAdjoint.star_eq] |
| congr 1 |
| calc |
| a * p * (p * star a) = a * (p * p) * star a := by simp only [mul_assoc] |
| _ = a * p * star a := by rw [hp.isIdempotentElem.eq] |
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| lemma norm_projection_mul_sq |
| {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
| [NonnegSpectrumClass ℝ A] (p a : A) (hp : IsStarProjection p) : |
| ‖p * a‖ ^ 2 = ‖star a * p * a‖ := by |
| rw [sq, ← CStarRing.norm_star_mul_self, star_mul, hp.isSelfAdjoint.star_eq] |
| congr 1 |
| calc |
| star a * p * (p * a) = star a * (p * p) * a := by simp only [mul_assoc] |
| _ = star a * p * a := by rw [hp.isIdempotentElem.eq] |
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| lemma norm_mul_projection_sq_le_sum {ι : Type*} [Fintype ι] |
| (a p : A) (hp : IsStarProjection p) (w : ι → ℝ) (r : ι → A) |
| (hw : ∀ i, 0 ≤ w i) (hdom : p ≤ ∑ i, w i • (r i * star (r i))) : |
| ‖a * p‖ ^ 2 ≤ ∑ i, w i * ‖a * r i‖ ^ 2 := by |
| rw [norm_mul_projection_sq a p hp] |
| have hcomp := star_right_conjugate_le_conjugate hdom a |
| have hpos : 0 ≤ a * p * star a := |
| star_right_conjugate_nonneg hp.nonneg a |
| calc |
| ‖a * p * star a‖ ≤ ‖a * (∑ i, w i • (r i * star (r i))) * star a‖ := |
| CStarAlgebra.norm_le_norm_of_le_of_nonneg hcomp hpos |
| _ = ‖∑ i, w i • ((a * r i) * star (a * r i))‖ := by |
| congr 1 |
| simp only [Finset.mul_sum, Finset.sum_mul, mul_smul_comm, smul_mul_assoc, |
| star_mul, mul_assoc] |
| _ ≤ ∑ i, ‖w i • ((a * r i) * star (a * r i))‖ := norm_sum_le _ _ |
| _ = ∑ i, w i * ‖a * r i‖ ^ 2 := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [norm_smul, Real.norm_of_nonneg (hw i), CStarRing.norm_self_mul_star, sq] |
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| lemma norm_projection_mul_sq_le_sum {ι : Type*} [Fintype ι] |
| (p a : A) (hp : IsStarProjection p) (w : ι → ℝ) (r : ι → A) |
| (hw : ∀ i, 0 ≤ w i) (hdom : p ≤ ∑ i, w i • (star (r i) * r i)) : |
| ‖p * a‖ ^ 2 ≤ ∑ i, w i * ‖r i * a‖ ^ 2 := by |
| have h := norm_mul_projection_sq_le_sum (star a) p hp w (fun i => star (r i)) hw |
| (by simpa using hdom) |
| have hab : ‖star a * p‖ = ‖p * a‖ := by |
| calc |
| ‖star a * p‖ = ‖star (p * a)‖ := by rw [star_mul, hp.isSelfAdjoint.star_eq] |
| _ = ‖p * a‖ := norm_star _ |
| have hr (i : ι) : ‖star a * star (r i)‖ = ‖r i * a‖ := by |
| rw [← star_mul, norm_star] |
| simpa only [hab, hr] using h |
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| lemma support_inverse_factorization |
| {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
| [NonnegSpectrumClass ℝ A] (r v t q k p : A) |
| (hvt : v * t = q) (hrq : r * q = r) |
| (htk : Commute t k) (htp : Commute t p) : |
| r * k * p = (r * v * k) * (p * t) := by |
| calc |
| r * k * p = (r * q) * k * p := by rw [hrq] |
| _ = (r * (v * t)) * k * p := by rw [hvt] |
| _ = (r * v * k) * (p * t) := by |
| simp only [mul_assoc] |
| rw [← mul_assoc t k, htk.eq, mul_assoc k, htp.eq] |
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| lemma norm_support_inverse_le (r v t q k p : A) |
| (hvt : v * t = q) (hrq : r * q = r) |
| (htk : Commute t k) (htp : Commute t p) : |
| ‖r * k * p‖ ^ 2 ≤ ‖r * v * k‖ ^ 2 * ‖p * t‖ ^ 2 := by |
| rw [support_inverse_factorization r v t q k p hvt hrq htk htp] |
| calc |
| ‖(r * v * k) * (p * t)‖ ^ 2 ≤ (‖r * v * k‖ * ‖p * t‖) ^ 2 := by |
| exact pow_le_pow_left₀ (norm_nonneg _) (norm_mul_le _ _) 2 |
| _ = _ := mul_pow _ _ _ |
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| lemma norm_projection_sqrt_sq (p t : A) (hp : IsStarProjection p) |
| (ht : IsSelfAdjoint t) (htp : Commute t p) : |
| ‖p * t‖ ^ 2 = ‖(t * t) * p‖ := by |
| rw [norm_projection_mul_sq p t hp, ht.star_eq] |
| congr 1 |
| calc |
| t * p * t = t * (p * t) := mul_assoc _ _ _ |
| _ = t * (t * p) := by rw [htp.eq] |
| _ = t * t * p := (mul_assoc _ _ _).symm |
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| lemma norm_three_projections_sq_le_one |
| {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
| [NonnegSpectrumClass ℝ A] (h k p : A) |
| (hh : IsStarProjection h) (hk : IsStarProjection k) (hp : IsStarProjection p) : |
| ‖h * k * p‖ ^ 2 ≤ 1 := by |
| have hn : ‖h * k * p‖ ≤ 1 := calc |
| _ ≤ (‖h‖ * ‖k‖) * ‖p‖ := (norm_mul_le _ _).trans <| |
| mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _) |
| _ ≤ (1 * 1) * 1 := mul_le_mul |
| (mul_le_mul hh.norm_le hk.norm_le (norm_nonneg _) zero_le_one) |
| hp.norm_le (norm_nonneg _) (by norm_num) |
| _ = 1 := by norm_num |
| calc |
| _ ≤ (1 : ℝ) ^ 2 := pow_le_pow_left₀ (norm_nonneg _) hn 2 |
| _ = 1 := by norm_num |
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| theorem three_projection_angle_of_density_mixtures |
| {ι κ : Type*} [Fintype ι] [Fintype κ] |
| (h k p : A) (hh : IsStarProjection h) (hk : IsStarProjection k) |
| (hp : IsStarProjection p) |
| (w : ι → ℝ) (z : κ → ℝ) (hw : ∀ i, 0 ≤ w i) (hz : ∀ j, 0 ≤ z j) |
| (r v t q : ι → A) (s : κ → A) |
| (hH : h ≤ ∑ i, w i • (star (r i) * r i)) |
| (hK : k ≤ ∑ j, z j • (s j * star (s j))) |
| (hvt : ∀ i, v i * t i = q i) (hrq : ∀ i, r i * q i = r i) |
| (htk : ∀ i, Commute (t i) k) (htp : ∀ i, Commute (t i) p) |
| (g phi : ℝ) (hg : 0 ≤ g) (hphi : 0 ≤ phi) |
| (hglobal : ∀ i, ‖p * t i‖ ^ 2 ≤ g) |
| (hcell : ∀ i j, ‖r i * v i * s j‖ ^ 2 ≤ phi) : |
| ‖h * k * p‖ ^ 2 ≤ min 1 ((∑ i, w i) * (∑ j, z j) * g * phi) := by |
| apply le_min (norm_three_projections_sq_le_one h k p hh hk hp) |
| have hs (i : ι) : ‖r i * v i * k‖ ^ 2 ≤ (∑ j, z j) * phi := by |
| calc |
| _ ≤ ∑ j, z j * ‖r i * v i * s j‖ ^ 2 := |
| norm_mul_projection_sq_le_sum (r i * v i) k hk z s hz hK |
| _ ≤ ∑ j, z j * phi := Finset.sum_le_sum fun j _ => |
| mul_le_mul_of_nonneg_left (hcell i j) (hz j) |
| _ = _ := (Finset.sum_mul _ _ _).symm |
| have hb (i : ι) : ‖r i * k * p‖ ^ 2 ≤ ((∑ j, z j) * phi) * g := calc |
| _ ≤ ‖r i * v i * k‖ ^ 2 * ‖p * t i‖ ^ 2 := |
| norm_support_inverse_le (r i) (v i) (t i) (q i) k p |
| (hvt i) (hrq i) (htk i) (htp i) |
| _ ≤ ((∑ j, z j) * phi) * g := by |
| by_cases hzero : ‖p * t i‖ ^ 2 = 0 |
| · rw [hzero, mul_zero] |
| exact mul_nonneg (mul_nonneg (Finset.sum_nonneg fun j _ => hz j) hphi) hg |
| · exact mul_le_mul (hs i) (hglobal i) (sq_nonneg _) |
| (mul_nonneg (Finset.sum_nonneg fun j _ => hz j) hphi) |
| calc |
| ‖h * k * p‖ ^ 2 = ‖h * (k * p)‖ ^ 2 := by rw [mul_assoc] |
| _ ≤ ∑ i, w i * ‖r i * (k * p)‖ ^ 2 := |
| norm_projection_mul_sq_le_sum h (k * p) hh w r hw hH |
| _ = ∑ i, w i * ‖r i * k * p‖ ^ 2 := by simp only [mul_assoc] |
| _ ≤ ∑ i, w i * (((∑ j, z j) * phi) * g) := |
| Finset.sum_le_sum fun i _ => mul_le_mul_of_nonneg_left (hb i) (hw i) |
| _ = (∑ i, w i) * (∑ j, z j) * g * phi := by rw [← Finset.sum_mul]; ring |
| |
| end SpinAngle.Analytic |
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| end |
| end |
| |
| end OAI |
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