| import Mathlib |
| import OAI.Analysis.Crouzeix.Density |
| import OAI.Analysis.Crouzeix.DiskEndpoint |
| import OAI.Analysis.Crouzeix.TensorDensity |
|
|
| namespace OAI |
|
|
| /-! Density Compatibility. -/ |
|
|
| noncomputable section |
|
|
| open MeasureTheory |
|
|
| open scoped InnerProductSpace TensorProduct Matrix Matrix.Norms.L2Operator MatrixOrder ComplexOrder |
|
|
| namespace CrouzeixHilbert |
|
|
| namespace Boundary |
|
|
| def toHSCLM (n : ℕ) : Coeff n →L[ℂ] HS n := |
| ({ toFun := toHS |
| map_add' := fun _ _ => by apply PiLp.ext; intro ij; rfl |
| map_smul' := fun _ _ => by apply PiLp.ext; intro ij; rfl } : |
| Coeff n →ₗ[ℂ] HS n).toContinuousLinearMap |
|
|
| @[simp] theorem toHSCLM_apply {n : ℕ} (X : Coeff n) : toHSCLM n X = toHS X := rfl |
|
|
| def weightedPartialTraceCLM {n : ℕ} (X Y : Coeff n) : Coeff n →L[ℂ] Coeff n := |
| ({ toFun E := weightedPartialTrace E X Y |
| map_add' E G := by simp [weightedPartialTrace, mul_add, add_mul] |
| map_smul' c E := by simp [weightedPartialTrace] } : |
| Coeff n →ₗ[ℂ] Coeff n).toContinuousLinearMap |
|
|
| @[simp] theorem weightedPartialTraceCLM_apply {n : ℕ} (E X Y : Coeff n) : |
| weightedPartialTraceCLM X Y E = weightedPartialTrace E X Y := rfl |
|
|
| theorem integral_weightedPartialTrace {α : Type*} [MeasurableSpace α] |
| {μ : Measure α} {n : ℕ} (E : α → Coeff n) (hE : Integrable E μ) (X Y : Coeff n) : |
| (∫ t, weightedPartialTrace (E t) X Y ∂μ) = |
| weightedPartialTrace (∫ t, E t ∂μ) X Y := |
| (weightedPartialTraceCLM X Y).integral_comp_comm hE |
|
|
| def densityContinuous {n : ℕ} (E : C(CircleSpace, Coeff n)) (X Y : Coeff n) : |
| C(CircleSpace, HS n) := |
| ⟨fun t => toHS (weightedPartialTrace (E t) X Y), |
| (toHSCLM n).continuous.comp ((weightedPartialTraceCLM X Y).continuous.comp E.continuous)⟩ |
|
|
| def densityLp {n : ℕ} (E : C(CircleSpace, Coeff n)) (X Y : Coeff n) : BoundaryL2 n := |
| ContinuousMap.toLp 2 circleMeasure ℂ (densityContinuous E X Y) |
|
|
| theorem densityLp_apply_ae {n : ℕ} (E : C(CircleSpace, Coeff n)) (X Y : Coeff n) : |
| densityLp E X Y =ᵐ[circleMeasure] fun t => toHS (weightedPartialTrace (E t) X Y) := |
| ContinuousMap.coeFn_toLp _ _ |
|
|
| theorem matrixMean_densityLp {n : ℕ} (E : C(CircleSpace, Coeff n)) (X Y : Coeff n) : |
| matrixMean (densityLp E X Y) = weightedPartialTrace (∫ t, E t ∂circleMeasure) X Y := by |
| rw [← integral_weightedPartialTrace E |
| (E.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))] |
| ext i j |
| change (∫ t, densityLp E X Y t (i,j) ∂circleMeasure) = _ |
| have hi : Integrable (fun t => weightedPartialTrace (E t) X Y) circleMeasure := |
| (weightedPartialTraceCLM X Y).integrable_comp |
| (E.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) |
| let e : Coeff n →L[ℂ] ℂ := |
| (PiLp.proj 2 (fun _ : Fin n × Fin n => ℂ) (i,j)).comp (toHSCLM n) |
| change (∫ t, densityLp E X Y t (i,j) ∂circleMeasure) = |
| e (∫ t, weightedPartialTrace (E t) X Y ∂circleMeasure) |
| rw [← e.integral_comp_comm hi] |
| apply integral_congr_ae |
| filter_upwards [densityLp_apply_ae E X Y] with t ht |
| rw [ht] |
| rfl |
|
|
| theorem matrixMean_densityLp_mass_one {n : ℕ} (E : C(CircleSpace, Coeff n)) |
| (hmass : ∫ t, E t ∂circleMeasure = 1) (X Y : Coeff n) : |
| matrixMean (densityLp E X Y) = (Xᴴ * Y)ᵀ := by |
| rw [matrixMean_densityLp, hmass, weightedPartialTrace, mul_one] |
|
|
| theorem densityLp_block_posSemidef {n : ℕ} (E : C(CircleSpace, Coeff n)) |
| (hE : ∀ t, (E t).PosSemidef) (X Y : Coeff n) : |
| ∀ᵐ t ∂circleMeasure, |
| (Matrix.fromBlocks (ofHS (densityLp E X X t)) |
| (ofHS (densityLp E X Y t))ᴴ (ofHS (densityLp E X Y t)) |
| (ofHS (densityLp E Y Y t))).PosSemidef := by |
| filter_upwards [densityLp_apply_ae E X X, densityLp_apply_ae E Y Y, |
| densityLp_apply_ae E X Y] with t hp hq hr |
| simp only [hp, hq, hr, ofHS_toHS] |
| have hs : (weightedPartialTrace (E t) X Y)ᴴ = weightedPartialTrace (E t) Y X := by |
| simp only [weightedPartialTrace, Matrix.conjTranspose_transpose_eq_transpose_conjTranspose, |
| Matrix.conjTranspose_mul, (hE t).isHermitian.eq, Matrix.conjTranspose_conjTranspose, |
| Matrix.mul_assoc] |
| rw [hs] |
| exact weightedPartialTrace_block_posSemidef (E t) (hE t) X Y |
|
|
| theorem densityLp_self_hermitian {n : ℕ} (E : C(CircleSpace, Coeff n)) |
| (hE : ∀ t, (E t).PosSemidef) (X : Coeff n) : |
| ∀ᵐ t ∂circleMeasure, (ofHS (densityLp E X X t)).IsHermitian := by |
| filter_upwards [densityLp_apply_ae E X X] with t ht |
| rw [ht, ofHS_toHS] |
| exact ((hE t).conjTranspose_mul_mul_same X).transpose.isHermitian |
|
|
| end Boundary |
|
|
| open Boundary |
|
|
| theorem inner_vectorization {n : ℕ} (X Y : Coeff n) : |
| ⟪vectorization X, vectorization Y⟫_ℂ = (Xᴴ * Y).trace := by |
| let e : Amplification (EuclideanSpace ℂ (Fin n)) n →ₗᵢ[ℂ] |
| EuclideanSpace ℂ (Fin n × Fin n) := |
| ⟨(matrixAmplificationCoordinates n n).toLinearMap, norm_matrixAmplificationCoordinates n n⟩ |
| rw [← e.inner_map_map] |
| change ⟪matrixAmplificationCoordinates n n (vectorization X), |
| matrixAmplificationCoordinates n n (vectorization Y)⟫_ℂ = _ |
| rw [vectorization_coordinates, vectorization_coordinates, inner_toHS] |
|
|
| theorem weightedPartialTrace_testing {n : ℕ} (E : Coeff n) (hE : E.IsHermitian) |
| (X Y U : Coeff n) : |
| ⟪toHS (weightedPartialTrace E X Y), toHS U⟫_ℂ = |
| ⟪vectorization Y, tensorOperator (Matrix.toEuclideanCLM (𝕜 := ℂ) E) U |
| (vectorization X)⟫_ℂ := by |
| rw [tensorOperator_vectorization, inner_vectorization, inner_toHS] |
| simp only [weightedPartialTrace, Matrix.conjTranspose_transpose_eq_transpose_conjTranspose, |
| Matrix.conjTranspose_mul, Matrix.conjTranspose_conjTranspose, hE.eq] |
| rw [← Matrix.trace_transpose (_ * U), Matrix.transpose_mul, |
| Matrix.transpose_transpose, Matrix.trace_mul_comm Uᵀ] |
| simp only [Matrix.mul_assoc] |
|
|
| end CrouzeixHilbert |
|
|
| end |
|
|
| end OAI |
|
|
