| import Mathlib |
| import OAI.Analysis.HardSphere.Density.DensityOperators |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped ComplexOrder InnerProductSpace NNReal ENNReal |
| namespace HardSphere.Density |
| open ContinuousLinearMap |
| variable {H K : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| [NormedAddCommGroup K] [InnerProductSpace ℂ K] [CompleteSpace K] |
|
|
| lemma Operator.compression_summable (T : Operator H) (A : H →L[ℂ] K) : |
| Summable (fun j : basisSet K ↦ |
| (inner ℂ (A.adjoint (basis K j)) (T.toCLM (A.adjoint (basis K j)))).re) := by |
| have hh := (T.entry_summable A (basis K)).prod_symm.prod |
| convert hh using 1 |
| funext j |
| change _ = ∑' i : basisSet H,T.entry A (basis K) i j |
| rw [(T.entry_col A (basis K) j).tsum_eq,T.root_norm_sq] |
|
|
| lemma Operator.compressionTrace_linear (T U V : Operator H) (A : H →L[ℂ] K) |
| (c d : ℝ) (hV : V.toCLM=c • T.toCLM+d • U.toCLM) : |
| V.compressionTrace A=c*T.compressionTrace A+d*U.compressionTrace A := by |
| unfold Operator.compressionTrace |
| simp only [hV,add_apply,smul_apply,inner_add_right,inner_smul_right_eq_smul, |
| Complex.add_re,Complex.smul_re,smul_eq_mul] |
| rw [((T.compression_summable A).mul_left c).tsum_add ((U.compression_summable A).mul_left d), |
| tsum_mul_left,tsum_mul_left] |
|
|
| omit [CompleteSpace H] in |
| lemma rankOne_diagonal (x y : H) : |
| (inner ℂ x ((InnerProductSpace.rankOne ℂ y y) x)).re=‖inner ℂ x y‖^2 := by |
| rw [InnerProductSpace.rankOne_apply,inner_smul_right,←inner_conj_symm y x] |
| rw [Complex.conj_mul'] |
| norm_cast |
|
|
| def pure (x : H) (hx : ‖x‖=1) : Operator H where |
| toCLM := InnerProductSpace.rankOne ℂ x x |
| positive := ContinuousLinearMap.nonneg_iff_isPositive.mpr (InnerProductSpace.isPositive_rankOne_self x) |
| trace_one := by |
| simpa only [rankOne_diagonal,hx,one_pow] using parseval (basis H) x |
|
|
| lemma pure_compression (x : H) (hx : ‖x‖=1) (A : H →L[ℂ] K) : |
| (pure x hx).compressionTrace A=‖A x‖^2 := by |
| unfold Operator.compressionTrace |
| change (∑' j : basisSet K,(inner ℂ (A.adjoint (basis K j)) |
| ((InnerProductSpace.rankOne ℂ x x) (A.adjoint (basis K j)))).re)=_ |
| calc |
| _ = ∑' j : basisSet K,‖inner ℂ (A.adjoint (basis K j)) x‖^2 := |
| tsum_congr fun j ↦ rankOne_diagonal _ _ |
| _ = ∑' j : basisSet K,‖inner ℂ (basis K j) (A x)‖^2 := by simp only [A.adjoint_inner_left] |
| _ = _ := (parseval (basis K) (A x)).tsum_eq |
|
|
| end HardSphere.Density |
|
|
| namespace HardSphere.Density |
| open ContinuousLinearMap |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| omit [CompleteSpace H] in |
| lemma projection_diagonal (V : Submodule ℂ H) [FiniteDimensional ℂ V] (x : H) : |
| (inner ℂ x (V.starProjection x)).re= |
| ∑j : Fin (Module.finrank ℂ V),‖inner ℂ x ((stdOrthonormalBasis ℂ V j : V):H)‖^2 := by |
| rw [(stdOrthonormalBasis ℂ V).starProjection_eq_sum_rankOne] |
| simp only [sum_apply,inner_sum,Complex.re_sum,rankOne_diagonal] |
|
|
| lemma projection_trace (V : Submodule ℂ H) [FiniteDimensional ℂ V] : |
| HasSum (fun i : basisSet H ↦ (inner ℂ (basis H i) (V.starProjection (basis H i))).re) |
| (Module.finrank ℂ V : ℝ) := by |
| have hj (j : Fin (Module.finrank ℂ V)) : |
| HasSum (fun i : basisSet H ↦ ‖inner ℂ (basis H i) ((stdOrthonormalBasis ℂ V j : V):H)‖^2) 1 := by |
| simpa only [Submodule.norm_coe,(stdOrthonormalBasis ℂ V).norm_eq_one,one_pow] using |
| parseval (basis H) ((stdOrthonormalBasis ℂ V j : V):H) |
| have hh := hasSum_sum (s:=Finset.univ) (fun j _ ↦ hj j) |
| simpa only [projection_diagonal,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_one] using hh |
|
|
| def normalizedProjection (V : Submodule ℂ H) [FiniteDimensional ℂ V] [Nontrivial V] : Operator H where |
| toCLM := ((Module.finrank ℂ V:ℝ)⁻¹) • V.starProjection |
| positive := smul_nonneg (inv_nonneg.mpr (Nat.cast_nonneg _)) isStarProjection_starProjection.nonneg |
| trace_one := by |
| have hd : (Module.finrank ℂ V:ℝ)≠0 := by exact_mod_cast (Module.finrank_pos (R:=ℂ) (M:=V)).ne' |
| have hh := (projection_trace V).mul_left ((Module.finrank ℂ V:ℝ)⁻¹) |
| simpa only [smul_apply,inner_smul_right_eq_smul,Complex.smul_re,smul_eq_mul,inv_mul_cancel₀ hd] using hh |
|
|
| lemma normalizedProjection_support (V : Submodule ℂ H) [FiniteDimensional ℂ V] [Nontrivial V] : |
| (normalizedProjection V).toCLM.kerᗮ=V := by |
| have hd : (Module.finrank ℂ V:ℝ)⁻¹≠0 := inv_ne_zero (by exact_mod_cast (Module.finrank_pos (R:=ℂ) (M:=V)).ne') |
| have hk : (normalizedProjection V).toCLM.ker=V.starProjection.ker := by |
| ext x |
| change ((Module.finrank ℂ V:ℝ)⁻¹) • V.starProjection x=0 ↔ V.starProjection x=0 |
| simp only [smul_eq_zero,hd,false_or] |
| rw [hk,V.ker_starProjection,V.orthogonal_orthogonal] |
|
|
| end HardSphere.Density |
| end |
|
|
| end OAI |
|
|
