| import Mathlib |
| import OAI.MathematicalPhysics.PEPSFilters.LocalOperators |
| import OAI.MathematicalPhysics.PEPSSubvolume.Pinning |
|
|
| namespace OAI |
|
|
| /-! Zero-padded spectra and pinned core/shell filters. -/ |
|
|
| noncomputable section |
| open scoped BigOperators ComplexOrder |
| open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator |
| open scoped BigOperators |
| open scoped Topology |
| open Filter |
| open scoped MatrixOrder |
| open scoped BigOperators Matrix.Norms.L2Operator |
| open scoped ComplexOrder BigOperators Matrix.Norms.L2Operator |
| open Matrix |
| open Filter Topology |
| open Set Filter Complex Complex.HadamardThreeLines |
| open PolynomialPEPS.PinnedEntropy |
|
|
| namespace PolynomialPEPS.Subvolume.PinnedSpectral |
| open Matrix Polynomial |
| variable {ι κ : Type*} [Fintype ι] [Fintype κ] [DecidableEq ι] [DecidableEq κ] |
| |
| theorem sum_eigen_stat_isometry (V : Matrix ι κ ℂ) (hV : V.conjTranspose * V = 1) |
| (A : Matrix κ κ ℂ) (hA : A.IsHermitian) (f : ℝ → ℝ) (hf : f 0 = 0) : |
| ∑ i, f ((Matrix.isHermitian_mul_mul_conjTranspose V hA).eigenvalues i) = |
| ∑ j, f (hA.eigenvalues j) := by |
| have hp := Matrix.charpoly_mul_comm' V (A * V.conjTranspose) |
| rw [Matrix.mul_assoc, hV, Matrix.mul_one] at hp |
| rw [← Matrix.mul_assoc] at hp |
| have hh := congrArg |
| (fun P : Polynomial ℂ => (P.roots.map (fun z => f z.re)).sum) hp |
| rw [Polynomial.roots_mul (mul_ne_zero (pow_ne_zero _ X_ne_zero) |
| (Matrix.charpoly_monic _).ne_zero), |
| Polynomial.roots_mul (mul_ne_zero (pow_ne_zero _ X_ne_zero) |
| (Matrix.charpoly_monic _).ne_zero)] at hh |
| rw [Multiset.map_add, Multiset.map_add, Multiset.sum_add, Multiset.sum_add, |
| Polynomial.roots_X_pow, Polynomial.roots_X_pow] at hh |
| simp only [Multiset.map_nsmul, Multiset.map_singleton, Multiset.sum_nsmul, Multiset.sum_singleton, Complex.zero_re, hf, |
| nsmul_zero, zero_add] at hh |
| have hm := Matrix.isHermitian_mul_mul_conjTranspose V hA |
| have hleft := hm.roots_charpoly_eq_eigenvalues |
| have hright := hA.roots_charpoly_eq_eigenvalues |
| rw [hleft, hright] at hh |
| simpa only [Multiset.map_map, Function.comp_def, RCLike.ofReal_eq_complex_ofReal, |
| Complex.ofReal_re, Finset.sum_map_val] using hh |
|
|
| end PolynomialPEPS.Subvolume.PinnedSpectral |
|
|
| namespace PolynomialPEPS.Subvolume.Pinning |
| open Matrix |
| variable {L q : ℕ} (C X : Finset (Vertex L)) (hCX : C ⊆ X) |
|
|
| def insertedMatrix (u : EuclideanSpace ℂ (RegionConfiguration q C)) |
| (b : Matrix (RegionConfiguration q (X \ C)) (RegionConfiguration q (X \ C)) ℂ) : |
| Matrix (RegionConfiguration q X) (RegionConfiguration q X) ℂ := |
| insertion C X hCX u * b * (insertion C X hCX u).conjTranspose |
|
|
| theorem insertedMatrix_posSemidef (u : EuclideanSpace ℂ (RegionConfiguration q C)) |
| (b : Matrix (RegionConfiguration q (X \ C)) (RegionConfiguration q (X \ C)) ℂ) |
| (hb : b.PosSemidef) : (insertedMatrix C X hCX u b).PosSemidef := |
| hb.mul_mul_conjTranspose_same _ |
|
|
| theorem insertedMatrix_tracePower (u : EuclideanSpace ℂ (RegionConfiguration q C)) |
| (hu : ‖u‖ = 1) |
| (b : Matrix (RegionConfiguration q (X \ C)) (RegionConfiguration q (X \ C)) ℂ) |
| (hb : b.PosSemidef) (p : ℝ) (hp : 0 < p) : |
| ∑ i, Real.rpow ((insertedMatrix_posSemidef C X hCX u b hb).isHermitian.eigenvalues i) p = |
| ∑ j, Real.rpow (hb.isHermitian.eigenvalues j) p := by |
| exact PinnedSpectral.sum_eigen_stat_isometry _ (insertion_isometry C X hCX u hu) |
| b hb.isHermitian (fun t => Real.rpow t p) (Real.zero_rpow (ne_of_gt hp)) |
|
|
| theorem insertedMatrix_join (u : EuclideanSpace ℂ (RegionConfiguration q C)) |
| (b : Matrix (RegionConfiguration q (X \ C)) (RegionConfiguration q (X \ C)) ℂ) |
| (x y : RegionConfiguration q C) (w z : RegionConfiguration q (X \ C)) : |
| insertedMatrix C X hCX u b (joinWithin C X x w) (joinWithin C X y z) = |
| u x * b w z * star (u y) := by |
| classical |
| simp [insertedMatrix, Matrix.mul_apply, Matrix.conjTranspose_apply, insertion_join, |
| apply_ite] |
|
|
| def coreMatrix (u : EuclideanSpace ℂ (RegionConfiguration q C)) : |
| Matrix (RegionConfiguration q C) (RegionConfiguration q C) ℂ := |
| fun x y => u x * star (u y) |
|
|
| theorem insertedMatrix_eq_product (u : EuclideanSpace ℂ (RegionConfiguration q C)) |
| (b : Matrix (RegionConfiguration q (X \ C)) (RegionConfiguration q (X \ C)) ℂ) : |
| insertedMatrix C X hCX u b = liftBetween hCX (coreMatrix C u) * |
| liftBetween (Finset.sdiff_subset : X \ C ⊆ X) b := by |
| classical |
| ext x y |
| obtain ⟨⟨xc,xs⟩,rfl⟩ := (splitEquiv (q := q) C X hCX).surjective x |
| obtain ⟨⟨yc,ys⟩,rfl⟩ := (splitEquiv (q := q) C X hCX).surjective y |
| change insertedMatrix C X hCX u b (joinWithin C X xc xs) (joinWithin C X yc ys) = _ |
| rw [insertedMatrix_join, Matrix.mul_apply, ← Equiv.sum_comp (splitEquiv (q := q) C X hCX)] |
| simp only [splitEquiv, Equiv.coe_fn_mk, Fintype.sum_prod_type, |
| liftBetween_core_join, liftBetween_shell_join C X hCX] |
| simp [coreMatrix, mul_comm, mul_assoc] |
|
|
| theorem liftBetween_core_shell_commute |
| (a : Matrix (RegionConfiguration q C) (RegionConfiguration q C) ℂ) |
| (b : Matrix (RegionConfiguration q (X \ C)) (RegionConfiguration q (X \ C)) ℂ) : |
| Commute (liftBetween hCX a) (liftBetween (Finset.sdiff_subset : X \ C ⊆ X) b) := by |
| classical |
| ext x y |
| obtain ⟨⟨xc,xs⟩,rfl⟩ := (splitEquiv (q := q) C X hCX).surjective x |
| obtain ⟨⟨yc,ys⟩,rfl⟩ := (splitEquiv (q := q) C X hCX).surjective y |
| simp only [Matrix.mul_apply] |
| rw [← Equiv.sum_comp (splitEquiv (q := q) C X hCX), |
| ← Equiv.sum_comp (splitEquiv (q := q) C X hCX)] |
| simp only [splitEquiv, Equiv.coe_fn_mk, Fintype.sum_prod_type, |
| liftBetween_core_join, liftBetween_shell_join C X hCX] |
| simp [mul_comm] |
|
|
| include hCX in |
| theorem lift_core_shell_commute |
| (a : Matrix (RegionConfiguration q C) (RegionConfiguration q C) ℂ) |
| (b : Matrix (RegionConfiguration q (X \ C)) (RegionConfiguration q (X \ C)) ℂ) : |
| Commute (liftLocal C a) (liftLocal (X \ C) b) := by |
| have h := congrArg (liftLocal X) (liftBetween_core_shell_commute C X hCX a b).eq |
| change _ = _ |
| simpa only [liftLocal_mul, liftLocal_liftBetween] using h |
|
|
| theorem lift_insertedMatrix (u : EuclideanSpace ℂ (RegionConfiguration q C)) |
| (b : Matrix (RegionConfiguration q (X \ C)) (RegionConfiguration q (X \ C)) ℂ) : |
| liftLocal X (insertedMatrix C X hCX u b) = |
| liftLocal C (coreMatrix C u) * liftLocal (X \ C) b := by |
| rw [insertedMatrix_eq_product, liftLocal_mul, liftLocal_liftBetween, liftLocal_liftBetween] |
|
|
| theorem coreMatrix_idempotent (u : EuclideanSpace ℂ (RegionConfiguration q C)) |
| (hu : ‖u‖ = 1) : coreMatrix C u * coreMatrix C u = coreMatrix C u := by |
| classical |
| have hnorm : ∑ i, star (u i) * u i = (1 : ℂ) := by |
| simpa only [PiLp.inner_apply, RCLike.inner_apply, hu, one_pow, |
| RCLike.ofReal_one, starRingEnd_apply, mul_comm] using (inner_self_eq_norm_sq_to_K (𝕜 := ℂ) u) |
| ext x y |
| simp only [Matrix.mul_apply, coreMatrix] |
| calc |
| _ = u x * (∑ i, star (u i) * u i) * star (u y) := by |
| simp only [Finset.mul_sum, Finset.sum_mul] |
| apply Finset.sum_congr rfl |
| intro i hi |
| ring |
| _ = _ := by rw [hnorm,mul_one] |
|
|
| theorem coreMatrix_isHermitian (u : EuclideanSpace ℂ (RegionConfiguration q C)) : |
| (coreMatrix C u).IsHermitian := by |
| ext x y |
| simp [coreMatrix,Matrix.conjTranspose_apply,mul_comm] |
|
|
| end PolynomialPEPS.Subvolume.Pinning |
|
|
| end |
|
|
| end OAI |
|
|
