| import Mathlib |
| import OAI.Probability.BinarySweep.TensorBounds.SignedCommutant |
| import OAI.Probability.BinarySweep.MatrixBounds.MatrixProjection |
| import OAI.Probability.BinarySweep.TensorBounds.SignedPinching |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Classical ComplexOrder |
| open Equiv Equiv.Perm Matrix Representation |
|
|
| namespace BinaryCoordinateSweeps.Signed |
| open Density Irrep |
|
|
| variable {A : Type*} [Fintype A] [DecidableEq A] {n : ℕ} |
|
|
| omit [DecidableEq A] in |
| lemma coeff_actHilbert (p : A → Bool) (g : Equiv.Perm (Fin n)) (v : TensorSpace A n) : |
| coeffEquiv A n (actHilbert p g v) = act p g (coeffEquiv A n v) := rfl |
|
|
| omit [DecidableEq A] in |
| lemma coeff_symm_act (p : A → Bool) (g : Equiv.Perm (Fin n)) (v : (Fin n → A) → ℂ) : |
| (coeffEquiv A n).symm (act p g v) = actHilbert p g ((coeffEquiv A n).symm v) := rfl |
|
|
| def projectionIntertwiner (p : A → Bool) (S : Submodule ℂ ((Fin n → A) → ℂ)) |
| (hS : ∀ g, S.map (act p g) = S) : |
| IntertwiningMap (tensorRep p n) (tensorRep p n) where |
| toLinearMap := (coeffEquiv A n).conj (hilbertSubspace S).starProjection.toLinearMap |
| isIntertwining' g := by |
| apply LinearMap.ext |
| intro v |
| change coeffEquiv A n ((hilbertSubspace S).starProjection ((coeffEquiv A n).symm (act p g v))) = |
| act p g (coeffEquiv A n ((hilbertSubspace S).starProjection ((coeffEquiv A n).symm v))) |
| rw [coeff_symm_act, ← coeff_actHilbert] |
| congr 1 |
| exact (projectionMatrix_commutes (hilbertSubspace S) (actionIsometry p g) |
| (hilbertSubspace_map p S hS g) ((coeffEquiv A n).symm v)).symm |
|
|
| lemma projectionMatrix_invariant (p : A → Bool) (S : Submodule ℂ ((Fin n → A) → ℂ)) |
| (hS : ∀ g, S.map (act p g) = S) : |
| Invariant p (projectionMatrix (hilbertSubspace S)) := by |
| have h := intertwiner_matrix_invariant p (projectionIntertwiner p S hS) |
| change Invariant p (LinearMap.toMatrix' ((coeffEquiv A n).conj |
| (hilbertSubspace S).starProjection.toLinearMap)) at h |
| rw [show coeffEquiv A n = WithLp.linearEquiv 2 ℂ ((Fin n → A) → ℂ) from rfl, |
| projectionMatrix_coeff] at h |
| exact h |
|
|
| variable {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] |
|
|
| def isotypicProjection (p : A → Bool) (ρ : Representation ℂ (Equiv.Perm (Fin n)) V) : |
| Matrix (Fin n → A) (Fin n → A) ℂ := |
| projectionMatrix (hilbertSubspace (isotypicSpan ρ (tensorRep p n))) |
|
|
| omit [FiniteDimensional ℂ V] in |
| lemma isotypicProjection_psd (p : A → Bool) (ρ : Representation ℂ (Equiv.Perm (Fin n)) V) : |
| (isotypicProjection p ρ).PosSemidef := projectionMatrix_psd _ |
|
|
| omit [FiniteDimensional ℂ V] in |
| lemma isotypicProjection_invariant (p : A → Bool) |
| (ρ : Representation ℂ (Equiv.Perm (Fin n)) V) : Invariant p (isotypicProjection p ρ) := by |
| apply projectionMatrix_invariant |
| exact isotypicSpan_map ρ (tensorRep p n) |
|
|
| omit [FiniteDimensional ℂ V] in |
| lemma isotypicProjection_trace (p : A → Bool) |
| (ρ : Representation ℂ (Equiv.Perm (Fin n)) V) : |
| (isotypicProjection p ρ).trace = |
| (Module.finrank ℂ (isotypicSpan ρ (tensorRep p n)) : ℂ) := by |
| rw [isotypicProjection, projectionMatrix_trace, hilbertSubspace_finrank] |
|
|
| lemma isotypicSpan_rank_bound (p : A → Bool) |
| (ρ : Representation ℂ (Equiv.Perm (Fin n)) V) [ρ.IsIrreducible] : |
| Module.finrank ℂ (isotypicSpan ρ (tensorRep p n)) ≤ |
| Module.finrank ℂ V * (n+1)^((Fintype.card A)^2) := by |
| exact (isotypicSpan_finrank ρ (tensorRep p n)).trans (Nat.mul_le_mul_left _ |
| ((multiplicity_le_commutant ρ (tensorRep p n)).trans (tensor_commutant_finrank p n))) |
|
|
| theorem isotypic_density_domination (n : ℕ) (hn : 0 < n) (p : A → Bool) |
| (ρ : Representation ℂ (Equiv.Perm (Fin n)) V) [ρ.IsIrreducible] : |
| (((Module.finrank ℂ V : ℂ) * (n+1 : ℂ)^(2*(Fintype.card A)^2)) • |
| (∑ c : WordType (MatchedPair p) n, ∑ t : MatchedPair p → ZMod (n+1), |
| matrixTensorPower n (reducedPhaseDensity n (matchedEmbedding p) c t)) - |
| isotypicProjection p ρ).PosSemidef := by |
| let Q := ∑ c : WordType (MatchedPair p) n, ∑ t : MatchedPair p → ZMod (n+1), |
| matrixTensorPower n (reducedPhaseDensity n (matchedEmbedding p) c t) |
| have hQ : Q.PosSemidef := by |
| apply Matrix.posSemidef_sum |
| intro c _ |
| apply Matrix.posSemidef_sum |
| intro t _ |
| rw [reducedPhaseDensity, ← tensor_gram] |
| exact gram_psd _ |
| have hd := signed_psd_postselection n hn p (isotypicProjection p ρ) |
| (isotypicProjection_psd p ρ) (isotypicProjection_invariant p ρ) |
| rw [isotypicProjection_trace] at hd |
| simp only [Nat.cast_add, Nat.cast_one] at hd |
| have hr : (Module.finrank ℂ (isotypicSpan ρ (tensorRep p n)) : ℂ) ≤ |
| (Module.finrank ℂ V : ℂ) * (n+1 : ℂ)^((Fintype.card A)^2) := by |
| exact_mod_cast isotypicSpan_rank_bound p ρ |
| have hc : (0 : ℂ) ≤ (n+1 : ℂ)^((Fintype.card A)^2) := by positivity |
| have hh := hQ.smul (mul_nonneg hc (sub_nonneg.mpr hr)) |
| have he : (n+1 : ℂ)^((Fintype.card A)^2) * |
| ((Module.finrank ℂ V : ℂ) * (n+1 : ℂ)^((Fintype.card A)^2) - |
| (Module.finrank ℂ (isotypicSpan ρ (tensorRep p n)) : ℂ)) + |
| (n+1 : ℂ)^((Fintype.card A)^2) * |
| (Module.finrank ℂ (isotypicSpan ρ (tensorRep p n)) : ℂ) = |
| (Module.finrank ℂ V : ℂ) * (n+1 : ℂ)^(2*(Fintype.card A)^2) := by |
| rw [two_mul, pow_add] |
| ring |
| have hi := hh.add hd |
| change (_ • Q + (_ • Q - isotypicProjection p ρ)).PosSemidef at hi |
| rw [← add_sub_assoc, ← add_smul, he] at hi |
| exact hi |
|
|
| end BinaryCoordinateSweeps.Signed |
|
|
| end |
|
|
| end OAI |
|
|
