| import Mathlib |
| import OAI.Analysis.FreeWords.RestrictedDerivationDefinitions |
| import OAI.Analysis.FreeWords.ScalarMatrixCommutator |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace KKMatrixHilbert |
|
|
| noncomputable section |
|
|
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
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|
| theorem controlled_lift_commutator {n d : ℕ} |
| (S : StarSubalgebra ℂ (H →L[ℂ] H)) |
| (ρ : Matrix (Fin d) (Fin d) ℂ →⋆ₐ[ℂ] (H →L[ℂ] H)) |
| (hρ : ∀ a, ρ a ∈ S) (h : H →L[ℂ] H) (_ : IsSelfAdjoint h) |
| (s : unitary (Matrix (Fin n) (Fin n) (H →L[ℂ] H))) |
| (hs : ∀ i j, s.val i j ∈ ρ.range) : |
| ‖matrixOperator (Matrix.diagonal fun _ : Fin n => h) * matrixOperator s.val - |
| matrixOperator s.val * matrixOperator (Matrix.diagonal fun _ : Fin n => h)‖ ≤ |
| 2 * ‖restrictedDerivation S h‖ := by |
| apply scalar_matrix_commutator_bound ρ h _ _ s hs |
| intro u |
| let a : S := ⟨ρ u.val,hρ u.val⟩ |
| have ha : a ∈ unitary S := by |
| constructor |
| · apply Subtype.ext |
| change star (ρ u.val) * ρ u.val = 1 |
| rw [←map_star,←map_mul,u.property.1,map_one] |
| · apply Subtype.ext |
| change ρ u.val * star (ρ u.val) = 1 |
| rw [←map_star,←map_mul,u.property.2,map_one] |
| have hnorm : ‖a‖ ≤ 1 := by |
| cases subsingleton_or_nontrivial S with |
| | inl h => have := h; simp [Subsingleton.elim a 0] |
| | inr h => have := h; exact (CStarRing.norm_of_mem_unitary ha).le |
| change ‖restrictedDerivation S h a‖ ≤ ‖restrictedDerivation S h‖ |
| calc |
| _ ≤ ‖restrictedDerivation S h‖ * ‖a‖ := (restrictedDerivation S h).le_opNorm a |
| _ ≤ ‖restrictedDerivation S h‖ * 1 := mul_le_mul_of_nonneg_left hnorm (norm_nonneg _) |
| _ = _ := mul_one _ |
|
|
| end |
|
|
| end KKMatrixHilbert |
| end |
|
|
| end OAI |
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|
