| import Mathlib |
| import OAI.Analysis.FreeWords.MatrixConditionalExpectation |
| import OAI.Analysis.FreeWords.ProjectionSmall |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace KKFreeLiftFrontier |
|
|
| open KKTraceTransfer KKMatrixExpectation |
|
|
| noncomputable section |
|
|
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| abbrev MatrixAlgebra (n : ℕ) (M : VonNeumannAlgebra H) := |
| Matrix (Fin n) (Fin n) (ConcreteAlgebra M) |
|
|
| def matrixTrace {M : VonNeumannAlgebra H} (τ : FaithfulNormalTrace M) (n : ℕ) |
| (x : MatrixAlgebra n M) : ℂ := (n : ℂ)⁻¹ * ∑ i, τ.map (x i i) |
|
|
| def FiniteAmplifiedNumericalModel : Prop := |
| ∀ (M : VonNeumannAlgebra H) (τ : FaithfulNormalTrace M), IsFactor M → IsDiffuse M → |
| ∀ n : ℕ, 0<n → ∀ (K : Finset ℤ) (F : Finset (List (ℤ × Matrix (Fin n) (Fin n) ℂ))), |
| (∀ w∈F, w≠[]) → (∀ w∈F, ∀ z∈w, z.1≠0 ∧ Matrix.trace z.2=0) → |
| ∀ ε : ℝ, 0<ε → ∃ d : ℕ, 0<d ∧ |
| ∃ ρ : Matrix (Fin d) (Fin d) ℂ →⋆ₐ[ℂ] ConcreteAlgebra M, |
| Function.Injective ρ ∧ ∃ U : unitary (MatrixAlgebra n M), |
| (∀i j,U.val i j∈ρ.range) ∧ |
| (∀k∈K,k≠0 → ‖matrixTrace τ n ((U^k).val)‖<ε) ∧ |
| (∀w∈F,‖matrixTrace τ n |
| ((w.map (fun z => (U^z.1).val*numericalHom (A := ConcreteAlgebra M) n z.2)).prod)‖<ε) |
|
|
| end |
|
|
| end KKFreeLiftFrontier |
| end |
|
|
| end OAI |
|
|
