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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 | 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
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