| import OAI.Analysis.FreeEntropy.TensorFreeness |
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| namespace OAI |
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| noncomputable section |
| open scoped ENNReal NNReal ComplexConjugate lp |
| open MeasureTheory Filter |
| namespace FiniteEntropySeparation |
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| def equidiscreteAtom (p : ℕ) (l : Fin p) : ℝ := -1 + 2 * l.val / (p - 1 : ℝ) |
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| theorem equidiscreteAtom_rev {p : ℕ} (hp : 2 ≤ p) (l : Fin p) : |
| equidiscreteAtom p l.rev = -equidiscreteAtom p l := by |
| have hd : (p : ℝ) - 1 ≠ 0 := by |
| have hp' : (2 : ℝ) ≤ p := by exact_mod_cast hp |
| linarith |
| have hl : (l.rev.val : ℝ) = (p : ℝ) - 1 - l.val := by |
| have hn : l.rev.val + 1 + l.val = p := by simp [Fin.rev]; omega |
| have hr : (l.rev.val : ℝ) + 1 + l.val = p := by exact_mod_cast hn |
| linarith |
| unfold equidiscreteAtom |
| rw [hl] |
| field_simp |
| ; ring |
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| theorem sum_equidiscreteAtom {p : ℕ} (hp : 2 ≤ p) : ∑ l : Fin p, equidiscreteAtom p l = 0 := by |
| have hs := Equiv.sum_comp (Fin.revPerm (n := p)) (equidiscreteAtom p) |
| simp only [Fin.revPerm_apply, equidiscreteAtom_rev hp, Finset.sum_neg_distrib] at hs |
| linarith |
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| theorem equidiscreteAtom_abs_le {p : ℕ} (hp : 2 ≤ p) (l : Fin p) : |
| |equidiscreteAtom p l| ≤ 1 := by |
| have hd : 0 < (p : ℝ) - 1 := by |
| have hp' : (2 : ℝ) ≤ p := by exact_mod_cast hp |
| linarith |
| have hl : (l.val : ℝ) ≤ (p : ℝ) - 1 := by |
| have hn : l.val + 1 ≤ p := l.isLt |
| exact_mod_cast (show (l.val : ℤ) ≤ (p : ℤ) - 1 by omega) |
| rw [abs_le, equidiscreteAtom] |
| constructor |
| · have h := div_nonneg (show 0 ≤ 2 * (l.val : ℝ) by positivity) hd.le |
| linarith |
| · have h : 2 * (l.val : ℝ) / ((p : ℝ) - 1) ≤ 2 := (div_le_iff₀ hd).mpr (by linarith) |
| linarith |
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| namespace FreeTensor |
| variable {B N : Type} [DecidableEq B] [DecidableEq N] (p : ℕ) [NeZero p] |
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| def dualIndex : AddChar (ZMod p) ℂ ≃ Fin p := |
| Fintype.equivFinOfCardEq (by simp) |
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| def equidiscreteSignal : Space B N (Multiplicative (ZMod p)) →L[ℂ] Space B N (Multiplicative (ZMod p)) := |
| spectralSignal (fun ψ => equidiscreteAtom p (dualIndex p ψ)) |
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| theorem equidiscreteSignal_selfAdjoint : IsSelfAdjoint (equidiscreteSignal (B := B) (N := N) p) := |
| spectralSignal_selfAdjoint _ |
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| theorem equidiscreteSignal_mem : equidiscreteSignal (B := B) (N := N) p ∈ |
| algebra B N (Multiplicative (ZMod p)) := spectralSignal_mem _ |
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| theorem equidiscreteSignal_norm_le (hp : 2 ≤ p) : |
| ‖equidiscreteSignal (B := B) (N := N) p‖ ≤ 1 := |
| spectralSignal_norm_le _ 1 zero_le_one (fun _character => equidiscreteAtom_abs_le hp _) |
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| theorem equidiscreteSignal_moments (k : ℕ) : |
| trace (tracialVector B N (Multiplicative (ZMod p))) (equidiscreteSignal p ^ k) = |
| (p : ℂ)⁻¹ * ∑ l : Fin p, (equidiscreteAtom p l : ℂ) ^ k := by |
| rw [equidiscreteSignal, trace_spectralSignal_pow] |
| simp only [ZMod.card] |
| congr 1 |
| exact (dualIndex p).sum_comp (fun l => (equidiscreteAtom p l : ℂ) ^ k) |
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| theorem trace_equidiscreteSignal (hp : 2 ≤ p) : |
| trace (tracialVector B N (Multiplicative (ZMod p))) (equidiscreteSignal p) = 0 := by |
| have h := equidiscreteSignal_moments (B := B) (N := N) p 1 |
| simpa [← Complex.ofReal_sum, sum_equidiscreteAtom hp] using h |
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| theorem characterProjection_mem_background {A : Type} [AddCommGroup A] [Fintype A] [DecidableEq A] |
| (J : Set N) (ψ : AddChar A ℂ) : |
| characterProjection (B := B) (N := N) ψ ∈ backgroundAlgebra J := by |
| apply (backgroundAlgebra J).smul_mem |
| apply (backgroundAlgebra J).sum_mem |
| intro a _ |
| apply (backgroundAlgebra J).smul_mem |
| exact Algebra.subset_adjoin (Or.inl (Or.inr ⟨Multiplicative.ofAdd a,rfl⟩)) |
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| theorem equidiscreteSignal_mem_background (J : Set N) : |
| equidiscreteSignal (B := B) (N := N) p ∈ backgroundAlgebra J := by |
| apply (backgroundAlgebra J).sum_mem |
| intro ψ _ |
| exact (backgroundAlgebra J).smul_mem (characterProjection_mem_background J ψ) _ |
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| theorem characterProjection_commutes_B {A : Type} [AddCommGroup A] [Fintype A] [DecidableEq A] |
| (ψ : AddChar A ℂ) (b : B) : |
| Commute (characterProjection (B := B) (N := N) ψ) (semicircularB b) := by |
| apply Commute.smul_left |
| apply Commute.sum_left |
| intro a _ |
| exact (unitaryG_commutes_semicircularB (Multiplicative.ofAdd a) b).smul_left _ |
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| theorem equidiscreteSignal_commutes_B (b : B) : |
| Commute (equidiscreteSignal (B := B) (N := N) p) (semicircularB b) := by |
| apply Commute.sum_left |
| intro ψ _ |
| exact (characterProjection_commutes_B ψ b).smul_left _ |
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| theorem tensor_characterProjection_moment {A : Type} [AddCommGroup A] [Fintype A] [DecidableEq A] |
| (w : List B) (ψ : AddChar A ℂ) : |
| trace (tracialVector B N (Multiplicative A)) |
| ((w.map semicircularB).prod * characterProjection ψ) = |
| trace (Fock.tracialVector B) ((w.map Fock.semicircular).prod) * (Fintype.card A : ℂ)⁻¹ := by |
| simp only [characterProjection, mul_smul_comm, Finset.mul_sum, trace_smul, trace_sum, |
| groupUnitary, tensor_word_moment] |
| have hi (a : A) : (Multiplicative.ofAdd a = (1 : Multiplicative A)) ↔ a = 0 := Iff.rfl |
| simp only [hi] |
| simp |
| ring |
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| theorem tensor_equidiscrete_moment (w : List B) (k : ℕ) : |
| trace (tracialVector B N (Multiplicative (ZMod p))) |
| ((w.map semicircularB).prod * equidiscreteSignal p ^ k) = |
| trace (Fock.tracialVector B) ((w.map Fock.semicircular).prod) * |
| ((p : ℂ)⁻¹ * ∑ l : Fin p, (equidiscreteAtom p l : ℂ) ^ k) := by |
| rw [equidiscreteSignal, spectralSignal_pow, Finset.mul_sum, trace_sum] |
| simp only [mul_smul_comm, trace_smul, tensor_characterProjection_moment, ZMod.card] |
| rw [← Finset.sum_mul] |
| have hs := (dualIndex p).sum_comp (fun l => (equidiscreteAtom p l : ℂ) ^ k) |
| change _ = _ at hs |
| rw [hs] |
| ring |
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| end FreeTensor |
| end FiniteEntropySeparation |
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| end |
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| end OAI |
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