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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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 | import OAI.Analysis.FreeEntropy.TensorFreeness
namespace OAI
noncomputable section
open scoped ENNReal NNReal ComplexConjugate lp
open MeasureTheory Filter
namespace FiniteEntropySeparation
def equidiscreteAtom (p : ℕ) (l : Fin p) : ℝ := -1 + 2 * l.val / (p - 1 : ℝ)
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
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
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
namespace FreeTensor
variable {B N : Type} [DecidableEq B] [DecidableEq N] (p : ℕ) [NeZero p]
def dualIndex : AddChar (ZMod p) ℂ ≃ Fin p :=
Fintype.equivFinOfCardEq (by simp)
def equidiscreteSignal : Space B N (Multiplicative (ZMod p)) →L[ℂ] Space B N (Multiplicative (ZMod p)) :=
spectralSignal (fun ψ => equidiscreteAtom p (dualIndex p ψ))
theorem equidiscreteSignal_selfAdjoint : IsSelfAdjoint (equidiscreteSignal (B := B) (N := N) p) :=
spectralSignal_selfAdjoint _
theorem equidiscreteSignal_mem : equidiscreteSignal (B := B) (N := N) p ∈
algebra B N (Multiplicative (ZMod p)) := spectralSignal_mem _
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 _)
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)
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
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⟩))
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) _
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 _
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 _
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
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
end FreeTensor
end FiniteEntropySeparation
end
end OAI
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