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9b6e014 | 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 | import Mathlib
import OAI.Probability.BinarySweep.Representations.IsotypicMoment
import OAI.Probability.BinarySweep.Conditional.Placement
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps
open Irrep Representation
variable {b h : ℕ} {bits : Fin b → ℕ} (H : PathFamily bits h) (z : ℝ)
def conditionalGroupLaw (a : Equiv.Perm (FreeSlot H 0)) : ℝ :=
∑ g : ConditionalChoices H, if remainingPerm H g.val g.property=a then
conditionalChoiceWeight H z g else 0
lemma conditionalGroupLaw_nonneg {z : ℝ} (hz : 0 ≤ z) (hz1 : z<1)
(a : Equiv.Perm (FreeSlot H 0)) : 0 ≤ conditionalGroupLaw H z a := by
apply Finset.sum_nonneg
intro g _
split_ifs
· exact conditionalChoiceWeight_nonneg H hz hz1 g
· rfl
lemma conditionalGroupLaw_sum {z : ℝ} (hz : 0 ≤ z) (hz1 : z<1) :
∑ a, conditionalGroupLaw H z a = 1 := by
simp only [conditionalGroupLaw]
rw [Finset.sum_comm]
simpa using conditionalChoiceWeight_sum H hz hz1
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
omit [FiniteDimensional ℂ V] in
lemma groupAverage_conditional (ρ : Representation ℂ (Equiv.Perm (FreeSlot H 0)) V) :
groupAverage ρ (fun a => (conditionalGroupLaw H z a : ℂ)) =
∑ g : ConditionalChoices H, (conditionalChoiceWeight H z g : ℂ) •
ρ (remainingPerm H g.val g.property) := by
simp only [groupAverage,conditionalGroupLaw,Complex.ofReal_sum,Finset.sum_smul]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro g _
simp only [apply_ite,Complex.ofReal_zero,ite_smul,zero_smul]
simp
omit [FiniteDimensional ℂ V] in
lemma groupAverage_conditional_grid (ρ : Representation ℂ (Equiv.Perm (FreeSlot H 0)) V) :
groupAverage ρ (fun a => (conditionalGroupLaw H z a : ℂ)) =
(conditionalNormalizer H z : ℂ)⁻¹ • ∑ g : GridChoices bits,
if hg : pathEvent H g then (gridWeight bits z g : ℂ) •
ρ (remainingPerm H g hg) else 0 := by
rw [groupAverage_conditional]
have hs : ∀ g : ConditionalChoices H,
(conditionalChoiceWeight H z g : ℂ) • ρ (remainingPerm H g.val g.property) =
(conditionalNormalizer H z : ℂ)⁻¹ •
((gridWeight bits z g.val : ℂ) • ρ (remainingPerm H g.val g.property)) := by
intro g
rw [conditionalChoiceWeight, Complex.ofReal_div, div_eq_mul_inv, mul_comm, mul_smul]
simp_rw [hs]
rw [← Finset.smul_sum]
congr 1
let F : GridChoices bits → (V →ₗ[ℂ] V) := fun g =>
if hg : pathEvent H g then (gridWeight bits z g : ℂ) • ρ (remainingPerm H g hg) else 0
have hF : ∑ g : ConditionalChoices H, F g.val = ∑ g, F g := by
rw [← Finset.sum_filter_of_ne (p := pathEvent H) (f := F) (by
intro g _ hg
by_contra hn
simp [F,hn] at hg)]
exact (Finset.sum_subtype _ (by simp) _).symm
simpa only [F,dite_eq_left (Subtype.property _)] using hF
lemma conditionalOperator_groupAverage {D : ℕ}
(ρ : Representation ℂ (Equiv.Perm (FreeSlot H 0)) (RepSpace D)) :
(conditionalOperator H z ρ).toLinearMap =
groupAverage ρ (fun a => (conditionalGroupLaw H z a : ℂ)) := by
rw [groupAverage_conditional_grid]
rfl
lemma conditionalMoment_evenMoment {D : ℕ}
(ρ : Representation ℂ (Equiv.Perm (FreeSlot H 0)) (RepSpace D)) (q : ℕ) :
traceMoment q (conditionalOperator H z ρ) =
evenMoment q (groupAverage ρ (fun a => (conditionalGroupLaw H z a : ℂ))) := by
rw [← conditionalOperator_groupAverage]
simp only [traceMoment,evenMoment,ContinuousLinearMap.toLinearMap_pow,
ContinuousLinearMap.toLinearMap_mul,ContinuousLinearMap.adjoint_toLinearMap]
end BinaryCoordinateSweeps
end
end OAI
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