| 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 | |