| import Mathlib |
| import OAI.Probability.SKGap.Localization.GraphPrediction |
|
|
| namespace OAI |
|
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| section |
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| noncomputable section |
| open scoped BigOperators |
| namespace SKGap.Noncrossing.Primary.Tensor |
| open Diagram |
| variable {ι : Type*} [Fintype ι] |
| abbrev Space := ι→List ι→ℝ |
|
|
| def tensorValue : List (ι→ℝ)→List ι→ℝ |
| | [],[] => 1 |
| | a::A,i::I => a i*tensorValue A I |
| | _,_ => 0 |
|
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| def realize : Primary.Space (ι:=ι)→ₗ[ℝ] Space (ι:=ι) := |
| Finsupp.linearCombination ℝ (fun s i I =>s.1 i*tensorValue s.2 I) |
| omit [Fintype ι] in |
| lemma realize_ket (s : Stack (ι:=ι)) (i : ι) (I : List ι) : |
| realize (ket s) i I=s.1 i*tensorValue s.2 I := by simp [realize,ket] |
|
|
| def diagonal (a : ι→ℝ) : Space (ι:=ι)→ₗ[ℝ] Space (ι:=ι) where |
| toFun v i I := a i*v i I |
| map_add' v w := by ext i I; simp [mul_add] |
| map_smul' c v := by ext i I; simp [mul_left_comm] |
| def creation : Space (ι:=ι)→ₗ[ℝ] Space (ι:=ι) where |
| toFun v i I := match I with | [] => 0 | q::Q => v q Q |
| map_add' v w := by ext i I; cases I <;> simp |
| map_smul' c v := by ext i I; cases I <;> simp |
| def annihilation : Space (ι:=ι)→ₗ[ℝ] Space (ι:=ι) where |
| toFun v i I := Diagram.mean (fun q =>v q (i::I)) |
| map_add' v w := by ext i I; simp [Diagram.mean,Finset.sum_add_distrib,add_div] |
| map_smul' c v := by ext i I; simp [Diagram.mean,←Finset.mul_sum,mul_div_assoc] |
| def noise (j : ℝ) : Space (ι:=ι)→ₗ[ℝ] Space (ι:=ι) := creation+j • annihilation |
| def project : Space (ι:=ι)→ₗ[ℝ] (ι→ℝ) where |
| toFun v i := v i [] |
| map_add' _ _ := rfl |
| map_smul' _ _ := rfl |
|
|
| def vacuum : Space (ι:=ι) := realize Primary.vacuum |
|
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| def letter (j : ℝ) : Letter (ι→ℝ)→Space (ι:=ι)→ₗ[ℝ] Space (ι:=ι) |
| | .diag a => diagonal a |
| | .noise => noise j |
| def word (j : ℝ) : List (Letter (ι→ℝ))→Space (ι:=ι)→ₗ[ℝ] Space (ι:=ι) |
| | [] => LinearMap.id |
| | a::F => (letter j a).comp (word j F) |
|
|
| omit [Fintype ι] in |
| lemma realize_diagonal (a : ι→ℝ) (v : Primary.Space (ι:=ι)) : |
| realize (Primary.diagonal a v)=diagonal a (realize v) := by |
| have h : realize.comp (Primary.diagonal a)=(diagonal a).comp realize := by |
| apply Finsupp.lhom_ext' |
| intro s |
| ext i I |
| change realize (Primary.diagonal a (ket s)) i I = diagonal a (realize (ket s)) i I |
| rw [Primary.diagonal_ket,realize_ket] |
| change (a i*s.1 i)*tensorValue s.2 I=a i*realize (ket s) i I |
| rw [realize_ket]; ring |
| exact LinearMap.congr_fun h v |
| omit [Fintype ι] in |
| lemma realize_creation (v : Primary.Space (ι:=ι)) : |
| realize (Primary.creation v)=creation (realize v) := by |
| have h : (realize (ι:=ι)).comp Primary.creation=creation.comp realize := by |
| apply Finsupp.lhom_ext' |
| intro s |
| ext i I |
| change realize (Primary.creation (ket s)) i I = creation (realize (ket s)) i I |
| rw [Primary.creation_ket,realize_ket] |
| cases I <;> simp [creation,tensorValue,realize_ket] |
| exact LinearMap.congr_fun h v |
| lemma realize_annihilation (v : Primary.Space (ι:=ι)) : |
| realize (Primary.annihilation v)=annihilation (realize v) := by |
| have h : (realize (ι:=ι)).comp Primary.annihilation=annihilation.comp realize := by |
| apply Finsupp.lhom_ext' |
| rintro ⟨a,L⟩ |
| ext i I |
| change realize (Primary.annihilation (ket (a,L))) i I = annihilation (realize (ket (a,L))) i I |
| cases L <;> simp [annihilation,realize_ket,tensorValue,Diagram.mean,←Finset.sum_mul,div_mul_eq_mul_div] |
| exact LinearMap.congr_fun h v |
| lemma realize_noise (j : ℝ) (v : Primary.Space (ι:=ι)) : |
| realize (Primary.noise j v)=noise j (realize v) := by |
| simp [Primary.noise,noise,realize_annihilation,realize_creation] |
| omit [Fintype ι] in |
| lemma project_realize (v : Primary.Space (ι:=ι)) : project (realize v)=Primary.project v := by |
| have h : (project (ι:=ι)).comp realize=Primary.project := by |
| apply Finsupp.lhom_ext' |
| rintro ⟨a,L⟩ |
| ext i |
| change project (realize (ket (a,L))) i = Primary.project (ket (a,L)) i |
| cases L <;> simp [project,realize_ket,tensorValue] |
| exact LinearMap.congr_fun h v |
| lemma realize_word (j : ℝ) (F : List (Letter (ι→ℝ))) (v : Primary.Space (ι:=ι)) : |
| realize (Primary.word j F v)=word j F (realize v) := by |
| induction F generalizing v with |
| | nil => rfl |
| | cons a F ih => |
| cases a <;> simp [Primary.word,Primary.letter,word,letter,realize_noise,realize_diagonal,ih] |
| omit [Fintype ι] in |
| lemma diagram_map_id (d : Diagram (ι→ℝ)) : d.map id=d := by |
| induction d <;> simp_all [Diagram.map] |
| lemma project_word (j : ℝ) (F : List (Letter (ι→ℝ))) (i : ι) : |
| project (word j F vacuum) i=WordSeries.ordinary j F i := by |
| rw [vacuum,←realize_word,project_realize,Primary.project_word] |
| simp [WordSeries.ordinary,Diagram.expect,Diagram.evaluate,Diagram.prediction,diagram_map_id] |
| lemma word_append (j : ℝ) (F G : List (Letter (ι→ℝ))) : |
| word j (F++G)=(word j F).comp (word j G) := by |
| induction F with |
| | nil => simp [word] |
| | cons a F ih => simp [word,ih,LinearMap.comp_assoc] |
| omit [Fintype ι] in |
| lemma diagonal_scale (c : ℝ) (a : ι→ℝ) : diagonal (fun i=>c*a i)=c • diagonal a := by |
| ext v i I |
| change (c*a i)*v i I=c*(a i*v i I) |
| ring |
| lemma annihilation_diagonal_creation (a : ι→ℝ) (v : Space (ι:=ι)) : |
| annihilation (diagonal a (creation v))=Diagram.mean a • v := by |
| ext i I |
| simp [annihilation,diagonal,creation,Diagram.mean,←Finset.sum_mul,mul_div_right_comm] |
| omit [Fintype ι] in |
| lemma project_diagonal_creation (a : ι→ℝ) (v : Space (ι:=ι)) : |
| project (diagonal a (creation v))=0 := by ext i; simp [project,diagonal,creation] |
| omit [Fintype ι] in |
| lemma project_creation (v : Space (ι:=ι)) : project (creation v)=0 := rfl |
| end SKGap.Noncrossing.Primary.Tensor |
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| end |
| end |
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| end OAI |
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