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feat: Contractions with pauli matrices

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jstoobysmith 2024-10-23 15:19:41 +00:00
parent ca55da6a34
commit 685c1b293c
4 changed files with 940 additions and 81 deletions
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60
HepLean/Tensors/Tree/NodeIdentities/Basic.lean
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@ -117,12 +117,26 @@ lemma perm_eq_of_eq_perm {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
/-!
## Additive identities
## Vector based identities
These identities are related to the fact that all the maps are linear.
-/
lemma smul_smul (t : TensorTree S c) (a b : S.k) :
(smul a (smul b t)).tensor = (smul (a * b) t).tensor := by
simp [smul_tensor]
exact _root_.smul_smul a b t.tensor
lemma smul_one (t : TensorTree S c) :
(smul 1 t).tensor = t.tensor := by
simp [smul_tensor]
lemma smul_eq_one (t : TensorTree S c) (a : S.k) (h : a = 1) :
(smul a t).tensor = t.tensor := by
rw [h]
exact smul_one _
/-- The addition node is commutative. -/
lemma add_comm (t1 t2 : TensorTree S c) : (add t1 t2).tensor = (add t2 t1).tensor := by
simp only [add_tensor]
@ -147,4 +161,48 @@ lemma add_eval {n : ℕ} {c : Fin n.succ → S.C} (i : Fin n.succ) (e : ℕ) (t
(add (eval i e t) (eval i e t1)).tensor = (eval i e (add t t1)).tensor := by
simp only [add_tensor, eval_tensor, Nat.succ_eq_add_one, map_add]
lemma contr_add {n : ℕ} {c : Fin n.succ.succ → S.C} {i : Fin n.succ.succ} {j : Fin n.succ}
{h : c (i.succAbove j) = S.τ (c i)} (t1 t2 : TensorTree S c) :
(contr i j h (add t1 t2)).tensor = (add (contr i j h t1) (contr i j h t2)).tensor := by
simp [contr_tensor, add_tensor]
lemma contr_smul {n : ℕ} {c : Fin n.succ.succ → S.C} {i : Fin n.succ.succ} {j : Fin n.succ}
{h : c (i.succAbove j) = S.τ (c i)} (a : S.k) (t : TensorTree S c) :
(contr i j h (smul a t)).tensor = (smul a (contr i j h t)).tensor := by
simp [contr_tensor, smul_tensor]
@[simp]
lemma add_prod {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
(t1 t2 : TensorTree S c) (t3 : TensorTree S c1) :
(prod (add t1 t2) t3).tensor = (add (prod t1 t3) (prod t2 t3)).tensor := by
simp only [prod_tensor, Functor.id_obj, mk_hom, Action.instMonoidalCategory_tensorObj_V,
Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
Action.FunctorCategoryEquivalence.functor_obj_obj, add_tensor, add_tmul, map_add]
@[simp]
lemma prod_add {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
(t1 : TensorTree S c) (t2 t3 : TensorTree S c1) :
(prod t1 (add t2 t3)).tensor = (add (prod t1 t2) (prod t1 t3)).tensor := by
simp only [prod_tensor, Functor.id_obj, mk_hom, Action.instMonoidalCategory_tensorObj_V,
Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
Action.FunctorCategoryEquivalence.functor_obj_obj, add_tensor, tmul_add, map_add]
lemma smul_prod {n m: ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
(a : S.k) (t1 : TensorTree S c) (t2 : TensorTree S c1) :
((prod (smul a t1) t2)).tensor = (smul a (prod t1 t2)).tensor := by
simp [prod_tensor, smul_tensor, tmul_smul, smul_tmul, map_smul]
lemma prod_smul {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
(a : S.k) (t1 : TensorTree S c) (t2 : TensorTree S c1) :
(prod t1 (smul a t2)).tensor = (smul a (prod t1 t2)).tensor := by
simp [prod_tensor, smul_tensor, tmul_smul, smul_tmul, map_smul]
lemma prod_add_both {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
(t1 t2 : TensorTree S c) (t3 t4 : TensorTree S c1) :
(prod (add t1 t2) (add t3 t4)).tensor = (add (add (prod t1 t3) (prod t1 t4))
(add (prod t2 t3) (prod t2 t4))).tensor := by
rw [add_prod]
rw [add_tensor_eq_fst (prod_add _ _ _)]
rw [add_tensor_eq_snd (prod_add _ _ _)]
end TensorTree