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import OAI.MathematicalPhysics.PEPSMove.GridStates
namespace OAI
noncomputable section
open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator MatrixOrder
open Matrix
namespace PolynomialPEPS.PhysicalMove
open Matrix
@[simp] theorem joinConfigurations_left {L q : ℕ} (A : Finset (Vertex L))
(x : RegionConfiguration q A) (z : RegionConfiguration q Aᶜ)
(v : {v : Vertex L // v ∈ A}) : joinConfigurations A x z v = x v := by
simp [joinConfigurations, v.property]
@[simp] theorem joinConfigurations_right {L q : ℕ} (A : Finset (Vertex L))
(x : RegionConfiguration q A) (z : RegionConfiguration q Aᶜ)
(v : {v : Vertex L // v ∈ Aᶜ}) : joinConfigurations A x z v = z v := by
simp [joinConfigurations, Finset.mem_compl.mp v.property]
def joinEquiv {L q : ℕ} (A : Finset (Vertex L)) :
(RegionConfiguration q A × RegionConfiguration q Aᶜ) ≃ Configuration L q where
toFun x := joinConfigurations A x.1 x.2
invFun x := (fun v => x v.val, fun v => x v.val)
left_inv x := by ext v <;> simp
right_inv x := by
funext v
simp only [joinConfigurations]
split_ifs <;> rfl
@[simp] theorem joinEquiv_apply {L q : ℕ} (A : Finset (Vertex L))
(x : RegionConfiguration q A × RegionConfiguration q Aᶜ) :
joinEquiv A x = joinConfigurations A x.1 x.2 := rfl
@[simp] theorem joinEquiv_symm_join {L q : ℕ} (A : Finset (Vertex L))
(x : RegionConfiguration q A) (z : RegionConfiguration q Aᶜ) :
(joinEquiv A).symm (joinConfigurations A x z) = (x,z) :=
(joinEquiv A).symm_apply_apply (x,z)
@[simp] theorem liftLocal_join {L q : ℕ} (A : Finset (Vertex L))
(a : Matrix (RegionConfiguration q A) (RegionConfiguration q A) ℂ)
(x y : RegionConfiguration q A) (z w : RegionConfiguration q Aᶜ) :
liftLocal A a (joinConfigurations A x z) (joinConfigurations A y w) =
a x y * (if z = w then 1 else 0) := by
classical
have hout : (∀ v, v ∉ A → joinConfigurations A x z v = joinConfigurations A y w v)
↔ z = w := by
constructor
· intro h
funext v
simpa [joinConfigurations, Finset.mem_compl.mp v.property] using
h v (Finset.mem_compl.mp v.property)
· intro h v hv
simp [joinConfigurations, hv, h]
simp only [liftLocal, hout]
have hx : (fun v : {v : Vertex L // v ∈ A} => joinConfigurations A x z v) = x := by
funext v
exact joinConfigurations_left A x z v
have hy : (fun v : {v : Vertex L // v ∈ A} => joinConfigurations A y w v) = y := by
funext v
exact joinConfigurations_left A y w v
rw [hx, hy]
split_ifs <;> simp
theorem liftLocal_eq_reindex {L q : ℕ} (A : Finset (Vertex L))
(a : Matrix (RegionConfiguration q A) (RegionConfiguration q A) ℂ) :
liftLocal A a = Matrix.reindex (joinEquiv A) (joinEquiv A)
(Matrix.kronecker a (1 : Matrix (RegionConfiguration q Aᶜ)
(RegionConfiguration q Aᶜ) ℂ)) := by
classical
apply Matrix.ext
intro x y
obtain ⟨⟨u,z⟩,rfl⟩ := (joinEquiv (q := q) A).surjective x
obtain ⟨⟨v,w⟩,rfl⟩ := (joinEquiv (q := q) A).surjective y
simp only [Matrix.reindex_apply]
simp [Matrix.kronecker, Matrix.kroneckerMap, Matrix.one_apply]
@[simp] theorem liftLocal_one {L q : ℕ} (A : Finset (Vertex L)) :
liftLocal A (1 : Matrix (RegionConfiguration q A) (RegionConfiguration q A) ℂ) = 1 := by
classical
rw [liftLocal_eq_reindex]
simp only [Matrix.kronecker, Matrix.one_kronecker_one]
exact (Matrix.reindexAlgEquiv ℂ ℂ (joinEquiv A)).map_one
@[simp] theorem liftLocal_mul {L q : ℕ} (A : Finset (Vertex L))
(a b : Matrix (RegionConfiguration q A) (RegionConfiguration q A) ℂ) :
liftLocal A (a*b) = liftLocal A a * liftLocal A b := by
classical
simp only [liftLocal_eq_reindex]
have hk := Matrix.mul_kronecker_mul a b
(1 : Matrix (RegionConfiguration q Aᶜ) (RegionConfiguration q Aᶜ) ℂ) (1 : Matrix (RegionConfiguration q Aᶜ)
(RegionConfiguration q Aᶜ) ℂ)
simp only [Matrix.one_mul] at hk
simp only [Matrix.kronecker]
rw [hk]
exact (Matrix.reindexAlgEquiv ℂ ℂ (joinEquiv A)).map_mul _ _
@[simp] theorem liftLocal_star {L q : ℕ} (A : Finset (Vertex L))
(a : Matrix (RegionConfiguration q A) (RegionConfiguration q A) ℂ) :
liftLocal A (star a) = star (liftLocal A a) := by
classical
simp only [liftLocal_eq_reindex, Matrix.star_eq_conjTranspose]
simp only [Matrix.kronecker]
rw [Matrix.conjTranspose_reindex, Matrix.conjTranspose_kronecker]
simp
@[simp] theorem liftLocal_zero {L q : ℕ} (A : Finset (Vertex L)) :
liftLocal A (0 : Matrix (RegionConfiguration q A) (RegionConfiguration q A) ℂ) = 0 := by
classical
ext x y
simp only [liftLocal, Matrix.zero_apply]
split_ifs <;> rfl
@[simp] theorem liftLocal_add {L q : ℕ} (A : Finset (Vertex L))
(a b : Matrix (RegionConfiguration q A) (RegionConfiguration q A) ℂ) :
liftLocal A (a+b) = liftLocal A a + liftLocal A b := by
classical
ext x y
simp only [liftLocal, Matrix.add_apply]
split_ifs <;> simp
@[simp] theorem liftLocal_smul {L q : ℕ} (A : Finset (Vertex L)) (c : ℂ)
(a : Matrix (RegionConfiguration q A) (RegionConfiguration q A) ℂ) :
liftLocal A (c • a) = c • liftLocal A a := by
classical
ext x y
simp only [liftLocal, Matrix.smul_apply, smul_eq_mul]
split_ifs <;> simp
def localLiftHom {L q : ℕ} (A : Finset (Vertex L)) :
Matrix (RegionConfiguration q A) (RegionConfiguration q A) ℂ →⋆ₐ[ℂ]
Operator L q where
toFun := liftLocal A
map_one' := liftLocal_one A
map_mul' := liftLocal_mul A
map_zero' := by ext x y; simp [liftLocal]
map_add' := liftLocal_add A
commutes' c := by
rw [Algebra.algebraMap_eq_smul_one, liftLocal_smul, liftLocal_one]
exact (Algebra.algebraMap_eq_smul_one c).symm
map_star' := liftLocal_star A
open scoped Matrix.Norms.L2Operator in
theorem opNorm_liftLocal_le {L q : ℕ} (A : Finset (Vertex L))
(a : Matrix (RegionConfiguration q A) (RegionConfiguration q A) ℂ) :
opNorm (liftLocal A a) ≤ ‖a‖ := by
change ‖Matrix.toEuclideanCLM (n := Configuration L q) (𝕜 := ℂ) (liftLocal A a)‖ ≤ ‖a‖
rw [Matrix.l2_opNorm_toEuclideanCLM]
exact NonUnitalStarAlgHom.norm_apply_le (localLiftHom A) a
@[simp] theorem asMap_liftLocal_join {L q : ℕ} (A : Finset (Vertex L))
(a : Matrix (RegionConfiguration q A) (RegionConfiguration q A) ℂ)
(Ω : State L q) (x : RegionConfiguration q A) (z : RegionConfiguration q Aᶜ) :
asMap (liftLocal A a) Ω (joinConfigurations A x z) =
∑ y, a x y * Ω (joinConfigurations A y z) := by
classical
change (∑ y, liftLocal A a (joinConfigurations A x z) y * Ω y) = _
rw [← Equiv.sum_comp (joinEquiv A)]
simp only [Fintype.sum_prod_type, joinEquiv_apply, liftLocal_join]
simp
theorem reducedDensity_posSemidef {L q : ℕ} (Ω : State L q)
(A : Finset (Vertex L)) : (reducedDensity Ω A).PosSemidef :=
Matrix.posSemidef_self_mul_conjTranspose _
theorem inner_liftLocal_trace {L q : ℕ} (A : Finset (Vertex L))
(a : Matrix (RegionConfiguration q A) (RegionConfiguration q A) ℂ)
(Ω : State L q) :
inner ℂ Ω (asMap (liftLocal A a) Ω) = Matrix.trace (a * reducedDensity Ω A) := by
classical
rw [PiLp.inner_apply, ← Equiv.sum_comp (joinEquiv A)]
simp only [Fintype.sum_prod_type, joinEquiv_apply, asMap_liftLocal_join,
RCLike.inner_apply, Finset.sum_mul]
simp only [Matrix.trace, Matrix.diag, Matrix.mul_apply, reducedDensity,
Matrix.conjTranspose_apply, coefficientMatrix, Finset.mul_sum]
apply Finset.sum_congr rfl
intro x hx
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro y hy
apply Finset.sum_congr rfl
intro z hz
simp only [starRingEnd_apply]
ring
def restrictRegion {L q : ℕ} {A B : Finset (Vertex L)} (h : A ⊆ B)
(x : RegionConfiguration q B) : RegionConfiguration q A :=
fun v => x ⟨v.val, h v.property⟩
def liftBetween {L q : ℕ} {A B : Finset (Vertex L)} (h : A ⊆ B)
(a : Matrix (RegionConfiguration q A) (RegionConfiguration q A) ℂ) :
Matrix (RegionConfiguration q B) (RegionConfiguration q B) ℂ := by
classical
exact fun x y => if ∀ v : {v : Vertex L // v ∈ B}, v.val ∉ A → x v = y v
then a (restrictRegion h x) (restrictRegion h y) else 0
theorem liftLocal_liftBetween {L q : ℕ} {A B : Finset (Vertex L)} (h : A ⊆ B)
(a : Matrix (RegionConfiguration q A) (RegionConfiguration q A) ℂ) :
liftLocal B (liftBetween h a) = liftLocal A a := by
classical
ext x y
have heq : (∀ v, v ∉ A → x v = y v) ↔
(∀ v, v ∉ B → x v = y v) ∧
(∀ v : {v : Vertex L // v ∈ B}, v.val ∉ A → x v.val = y v.val) := by
constructor
· intro ha
exact ⟨fun v hv => ha v (fun hva => hv (h hva)), fun v hv => ha v hv⟩
· rintro ⟨hb, hi⟩ v hv
by_cases hvb : v ∈ B
· exact hi ⟨v,hvb⟩ hv
· exact hb v hvb
simp only [liftLocal, liftBetween, heq]
split_ifs <;> simp_all
rfl
theorem liftLocal_injective {L q : ℕ} (hq : 0 < q) (A : Finset (Vertex L)) :
Function.Injective (liftLocal (q := q) A) := by
classical
intro a b hab
ext x y
let z : RegionConfiguration q Aᶜ := fun _ => ⟨0,hq⟩
have h := congrArg (fun M => M (joinConfigurations A x z)
(joinConfigurations A y z)) hab
simpa only [liftLocal_join, ite_true, mul_one] using h
def liftBetweenHom {L q : ℕ} (hq : 0 < q) {A B : Finset (Vertex L)} (h : A ⊆ B) :
Matrix (RegionConfiguration q A) (RegionConfiguration q A) ℂ →⋆ₐ[ℂ]
Matrix (RegionConfiguration q B) (RegionConfiguration q B) ℂ where
toFun := liftBetween h
map_one' := by
apply liftLocal_injective hq B
simp [liftLocal_liftBetween]
map_mul' a b := by
apply liftLocal_injective hq B
simp [liftLocal_liftBetween]
map_zero' := by
apply liftLocal_injective hq B
simp [liftLocal_liftBetween]
map_add' a b := by
apply liftLocal_injective hq B
simp [liftLocal_liftBetween]
commutes' c := by
apply liftLocal_injective hq B
simp only [liftLocal_liftBetween, Algebra.algebraMap_eq_smul_one,
liftLocal_smul, liftLocal_one]
map_star' a := by
apply liftLocal_injective hq B
simp [liftLocal_liftBetween]
end PolynomialPEPS.PhysicalMove
end
end OAI
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