File size: 4,628 Bytes
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 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 | import OAI.Probability.ClassicalON.ObservableBounds
namespace OAI
universe uE uI uV uΩ
noncomputable section
open MeasureTheory
open scoped BigOperators InnerProductSpace ComplexConjugate
namespace ClassicalON
variable {I : Type uI} {E : Type uE} [Fintype I] [Fintype E]
def familyKernel (u : I → E → ℂ) (e l : E) : ℂ := ∑ i, u i e * conj (u i l)
def rowSquare (u : I → E → ℂ) (e : E) : ℝ := ∑ l, ‖familyKernel u e l‖^2
def familyHSSquare (u : I → E → ℂ) : ℝ := ∑ e, rowSquare u e
omit [Fintype E] in
@[simp] theorem familyKernel_conj (u : I → E → ℂ) (e l : E) :
familyKernel (fun i e => conj (u i e)) e l = conj (familyKernel u e l) := by
simp [familyKernel]
theorem rowSquare_nonneg (u : I → E → ℂ) (e : E) : 0 ≤ rowSquare u e :=
Finset.sum_nonneg fun _ _ => sq_nonneg _
theorem familyHSSquare_nonneg (u : I → E → ℂ) : 0 ≤ familyHSSquare u :=
Finset.sum_nonneg fun e _ => rowSquare_nonneg u e
theorem kernel_diag_sq_le_row (u : I → E → ℂ) (e : E) :
‖familyKernel u e e‖^2 ≤ rowSquare u e :=
Finset.single_le_sum (fun l _ => sq_nonneg ‖familyKernel u e l‖) (Finset.mem_univ e)
theorem kernel_diag_norm_le_sqrt (u : I → E → ℂ) (e : E) :
‖familyKernel u e e‖ ≤ Real.sqrt (rowSquare u e) :=
Real.le_sqrt_of_sq_le (kernel_diag_sq_le_row u e)
namespace SpinSystem
variable {n : ℕ} {V : Type uV} [Fintype V]
omit [Fintype V] [Fintype I] in
def coefficientObservable (S : SpinSystem n V E) (M : SpinOperator n) (e : E) :
C((V → Spin n), ℂ) := ⟨fun σ => (S.localCoefficient M σ e : ℂ), by
unfold localCoefficient
fun_prop⟩
omit [Fintype V] [Fintype I] in
def currentObservable (S : SpinSystem n V E) (M : SpinOperator n) (u : E → ℂ) :
C((V → Spin n), ℂ) := ⟨_, S.continuous_complexEnergy M u⟩
omit [Fintype V] [Fintype I] [Fintype E] in
def conjugateObservable {Ω : Type uΩ} [TopologicalSpace Ω] (f : C(Ω, ℂ)) : C(Ω, ℂ) :=
⟨fun σ => conj (f σ), Complex.continuous_conj.comp f.continuous⟩
omit [Fintype V] [Fintype I] [Fintype E] in
@[simp] theorem conjugateObservable_apply {Ω : Type uΩ} [TopologicalSpace Ω]
(f : C(Ω, ℂ)) (σ : Ω) : conjugateObservable f σ = conj (f σ) := rfl
omit [Fintype V] [Fintype I] [Fintype E] in
@[simp] theorem normObservable_conjugate {Ω : Type uΩ} [TopologicalSpace Ω]
(f : C(Ω, ℂ)) : normObservable (conjugateObservable f) = normObservable f := by
ext σ
exact Complex.norm_conj (f σ)
omit [Fintype V] [Fintype I] in
@[simp] theorem currentObservable_conj (S : SpinSystem n V E) (M : SpinOperator n)
(u : E → ℂ) : S.currentObservable M (fun e => conj (u e)) =
conjugateObservable (S.currentObservable M u) := by
ext σ
exact S.complexEnergy_conj M u σ
omit [Fintype V] [Fintype I] [Fintype E] in
@[simp] theorem coefficientObservable_conj (S : SpinSystem n V E) (M : SpinOperator n)
(e : E) : conjugateObservable (S.coefficientObservable M e) =
S.coefficientObservable M e := by
ext σ
exact Complex.conj_ofReal _
omit [Fintype V] [Fintype E] in
theorem familyKernel_differential (S : SpinSystem n V E) (f : I → V → ℂ) (e : E) :
familyKernel (fun i => S.differential (f i)) e =
S.differential (fun x => ∑ i, S.differential (f i) e * conj (f i x)) := by
funext l
simp only [familyKernel, differential, map_sub, mul_sub, Finset.sum_sub_distrib]
theorem row_current_moment [NeZero n] (S : SpinSystem n V E) (β : ℝ)
(hb : ∀ e, 0 ≤ S.strength e ∧ S.strength e ≤ β)
(M : SpinOperator n) (hM : M ∈ skewAdjoint (SpinOperator n)) (hMn : ‖M‖ ≤ 1)
(f : I → V → ℂ) (hP : ∀ i x s, S.pin x = some s → f i x = 0) (e : E) :
S.averageLinear (normObservable (S.currentObservable M
(familyKernel (fun i => S.differential (f i)) e)) *
normObservable (S.currentObservable M
(familyKernel (fun i => S.differential (f i)) e))) ≤
β * rowSquare (fun i => S.differential (f i)) e := by
have hp : ∀ x s, S.pin x = some s →
(∑ i, S.differential (f i) e * conj (f i x)) = 0 := by
intro x s hs
simp [hP _ x s hs]
have h := S.complex_gradient_current_bound β hb M hM hMn
(fun x => ∑ i, S.differential (f i) e * conj (f i x)) hp
rw [← S.familyKernel_differential f e] at h
change S.average (fun σ =>
‖S.complexEnergy M (familyKernel (fun i => S.differential (f i)) e) σ‖ *
‖S.complexEnergy M (familyKernel (fun i => S.differential (f i)) e) σ‖) ≤ _
simpa only [rowSquare, pow_two] using h
end SpinSystem
end ClassicalON
end
end OAI
|