| import OAI.Probability.ClassicalON.ObservableBounds |
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| namespace OAI |
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| universe uE uI uV uΩ |
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| noncomputable section |
| open MeasureTheory |
| open scoped BigOperators InnerProductSpace ComplexConjugate |
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| namespace ClassicalON |
| variable {I : Type uI} {E : Type uE} [Fintype I] [Fintype E] |
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| def familyKernel (u : I → E → ℂ) (e l : E) : ℂ := ∑ i, u i e * conj (u i l) |
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| def rowSquare (u : I → E → ℂ) (e : E) : ℝ := ∑ l, ‖familyKernel u e l‖^2 |
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| def familyHSSquare (u : I → E → ℂ) : ℝ := ∑ e, rowSquare u e |
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| 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] |
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| theorem rowSquare_nonneg (u : I → E → ℂ) (e : E) : 0 ≤ rowSquare u e := |
| Finset.sum_nonneg fun _ _ => sq_nonneg _ |
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| theorem familyHSSquare_nonneg (u : I → E → ℂ) : 0 ≤ familyHSSquare u := |
| Finset.sum_nonneg fun e _ => rowSquare_nonneg u e |
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| 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) |
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| 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) |
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| namespace SpinSystem |
| variable {n : ℕ} {V : Type uV} [Fintype V] |
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| 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⟩ |
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| 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⟩ |
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| 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⟩ |
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| 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 |
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| 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) |
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| 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 |
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| 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 _ |
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| 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] |
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| 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 |
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| end SpinSystem |
| end ClassicalON |
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| end |
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| end OAI |
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