| import OAI.Probability.EntangledGames.EffectExposure |
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| namespace OAI |
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| universe u_X u_Y u_I u_m u_n u_J u_K |
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| noncomputable section |
| open scoped BigOperators MatrixOrder ComplexOrder |
| open Matrix |
| namespace ThresholdParallelRepetition.MixedExposure |
| open QuantumSampling Resolvent OperatorEntropy FiniteProbability Law |
| variable {X : Type u_X} {Y : Type u_Y} {I : Type u_I} {m : Type u_m} {n : Type u_n} {J : Type u_J} {K : Type u_K} [Fintype X] [Fintype Y] [Fintype I] |
| [DecidableEq X] [DecidableEq Y] [DecidableEq I] |
| [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n] |
| [Fintype J] [DecidableEq J] [Nonempty J] [Fintype K] [DecidableEq K] [Nonempty K] |
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| lemma step_left (μ : Law (X×Y)) (x₀ : X) (y₀ : Y) (i : I) |
| (C : Matrix m n ℂ) (A : J → Matrix m m ℂ) (B : K → Matrix n n ℂ) |
| (hA : ∀ j, (A j).PosSemidef) (hA1 : ∀ j, A j ≤ 1) (hB : ∀ j, (B j).PosSemidef) |
| (F : Profile I X Y → Matrix m m ℂ) (H : Profile I X Y → Matrix n n ℂ) |
| (hF : ∀ z, F z ∈ Set.range A) (hF0 : ∀ z, μ.left x₀ i F z ∈ Set.range A) |
| (hH : ∀ z, H z ∈ Set.range B) (hFi : IgnoresRight F i) (hHi : IgnoresLeft H i) |
| (θ : Profile I X Y → ℝ) (hθ : ∀ z, 0 ≤ θ z) |
| (hθL : IgnoresLeft θ i) (hθR : IgnoresRight θ i) : |
| (pi (fun _ : I => μ)).avg (fun z => θ z * hsSq |
| ((commonMap A (F z) - commonMap A (μ.left x₀ i F z))*C*(commonMap B (H z))ᵀ)) ≤ |
| (pi (fun _ : I => μ)).avg (fun z => θ z * prob C (effectEntropy (F z)) (μ.right y₀ i H z)) - |
| (pi (fun _ : I => μ)).avg (fun z => θ z * prob C (effectEntropy (μ.left x₀ i F z)) (H z)) := by |
| let ρ := pi (fun _ : I => μ) |
| let T := μ.left x₀ i F |
| let e : Profile I X Y → ℝ := fun z => θ z * hsSq |
| ((commonMap A (F z)-commonMap A (T z))*C*(commonMap B (H z))ᵀ) |
| have hTi : IgnoresLeft T i := left_ignoresLeft μ x₀ i F |
| have hstep (z : Profile I X Y) : μ.left x₀ i e z ≤ |
| θ z * (prob C (μ.left x₀ i (fun z => effectEntropy (F z)) z) (H z) - |
| prob C (effectEntropy (T z)) (H z)) := by |
| have h := commonMap_error_left A hA hA1 (μ.givenSecond x₀ (z i).2) |
| (fun x => F (updateLeft z i x)) (fun x => hF _) (hF0 z) C (commonMap B (H z)) |
| rw [commonMap_gram B hB (hH z)] at h |
| have hl : μ.left x₀ i e z = θ z * |
| (μ.givenSecond x₀ (z i).2).avg (fun x => hsSq |
| ((commonMap A (F (updateLeft z i x))-commonMap A (T z))*C*(commonMap B (H z))ᵀ)) := by |
| simp only [left, e, hθL z, hTi z, hHi z, Law.avg, smul_eq_mul] |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro x _ |
| ring |
| rw [hl] |
| apply mul_le_mul_of_nonneg_left _ (hθ z) |
| have heq := (μ.givenSecond x₀ (z i).2).avg_linear |
| (fun x => effectEntropy (F (updateLeft z i x))) (probLeft C (H z)) |
| change prob C (μ.left x₀ i (fun z => effectEntropy (F z)) z) (H z) = _ at heq |
| rw [heq] |
| exact h |
| have havg := ρ.avg_mono hstep |
| have hPairL : ρ.avg (fun z => θ z * prob C (μ.left x₀ i (fun z => effectEntropy (F z)) z) (H z)) = |
| ρ.avg (fun z => θ z * prob C (effectEntropy (F z)) (H z)) := by |
| apply pi_left_pairing μ x₀ i (fun z => effectEntropy (F z)) |
| (fun z => θ z • probLeft C (H z)) |
| intro z x |
| simp only [hθL z, hHi z] |
| have hPairR : ρ.avg (fun z => θ z * prob C (effectEntropy (F z)) (μ.right y₀ i H z)) = |
| ρ.avg (fun z => θ z * prob C (effectEntropy (F z)) (H z)) := by |
| apply pi_right_pairing μ y₀ i H (fun z => θ z • probRight C (effectEntropy (F z))) |
| intro z y |
| simp only [hθR z, hFi z] |
| have he : ρ.avg (μ.left x₀ i e) = ρ.avg e := pi_avg_left μ x₀ i e |
| rw [he] at havg |
| have hr : ρ.avg (fun z => θ z * (prob C (μ.left x₀ i (fun z => effectEntropy (F z)) z) (H z) - |
| prob C (effectEntropy (T z)) (H z))) = |
| ρ.avg (fun z => θ z*prob C (effectEntropy (F z)) (μ.right y₀ i H z)) - |
| ρ.avg (fun z => θ z*prob C (effectEntropy (T z)) (H z)) := by |
| simp only [mul_sub, Law.avg_sub] |
| rw [hPairL, hPairR] |
| exact havg.trans_eq hr |
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| lemma step_right (μ : Law (X×Y)) (x₀ : X) (y₀ : Y) (i : I) |
| (C : Matrix m n ℂ) (A : J → Matrix m m ℂ) (B : K → Matrix n n ℂ) |
| (hA : ∀ j, (A j).PosSemidef) (hB : ∀ j, (B j).PosSemidef) (hB1 : ∀ j, B j ≤ 1) |
| (F : Profile I X Y → Matrix m m ℂ) (H : Profile I X Y → Matrix n n ℂ) |
| (hF : ∀ z, F z ∈ Set.range A) (hH : ∀ z, H z ∈ Set.range B) |
| (hH1 : ∀ z, μ.right y₀ i H z ∈ Set.range B) |
| (hFi : IgnoresRight F i) (hHi : IgnoresLeft H i) |
| (θ : Profile I X Y → ℝ) (hθ : ∀ z, 0 ≤ θ z) |
| (hθL : IgnoresLeft θ i) (hθR : IgnoresRight θ i) : |
| (pi (fun _ : I => μ)).avg (fun z => θ z * hsSq |
| (commonMap A (F z)*C*(commonMap B (H z) - commonMap B (μ.right y₀ i H z))ᵀ)) ≤ |
| (pi (fun _ : I => μ)).avg (fun z => θ z * prob C (μ.left x₀ i F z) (effectEntropy (H z))) - |
| (pi (fun _ : I => μ)).avg (fun z => θ z * prob C (F z) (effectEntropy (μ.right y₀ i H z))) := by |
| let ρ := pi (fun _ : I => μ) |
| let T := μ.right y₀ i H |
| let e : Profile I X Y → ℝ := fun z => θ z * hsSq |
| (commonMap A (F z)*C*(commonMap B (H z)-commonMap B (T z))ᵀ) |
| have hTi : IgnoresRight T i := right_ignoresRight μ y₀ i H |
| have hstep (z : Profile I X Y) : μ.right y₀ i e z ≤ |
| θ z * (prob C (F z) (μ.right y₀ i (fun z => effectEntropy (H z)) z) - |
| prob C (F z) (effectEntropy (T z))) := by |
| have h := commonMap_error_right B hB hB1 (μ.givenFirst y₀ (z i).1) |
| (fun y => H (updateRight z i y)) (fun y => hH _) (hH1 z) C (commonMap A (F z)) |
| rw [commonMap_gram A hA (hF z)] at h |
| have hl : μ.right y₀ i e z = θ z * |
| (μ.givenFirst y₀ (z i).1).avg (fun y => hsSq |
| (commonMap A (F z)*C*(commonMap B (H (updateRight z i y))-commonMap B (T z))ᵀ)) := by |
| simp only [right, e, hθR z, hTi z, hFi z, Law.avg, smul_eq_mul] |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro x _ |
| ring |
| rw [hl] |
| apply mul_le_mul_of_nonneg_left _ (hθ z) |
| have heq := (μ.givenFirst y₀ (z i).1).avg_linear |
| (fun y => effectEntropy (H (updateRight z i y))) (probRight C (F z)) |
| change prob C (F z) (μ.right y₀ i (fun z => effectEntropy (H z)) z) = _ at heq |
| rw [heq] |
| exact h |
| have havg := ρ.avg_mono hstep |
| have hPairR : ρ.avg (fun z => θ z * prob C (F z) (μ.right y₀ i (fun z => effectEntropy (H z)) z)) = |
| ρ.avg (fun z => θ z * prob C (F z) (effectEntropy (H z))) := by |
| apply pi_right_pairing μ y₀ i (fun z => effectEntropy (H z)) |
| (fun z => θ z • probRight C (F z)) |
| intro z y |
| simp only [hθR z, hFi z] |
| have hPairL : ρ.avg (fun z => θ z * prob C (μ.left x₀ i F z) (effectEntropy (H z))) = |
| ρ.avg (fun z => θ z * prob C (F z) (effectEntropy (H z))) := by |
| apply pi_left_pairing μ x₀ i F (fun z => θ z • probLeft C (effectEntropy (H z))) |
| intro z x |
| simp only [hθL z, hHi z] |
| have he : ρ.avg (μ.right y₀ i e) = ρ.avg e := pi_avg_right μ y₀ i e |
| rw [he] at havg |
| have hr : ρ.avg (fun z => θ z * (prob C (F z) (μ.right y₀ i (fun z => effectEntropy (H z)) z) - |
| prob C (F z) (effectEntropy (T z)))) = |
| ρ.avg (fun z => θ z*prob C (μ.left x₀ i F z) (effectEntropy (H z))) - |
| ρ.avg (fun z => θ z*prob C (F z) (effectEntropy (T z))) := by |
| simp only [mul_sub, Law.avg_sub] |
| rw [hPairR, hPairL] |
| exact havg.trans_eq hr |
| end ThresholdParallelRepetition.MixedExposure |
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| end |
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|
| noncomputable section |
| open scoped BigOperators MatrixOrder ComplexOrder |
| open Matrix |
| namespace ThresholdParallelRepetition.MixedExposure |
| open QuantumSampling Resolvent OperatorEntropy FiniteProbability Law |
| variable {X : Type u_X} {Y : Type u_Y} {I : Type u_I} {m : Type u_m} {n : Type u_n} [Fintype X] [Fintype Y] [Fintype I] |
| [DecidableEq X] [DecidableEq Y] [DecidableEq I] |
| [Fintype m] [Fintype n] |
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| lemma mixed_pairing (μ : Law (X×Y)) (x₀ : X) (y₀ : Y) (l r : List I) |
| (hlr : ∀ i ∈ l, i ∉ r) (C : Matrix m n ℂ) |
| (F : Profile I X Y → Matrix m m ℂ) (H : Profile I X Y → Matrix n n ℂ) |
| (hF : ∀ i ∈ r, IgnoresRight F i) (hH : ∀ i ∈ l, IgnoresLeft H i) |
| (θ : Profile I X Y → ℝ) (hθL : ∀ i ∈ l, IgnoresLeft θ i) (hθR : ∀ i ∈ r, IgnoresRight θ i) : |
| (pi (fun _ : I => μ)).avg (fun z => θ z * prob C (μ.lefts x₀ l F z) (μ.rights y₀ r H z)) = |
| (pi (fun _ : I => μ)).avg (fun z => θ z * prob C (F z) (H z)) := by |
| calc |
| _ = (pi (fun _ : I => μ)).avg (fun z => θ z * prob C (F z) (μ.rights y₀ r H z)) := by |
| apply pi_lefts_pairing μ x₀ l F (fun z => θ z • probLeft C (μ.rights y₀ r H z)) |
| intro i hi z x |
| simp only [hθL i hi z, rights_ignoresLeft_notMem μ y₀ r (hlr i hi) (hH i hi) z] |
| _ = _ := by |
| apply pi_rights_pairing μ y₀ r H (fun z => θ z • probRight C (F z)) |
| intro i hi z y |
| simp only [hθR i hi z, hF i hi z] |
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| lemma fixed_mixed_pairing (μ : Law (X×Y)) (x₀ : X) (y₀ : Y) (l r : List I) |
| (hlr : ∀ i ∈ l, i ∉ r) (C : Matrix m n ℂ) |
| (F : Profile I X Y → Matrix m m ℂ) (H : Profile I X Y → Matrix n n ℂ) |
| (hF : ∀ i ∈ r, IgnoresRight F i) (hH : ∀ i ∈ l, IgnoresLeft H i) |
| (θ : Profile I X Y → ℝ) (hθL : ∀ i ∈ l, IgnoresLeft θ i) (hθR : ∀ i ∈ r, IgnoresRight θ i) |
| (i : I) (hiL : i ∉ l) (hiR : i ∉ r) (q : X×Y) : |
| (pi (fun _ : I => μ)).avg (fun z => θ (Function.update z i q) * prob C |
| (μ.lefts x₀ l F (Function.update z i q)) (μ.rights y₀ r H (Function.update z i q))) = |
| (pi (fun _ : I => μ)).avg (fun z => θ (Function.update z i q) * |
| prob C (F (Function.update z i q)) (H (Function.update z i q))) := by |
| calc |
| _ = (pi (fun _ : I => μ)).avg (fun z => θ (Function.update z i q) * |
| prob C (F (Function.update z i q)) (μ.rights y₀ r H (Function.update z i q))) := by |
| apply pi_fixed_lefts_pairing μ x₀ l hiL q F (fun z => θ z • probLeft C (μ.rights y₀ r H z)) |
| intro j hj z x |
| simp only [hθL j hj z, rights_ignoresLeft_notMem μ y₀ r (hlr j hj) (hH j hj) z] |
| _ = _ := by |
| apply pi_fixed_rights_pairing μ y₀ r hiR q H (fun z => θ z • probRight C (F z)) |
| intro j hj z y |
| simp only [hθR j hj z, hF j hj z] |
| end ThresholdParallelRepetition.MixedExposure |
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| end |
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| end OAI |
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