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f61e547 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 | import OAI.Probability.EntangledGames.EffectExposure
namespace OAI
universe u_X u_Y u_I u_m u_n u_J u_K
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]
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
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
end
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]
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]
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
end
end OAI
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