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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 | import OAI.Probability.EntangledGames.Model
namespace OAI
noncomputable section
open scoped BigOperators ComplexOrder
namespace ThresholdParallelRepetition
open scoped Kronecker
open Matrix
namespace Strategy
variable {X Y A B : Type} [Fintype A] [Fintype B]
lemma born_eq_quadratic (S : Strategy X Y A B) (x : X) (y : Y) (a : A) (b : B) :
S.born x y a b =
(star S.state ⬝ᵥ ((S.alice x a ⊗ₖ S.bob y b) *ᵥ S.state)).re := by
simp only [born, dotProduct, Matrix.mulVec, Matrix.kroneckerMap_apply, Pi.star_apply,
Finset.mul_sum, ← mul_assoc, Complex.re_sum]
lemma born_nonneg (S : Strategy X Y A B) (x : X) (y : Y) (a : A) (b : B) :
0 ≤ S.born x y a b := by
rw [born_eq_quadratic]
exact (Complex.nonneg_iff.mp
(((S.alice_pos x a).kronecker (S.bob_pos y b)).dotProduct_mulVec_nonneg S.state)).1
lemma total_effect (S : Strategy X Y A B) (x : X) (y : Y) :
∑ a, ∑ b, S.alice x a ⊗ₖ S.bob y b = 1 := by
classical
calc
_ = (∑ a, S.alice x a) ⊗ₖ (∑ b, S.bob y b) := by
ext i j
simp only [Matrix.sum_apply, Matrix.kroneckerMap_apply,
Finset.sum_mul, Finset.mul_sum]
rw [Finset.sum_comm]
_ = 1 := by rw [S.alice_total, S.bob_total, Matrix.one_kronecker_one]
lemma born_sum (S : Strategy X Y A B) (x : X) (y : Y) :
∑ a, ∑ b, S.born x y a b = 1 := by
simp_rw [born_eq_quadratic]
simp_rw [← Complex.re_sum, ← dotProduct_sum, ← Matrix.sum_mulVec]
rw [total_effect, Matrix.one_mulVec]
rw [dotProduct, Complex.re_sum]
convert S.state_unit using 1
apply Finset.sum_congr rfl
intro i _
simp [Complex.normSq, Complex.mul_re]
lemma born_le_one (S : Strategy X Y A B) (x : X) (y : Y) (a : A) (b : B) :
S.born x y a b ≤ 1 := by
classical
rw [← born_sum S x y]
exact (Finset.single_le_sum (fun b' _ => born_nonneg S x y a b') (Finset.mem_univ b)).trans
(Finset.single_le_sum (fun a' _ => Finset.sum_nonneg
(fun b' _ => born_nonneg S x y a' b')) (Finset.mem_univ a))
end Strategy
namespace Strategy
def deterministic {X Y A B : Type} [Fintype A] [Fintype B]
[DecidableEq A] [DecidableEq B] (a₀ : A) (b₀ : B) : Strategy X Y A B where
dimA := 0
dimB := 0
state := fun _ => 1
state_unit := by simp
alice := fun _ a => if a = a₀ then 1 else 0
bob := fun _ b => if b = b₀ then 1 else 0
alice_pos := by
intro _ a
split_ifs
· exact Matrix.PosSemidef.one
· exact Matrix.PosSemidef.zero
bob_pos := by
intro _ b
split_ifs
· exact Matrix.PosSemidef.one
· exact Matrix.PosSemidef.zero
alice_total := by intro _; simp
bob_total := by intro _; simp
end Strategy
lemma successProbability_nonneg {x y a b : ℕ} (G : Game x y a b)
(S : SingleStrategy x y a b) : 0 ≤ successProbability G S := by
apply Finset.sum_nonneg
intro z _
apply mul_nonneg (G.questionProb_nonneg z)
apply Finset.sum_nonneg
intro w _
split_ifs
· exact S.born_nonneg _ _ _ _
· exact le_rfl
lemma successProbability_le_one {x y a b : ℕ} (G : Game x y a b)
(S : SingleStrategy x y a b) : successProbability G S ≤ 1 := by
calc
successProbability G S ≤ ∑ z, G.questionProb z *
∑ w : Fin (a + 1) × Fin (b + 1), S.born z.1 z.2 w.1 w.2 := by
apply Finset.sum_le_sum
intro z _
apply mul_le_mul_of_nonneg_left _ (G.questionProb_nonneg z)
apply Finset.sum_le_sum
intro w _
split_ifs
· exact le_rfl
· exact S.born_nonneg _ _ _ _
_ = 1 := by
simp_rw [Fintype.sum_prod_type, S.born_sum, mul_one]
simpa only [Fintype.sum_prod_type] using G.questionProb_sum
lemma entangledValue_nonneg {x y a b : ℕ} (G : Game x y a b) :
0 ≤ entangledValue G := by
let S : SingleStrategy x y a b := Strategy.deterministic 0 0
exact (successProbability_nonneg G S).trans (le_csSup
(show BddAbove (Set.range (successProbability G)) from
⟨1, by rintro _ ⟨T, rfl⟩; exact successProbability_le_one G T⟩)
(Set.mem_range_self S))
lemma successProbability_le_value {x y a b : ℕ} (G : Game x y a b)
(S : SingleStrategy x y a b) : successProbability G S ≤ entangledValue G := by
exact le_csSup ⟨1, by rintro _ ⟨T, rfl⟩; exact successProbability_le_one G T⟩
(Set.mem_range_self S)
lemma entangledValue_le_one {x y a b : ℕ} (G : Game x y a b) :
entangledValue G ≤ 1 := by
apply csSup_le
· exact ⟨_, Set.mem_range_self (Strategy.deterministic (X := Fin (x + 1))
(Y := Fin (y + 1)) (0 : Fin (a + 1)) (0 : Fin (b + 1)))⟩
· rintro _ ⟨S, rfl⟩
exact successProbability_le_one G S
end ThresholdParallelRepetition
end
end OAI
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