| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators ComplexOrder |
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| namespace ThresholdParallelRepetition |
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|
| structure Game (x y a b : ℕ) where |
| questionProb : Fin (x + 1) × Fin (y + 1) → ℝ |
| questionProb_nonneg : ∀ z, 0 ≤ questionProb z |
| questionProb_sum : ∑ z, questionProb z = 1 |
| accepts : Fin (x + 1) → Fin (y + 1) → Fin (a + 1) → Fin (b + 1) → Bool |
| |
| structure Strategy (X Y A B : Type) [Fintype A] [Fintype B] where |
| dimA : ℕ |
| dimB : ℕ |
| state : Fin (dimA + 1) × Fin (dimB + 1) → ℂ |
| state_unit : ∑ i, Complex.normSq (state i) = 1 |
| alice : X → A → Matrix (Fin (dimA + 1)) (Fin (dimA + 1)) ℂ |
| bob : Y → B → Matrix (Fin (dimB + 1)) (Fin (dimB + 1)) ℂ |
| alice_pos : ∀ x a, (alice x a).PosSemidef |
| bob_pos : ∀ y b, (bob y b).PosSemidef |
| alice_total : ∀ x, ∑ a, alice x a = 1 |
| bob_total : ∀ y, ∑ b, bob y b = 1 |
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|
| def Strategy.born {X Y A B : Type} [Fintype A] [Fintype B] |
| (S : Strategy X Y A B) (x : X) (y : Y) (a : A) (b : B) : ℝ := |
| ∑ i, ∑ j, (star (S.state i) * |
| (S.alice x a i.1 j.1 * S.bob y b i.2 j.2) * S.state j).re |
| |
| abbrev SingleStrategy (x y a b : ℕ) := |
| Strategy (Fin (x + 1)) (Fin (y + 1)) (Fin (a + 1)) (Fin (b + 1)) |
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| def successProbability {x y a b : ℕ} (G : Game x y a b) (S : SingleStrategy x y a b) : ℝ := |
| ∑ z : Fin (x + 1) × Fin (y + 1), G.questionProb z * |
| ∑ w : Fin (a + 1) × Fin (b + 1), |
| if G.accepts z.1 z.2 w.1 w.2 then S.born z.1 z.2 w.1 w.2 else 0 |
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| def entangledValue {x y a b : ℕ} (G : Game x y a b) : ℝ := |
| sSup (Set.range (successProbability G)) |
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|
| abbrev RepeatedStrategy (x y a b k : ℕ) := |
| Strategy (Fin k → Fin (x + 1)) (Fin k → Fin (y + 1)) |
| (Fin k → Fin (a + 1)) (Fin k → Fin (b + 1)) |
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| def wins {x y a b k : ℕ} (G : Game x y a b) |
| (xs : Fin k → Fin (x + 1)) (ys : Fin k → Fin (y + 1)) |
| (as : Fin k → Fin (a + 1)) (bs : Fin k → Fin (b + 1)) : ℕ := |
| ∑ i, if G.accepts (xs i) (ys i) (as i) (bs i) then 1 else 0 |
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| def thresholdProbability {x y a b k : ℕ} (G : Game x y a b) (δ : ℝ) |
| (S : RepeatedStrategy x y a b k) : ℝ := |
| ∑ xs : Fin k → Fin (x + 1), ∑ ys : Fin k → Fin (y + 1), |
| (∏ i, G.questionProb (xs i, ys i)) * |
| ∑ as : Fin k → Fin (a + 1), ∑ bs : Fin k → Fin (b + 1), |
| if ⌈(entangledValue G + δ) * (k : ℝ)⌉₊ ≤ wins G xs ys as bs |
| then S.born xs ys as bs else 0 |
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| def thresholdValue {x y a b k : ℕ} (G : Game x y a b) (δ : ℝ) : ℝ := |
| sSup (Set.range (thresholdProbability (k := k) G δ)) |
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| end ThresholdParallelRepetition |
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| end |
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| end OAI |
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