File size: 2,602 Bytes
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 | import Mathlib
namespace OAI
noncomputable section
open scoped BigOperators ComplexOrder
namespace ThresholdParallelRepetition
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
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))
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
def entangledValue {x y a b : ℕ} (G : Game x y a b) : ℝ :=
sSup (Set.range (successProbability G))
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))
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
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
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))
end ThresholdParallelRepetition
end
end OAI
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