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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 | import OAI.Probability.EntangledGames.ExponentialBounds
namespace OAI
universe u_I u_X u_Y
noncomputable section
open scoped BigOperators MatrixOrder ComplexOrder
open Matrix
namespace ThresholdParallelRepetition
open QuantumSampling FiniteProbability Law
namespace Strategy
variable {X Y A B : Type} [Fintype A] [Fintype B]
def coeff (S : Strategy X Y A B) : Matrix (Fin (S.dimA+1)) (Fin (S.dimB+1)) ℂ := fun i j => S.state (i,j)
lemma coeff_unit (S : Strategy X Y A B) : hsSq S.coeff = 1 := by
simpa only [hsSq, coeff, Fintype.sum_prod_type] using S.state_unit
lemma born_eq_prob (S : Strategy X Y A B) (x : X) (y : Y) (a : A) (b : B) :
S.born x y a b = prob S.coeff (S.alice x a) (S.bob y b) := by
rw [S.born_eq_quadratic]
rfl
def repetitionModel {x y a b k : ℕ} (S : RepeatedStrategy x y a b k) (G : Game x y a b) :
RepeatedModel (Fin (x+1)) (Fin (y+1)) (Fin (a+1)) (Fin (b+1)) (Fin k)
(Fin (S.dimA+1)) (Fin (S.dimB+1)) where
μ := G.questionLaw
C := S.coeff
hC := S.coeff_unit
P xs := ⟨S.alice xs, S.alice_pos xs, S.alice_total xs⟩
Q ys := ⟨S.bob ys, S.bob_pos ys, S.bob_total ys⟩
V := G.accepts
end Strategy
lemma sum_profiles {I : Type u_I} {X : Type u_X} {Y : Type u_Y} [Fintype I] [DecidableEq I] [Fintype X] [Fintype Y]
(f : (I→X×Y) → ℝ) :
(∑ z, f z) = ∑ xs : I → X, ∑ ys : I → Y, f (fun i => (xs i,ys i)) := by
let e : (I→X×Y) ≃ (I→X)×(I→Y) := {
toFun := fun z => (fun i => (z i).1, fun i => (z i).2)
invFun := fun t i => (t.1 i,t.2 i)
left_inv := fun z => by funext i; exact Prod.eta _
right_inv := fun t => by rcases t with ⟨xs,ys⟩; rfl }
calc
(∑ z, f z) = ∑ t : (I→X)×(I→Y), f (e.symm t) := (e.symm.sum_comp f).symm
_ = _ := by rw [Fintype.sum_prod_type]; rfl
namespace Strategy
variable {x y a b k : ℕ} (S : RepeatedStrategy x y a b k) (G : Game x y a b)
lemma rate_eq_wins [Nonempty (Fin k)] (z : Profile (Fin k) (Fin (x+1)) (Fin (y+1)))
(as : Fin k → Fin (a+1)) (bs : Fin k → Fin (b+1)) :
Tests.rate (S.repetitionModel G).winsAt (z,as,bs) =
(wins G (fun i => (z i).1) (fun i => (z i).2) as bs : ℝ)/(k : ℝ) := by
rw [Tests.rate, uniform_avg, Fintype.card_fin, one_div, inv_mul_eq_div]
congr 1
unfold wins
rw [Nat.cast_sum]
apply Finset.sum_congr rfl; intro i _
change (if G.accepts (z i).1 (z i).2 (as i) (bs i) then (1:ℝ) else 0) = _
split_ifs <;> norm_num
lemma threshold_eq_model [Nonempty (Fin k)] (hk : 0 < k) (δ : ℝ) :
thresholdProbability G δ S =
Tests.threshold (S.repetitionModel G).joint
(S.repetitionModel G).winsAt (entangledValue G+δ) := by
have hk' : (0:ℝ) < k := by exact_mod_cast hk
symm
unfold Tests.threshold
rw [RepeatedModel.joint_avg]
simp only [S.rate_eq_wins G, le_div_iff₀ hk', ← Nat.ceil_le]
simp only [avg, pi, RepeatedModel.answers, outcomeLaw, Strategy.repetitionModel,
smul_eq_mul, Fintype.sum_prod_type]
rw [sum_profiles]
unfold thresholdProbability
apply Finset.sum_congr rfl; intro xs _
apply Finset.sum_congr rfl; intro ys _
congr 1
apply Finset.sum_congr rfl; intro as _
apply Finset.sum_congr rfl; intro bs _
rw [S.born_eq_prob]
split_ifs <;> simp
end Strategy
end ThresholdParallelRepetition
end
end OAI
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