File size: 7,346 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 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 | import OAI.Probability.EntangledGames.FreshStates
namespace OAI
universe u_X u_Y u_I u_R u_m u_n
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) (R : Type u_R) (m : Type u_m) (n : Type u_n) [Fintype X] [Fintype Y] [Fintype I]
[DecidableEq X] [DecidableEq Y] [DecidableEq I]
[Fintype R] [DecidableEq R] [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n]
structure EventSystem where
μ : Law (X×Y)
x₀ : X
y₀ : Y
l : List I
nodup : l.Nodup
C : Matrix m n ℂ
hC : hsSq C = 1
F : R → Profile I X Y → Matrix m m ℂ
H : R → Profile I X Y → Matrix n n ℂ
hF : ∀ r z, (F r z).PosSemidef
hF1 : ∀ r z, F r z ≤ 1
hH : ∀ r z, (H r z).PosSemidef
hH1 : ∀ r z, H r z ≤ 1
localF : ∀ r i, IgnoresRight (F r) i
localH : ∀ r i, IgnoresLeft (H r) i
θ : R → Profile I X Y → ℝ
hθ : ∀ r z, 0 ≤ θ r z
hθ1 : ∀ r z, θ r z ≤ 1
localθL : ∀ r i, i ∈ l → IgnoresLeft (θ r) i
localθR : ∀ r i, i ∈ l → IgnoresRight (θ r) i
event_le_one : ∀ z, (∑ r, θ r z*prob C (F r z) (H r z)) ≤ 1
p : ℝ
hp : 0 < p
mass : (∑ r, (pi (fun _ : I => μ)).avg (fun z => θ r z*prob C (F r z) (H r z))) = p
variable {X Y I R m n} [Nonempty X] [Nonempty Y] [Nonempty R]
namespace EventSystem
variable (S : EventSystem X Y I R m n)
abbrev J := ExposureIndex (R := R) (X := X) (Y := Y) S.l
abbrev f := chainLeft S.μ S.x₀ S.l S.F
abbrev h := chainRight S.μ S.y₀ S.l S.H
abbrev A := chainLeftFamily S.μ S.x₀ S.l S.F
abbrev B := chainRightFamily S.μ S.y₀ S.l S.H
def U (j : ℕ) (r : R) (z : Profile I X Y) : Matrix (S.J×m) m ℂ := commonMap S.A (S.f j r z)
def V (j : ℕ) (r : R) (z : Profile I X Y) : Matrix (S.J×n) n ℂ := commonMap S.B (S.h j r z)
omit [DecidableEq Y] [DecidableEq R] [Nonempty X] [Nonempty Y] [Nonempty R] in
lemma f_pos (j : ℕ) (r : R) (z : Profile I X Y) : (S.f j r z).PosSemidef :=
lefts_posSemidef S.μ S.x₀ _ _ (S.hF r) z
omit [DecidableEq X] [DecidableEq R] [Nonempty X] [Nonempty Y] [Nonempty R] in
lemma h_pos (j : ℕ) (r : R) (z : Profile I X Y) : (S.h j r z).PosSemidef :=
rights_posSemidef S.μ S.y₀ _ _ (S.hH r) z
lemma U_gram {j : ℕ} (hj : j ≤ S.l.length) (r : R) (z : Profile I X Y) :
(S.U j r z)ᴴ*S.U j r z = S.f j r z :=
commonMap_gram S.A (fun t => S.f_pos _ _ _) ⟨(⟨j, by omega⟩,r,z),rfl⟩
lemma V_gram {j : ℕ} (hj : j ≤ S.l.length) (r : R) (z : Profile I X Y) :
(S.V j r z)ᴴ*S.V j r z = S.h j r z :=
commonMap_gram S.B (fun t => S.h_pos _ _ _) ⟨(⟨j, by omega⟩,r,z),rfl⟩
def D (j : Fin S.l.length) : X×Y → Matrix ((S.J×m)×(R×Profile I X Y)) ((S.J×n)×(R×Profile I X Y)) ℂ :=
stateAt S.μ S.θ S.p S.l[j.val] S.C (S.U (j+1)) (S.V j)
def rawA (j : Fin S.l.length) : X → Matrix ((S.J×m)×(R×Profile I X Y)) ((S.J×n)×(R×Profile I X Y)) ℂ :=
fun x => stateAt S.μ S.θ S.p S.l[j.val] S.C (S.U (j+1)) (S.V (j+1)) (x,S.y₀)
def rawB (j : Fin S.l.length) : Y → Matrix ((S.J×m)×(R×Profile I X Y)) ((S.J×n)×(R×Profile I X Y)) ℂ :=
fun y => stateAt S.μ S.θ S.p S.l[j.val] S.C (S.U j) (S.V j) (S.x₀,y)
lemma rawA_eq (j : Fin S.l.length) (q : X×Y) : S.rawA j q.1 =
stateAt S.μ S.θ S.p S.l[j.val] S.C (S.U (j+1)) (S.V (j+1)) q := by
apply stateAt_right_local
· intro r z y
exact congrArg (commonMap S.A) (chainLeft_next_local S.μ S.x₀ S.l S.nodup S.F S.localF j j.isLt r z y)
· intro r z y
apply congrArg (commonMap S.B)
rw [h, chainRight_step S.μ S.y₀ S.l S.H j j.isLt r]
exact right_ignoresRight S.μ S.y₀ _ _ z y
lemma rawB_eq (j : Fin S.l.length) (q : X×Y) : S.rawB j q.2 =
stateAt S.μ S.θ S.p S.l[j.val] S.C (S.U j) (S.V j) q := by
apply stateAt_left_local
· intro r z x
apply congrArg (commonMap S.A)
rw [f, chainLeft_step S.μ S.x₀ S.l S.F j j.isLt r]
exact left_ignoresLeft S.μ S.x₀ _ _ z x
· intro r z x
exact congrArg (commonMap S.B) (chainRight_prev_local S.μ S.y₀ S.l S.nodup S.H S.localH j j.isLt r z x)
def density : Profile I X Y → ℝ := eventDensity S.θ S.p S.C S.F S.H
omit [DecidableEq X] [DecidableEq Y] [DecidableEq R] [Nonempty X] [Nonempty Y] [Nonempty R] in
lemma density_nonneg : ∀ z, 0 ≤ S.density z := eventDensity_nonneg S.θ S.hθ S.hp S.C S.F S.H S.hF S.hH
omit [DecidableEq X] [DecidableEq Y] [DecidableEq R] [Nonempty X] [Nonempty Y] [Nonempty R] in
lemma density_total : (pi (fun _ : I => S.μ)).avg S.density = 1 :=
eventDensity_total S.μ S.θ S.hp S.C S.F S.H S.mass
omit [DecidableEq X] [DecidableEq Y] [DecidableEq R] [Nonempty X] [Nonempty Y] [Nonempty R] in
lemma density_le (z : Profile I X Y) : S.density z ≤ 1/S.p := by
exact (mul_le_mul_of_nonneg_left (S.event_le_one z) (div_nonneg zero_le_one S.hp.le)).trans_eq (mul_one _)
lemma D_norm (j : Fin S.l.length) (q : X×Y) : hsSq (S.D j q) = coordDensity (fun _ : I => S.μ) S.density S.l[j.val] q := by
rw [D, stateAt_norm_gram S.μ S.θ S.hθ S.hp _ S.C _ _ (S.f (j+1)) (S.h j)
(S.U_gram (by omega)) (S.V_gram (by omega))]
change massAt S.μ S.θ S.p S.l[j.val] S.C (S.f (j+1)) (S.h j) q = _
rw [chain_fixed_mass S.μ S.x₀ S.y₀ S.l S.nodup S.C S.F S.H S.localF S.localH S.θ S.localθL S.localθR]
exact massAt_eq_coordDensity S.μ S.θ S.p S.C S.F S.H _
(fun r => S.localθL r _ (List.getElem_mem j.isLt)) (fun r => S.localθR r _ (List.getElem_mem j.isLt)) q
lemma rawA_norm (j : Fin S.l.length) (x : X) : hsSq (S.rawA j x) =
(S.μ.givenFirst S.y₀ x).avg (fun y => coordDensity (fun _ : I => S.μ) S.density S.l[j.val] (x,y)) := by
rw [rawA, stateAt_norm_gram S.μ S.θ S.hθ S.hp _ S.C _ _ (S.f (j+1)) (S.h (j+1))
(S.U_gram (by omega)) (S.V_gram (by omega))]
change massAt S.μ S.θ S.p S.l[j.val] S.C (S.f (j+1)) (S.h (j+1)) (x,S.y₀) = _
have he : S.h (j+1) = fun r => S.μ.right S.y₀ S.l[j.val] (S.h j r) := by
funext r; exact chainRight_step S.μ S.y₀ S.l S.H j j.isLt r
rw [he, massAt_right S.μ S.y₀ S.θ S.p _ S.C _ _
(chainLeft_next_local S.μ S.x₀ S.l S.nodup S.F S.localF j j.isLt)]
congr 1; funext y
rw [← S.D_norm j (x,y), D, stateAt_norm_gram S.μ S.θ S.hθ S.hp _ S.C _ _ (S.f (j+1)) (S.h j)
(S.U_gram (by omega)) (S.V_gram (by omega))]
rfl
lemma rawB_norm (j : Fin S.l.length) (y : Y) : hsSq (S.rawB j y) =
(S.μ.givenSecond S.x₀ y).avg (fun x => coordDensity (fun _ : I => S.μ) S.density S.l[j.val] (x,y)) := by
rw [rawB, stateAt_norm_gram S.μ S.θ S.hθ S.hp _ S.C _ _ (S.f j) (S.h j)
(S.U_gram (by omega)) (S.V_gram (by omega))]
change massAt S.μ S.θ S.p S.l[j.val] S.C (S.f j) (S.h j) (S.x₀,y) = _
have he : S.f j = fun r => S.μ.left S.x₀ S.l[j.val] (S.f (j+1) r) := by
funext r; exact chainLeft_step S.μ S.x₀ S.l S.F j j.isLt r
rw [he, massAt_left S.μ S.x₀ S.θ S.p _ S.C _ _
(chainRight_prev_local S.μ S.y₀ S.l S.nodup S.H S.localH j j.isLt)]
congr 1; funext x
rw [← S.D_norm j (x,y), D, stateAt_norm_gram S.μ S.θ S.hθ S.hp _ S.C _ _ (S.f (j+1)) (S.h j)
(S.U_gram (by omega)) (S.V_gram (by omega))]
rfl
end EventSystem
end ThresholdParallelRepetition.MixedExposure
end
end OAI
|