| import OAI.Probability.EntangledGames.FreshStates |
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| namespace OAI |
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| universe u_X u_Y u_I u_R u_m u_n |
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| 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 |
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| 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) |
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| 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⟩ |
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| 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) |
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| 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) |
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| 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 _) |
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| 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 |
|
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| 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 |
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| end |
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| end OAI |
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