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f1f7953 | 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 | import Mathlib
import OAI.Analysis.TracialSplitting.PreparedBlocks
import OAI.Analysis.JiangSu.Comparison.PackingRounds
namespace OAI
noncomputable section
open Pureness
open scoped ComplexOrder
namespace DirectPacking
open CurrentMain
variable {D : Type} [CStarAlgebra D] [PartialOrder D] [StarOrderedRing D] [TraceFamily D]
lemma PackingStepBounds.mono_quality {p z y t : D} {c σ η δ : ℝ}
{K : Finset (unitary D)} (H : PackingStepBounds p z y t c σ δ K)
(hp : IsStarProjection p) (hy : IsStarProjection y) (hc : 0 ≤ c) (hση : σ ≤ η) :
PackingStepBounds p z y t c η δ K := by
refine { H with sampled_linear := ?_, sampled_energy := ?_ }
· intro x hx τ
exact (H.sampled_linear x hx τ).trans (by
gcongr
exact add_nonneg (τ.functional.map_nonneg hy.nonneg).1
(mul_nonneg hc (τ.functional.map_nonneg hp.nonneg).1))
· intro x hx τ
exact (H.sampled_energy x hx τ).trans (by
gcongr
exact add_nonneg (τ.functional.map_nonneg hy.nonneg).1
(mul_nonneg hc (τ.functional.map_nonneg hp.nonneg).1))
lemma exists_precise_packing_round (hcoef : CoefficientSupply (D := D)) (hinsert : InsertionSupply (D := D))
{z₀ r : D} (hz₀ : IsStarProjection z₀) (F : WeightedBlockFamily r)
(K : Finset (unitary D)) {η σ δ : ℝ} (hη : 0 < η) (hη1 : η < 1)
(hση : σ ≤ η) (hδ : 0 < δ)
(happrox : ∀ j, ∀ x ∈ K, ∃ H : Matrix (Fin (F.columns j)) (Fin (F.columns j)) ℂ,
‖F.range j*(x : D)*F.range j-(F.row j).matrix (F.base j) H‖ < σ) :
∃ R : SampledPackingRound F z₀ K η δ, ∀ a,
PackingStepBounds (F.range a.1) (R.occupied a) (R.sampled a) (R.inserted a)
(η*F.weight a.1/2) σ δ K := by
classical
let e := Fintype.equivFin (F.Index × Bool)
let Q := fun (a : Fin (Fintype.card (F.Index × Bool))) (z y t : D) =>
PackingStepBounds (F.range (e.symm a).1) z y t (η*F.weight (e.symm a).1/2) σ δ K
have hc (a : F.Index × Bool) : η*F.weight a.1/2 ∈ Set.Icc (0 : ℝ) 1 := by
have hw := F.weight_le_one a.1
have hw0 := F.weight_pos a.1
constructor
· positivity
· nlinarith
have hstep (a : Fin (Fintype.card (F.Index × Bool))) (z : D) (hz : IsStarProjection z) :
∃ y t, IsStarProjection y ∧ IsStarProjection t ∧ t ≤ 1-z ∧ Q a z y t :=
exists_packing_step (F.row (e.symm a).1) hcoef hinsert hz K (hc _) (hση.trans_lt hη1) hδ (happrox _)
obtain ⟨S,hS⟩ := exists_projection_insertion _ hz₀ Q hstep
let R : SampledPackingRound F z₀ K η δ := {
final := S.occupied (Fin.last _)
occupied := fun a => S.occupied (e a).castSucc
sampled := fun a => S.sampled (e a)
inserted := fun a => S.inserted (e a)
final_projection := S.occupied_projection _
occupied_projection := fun a => S.occupied_projection _
sampled_projection := fun a => S.sampled_projection _
inserted_projection := fun a => S.inserted_projection _
occupied_le_final := fun a => S.monotone (Fin.le_last _)
inserted_orthogonal := fun a b hab => S.orthogonal (e.injective.ne hab)
initial_orthogonal := fun a => S.initial_orthogonal _
final_eq := by rw [e.sum_comp S.inserted]; exact S.final_eq
bounds := by
intro a
have h : PackingStepBounds (F.range a.1) (S.occupied (e a).castSucc)
(S.sampled (e a)) (S.inserted (e a)) (η*F.weight a.1/2) σ δ K := by
simpa only [Q,Equiv.symm_apply_apply] using hS (e a)
exact h.mono_quality (F.range_projection a.1) (S.sampled_projection _) (hc a).1 hση }
refine ⟨R,?_⟩
intro a
simpa only [Q,Equiv.symm_apply_apply] using hS (e a)
namespace SampledPackingRound
variable {z₀ r : D} {F : WeightedBlockFamily r} {K : Finset (unitary D)} {η σ δ : ℝ}
variable (R : SampledPackingRound F z₀ K η δ)
lemma precise_sampled_energy_sum
(H : ∀ a, PackingStepBounds (F.range a.1) (R.occupied a) (R.sampled a) (R.inserted a)
(η*F.weight a.1/2) σ δ K) (τ : RestrictedTracialState D) (x : unitary D) (hx : x ∈ K) :
(∑ a, traceTwoNorm τ ((x : D)*R.sampled a-R.sampled a*(x : D))^2) ≤
η*(∑ j, F.weight j*traceTwoNorm τ ((x : D)*F.range j-F.range j*(x : D))^2)+
6*σ*(R.sampledMass τ+R.targetMass τ)+10*R.totalError := by
have h := Finset.sum_le_sum fun a (_ : a ∈ Finset.univ) => (H a).sampled_energy x hx τ
apply h.trans_eq
simp only [Finset.sum_add_distrib,← Finset.mul_sum,
Finset.sum_const,Finset.card_univ,nsmul_eq_mul,sampledMass,targetMass,totalError]
rw [signed_weighted_sum F η (fun j => traceTwoNorm τ ((x : D)*F.range j-F.range j*(x : D))^2),
signed_weighted_sum F η (fun j => (τ.functional (F.range j)).re)]
lemma inserted_row_energy (V : BistochasticTest D) (τ : RestrictedTracialState D) :
(∑ a, V.energy τ (R.inserted a)) ≤
2*(∑ a, V.energy τ (R.sampled a))+40*R.overlap τ := by
have hd := Finset.sum_le_sum fun a (_ : a ∈ Finset.univ) => (R.bounds a).inserted_distance τ
have he := Finset.sum_le_sum fun a (_ : a ∈ Finset.univ) => V.energy_perturb τ (R.inserted a) (R.sampled a)
simp only [Finset.sum_add_distrib,← Finset.mul_sum] at hd he
dsimp [overlap]
linarith
end SampledPackingRound
end DirectPacking
end
noncomputable section
namespace DirectPacking
open CurrentMain
variable {A D : Type*} [CStarAlgebra A] [CStarAlgebra D] [PartialOrder D] [StarOrderedRing D] [TraceFamily D]
namespace OrthogonalProjectionPair
def zero : OrthogonalProjectionPair D where
left := 0
right := 0
left_projection := IsStarProjection.zero D
right_projection := IsStarProjection.zero D
orthogonal := mul_zero 0
end OrthogonalProjectionPair
structure PackingBounds (ι : A →⋆ₐ[ℂ] D) (Q : OrthogonalProjectionPair D)
(K : Finset (unitary A)) (η : ℝ) (n : ℕ) : Prop where
energy : ∀ x ∈ K, ∀ τ : RestrictedTracialState D, Q.energy τ (ι (x : A)) ≤ packingEnergy η n
remaining : ∀ τ : RestrictedTracialState D, Q.remaining τ ≤ packingRemainder η n
balance : ∀ x ∈ K, ∀ τ : RestrictedTracialState D, Q.balance τ (ι (x : A)) ≤ packingBalance η n
end DirectPacking
end
end OAI
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