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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 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 | import OAI.Probability.EntangledGames.SpectralEntropy
namespace OAI
universe u_n u_α u_m u_p u_J u_R
open scoped BigOperators ComplexOrder
open scoped MatrixOrder
open Matrix
open MeasureTheory Filter Set
open scoped Topology
open scoped Matrix.Norms.Elementwise
noncomputable section
open scoped BigOperators ComplexOrder MatrixOrder Matrix.Norms.Elementwise Topology
open Matrix MeasureTheory Filter Set
namespace ThresholdParallelRepetition.Resolvent
attribute [local instance] matrixContinuousENorm matrixPreorder matrixClosedIci rectContinuousENorm
variable {n : Type u_n} [Fintype n] [DecidableEq n]
lemma integral_matrix_apply {α : Type u_α} [MeasurableSpace α] {μ : Measure α}
{m : Type u_m} {p : Type u_p} [Fintype m] [Fintype p] {f : α → Matrix m p ℂ}
(hf : Integrable f μ) (i : m) (j : p) :
(∫ s, f s ∂μ) i j = ∫ s, f s i j ∂μ := by
erw [eval_integral (fun i => hf.eval i), eval_integral (fun j => (hf.eval i).eval j)]
variable {J : Type u_J} [Fintype J] [DecidableEq J]
def commonKernel (A : J → Matrix n n ℂ) (s : ℝ) : Matrix n (J × n) ℂ :=
fun r v => purificationKernel (A v.1) s r v.2
def commonGram (A : J → Matrix n n ℂ) : Matrix (J × n) (J × n) ℂ :=
∫ s : ℝ in Ioi 0, (commonKernel A s)ᴴ * commonKernel A s
omit [DecidableEq J] in
lemma commonGram_integrable (A : J → Matrix n n ℂ) (hA : ∀ j, (A j).PosSemidef) :
IntegrableOn (fun s : ℝ => (commonKernel A s)ᴴ * commonKernel A s) (Ioi 0) := by
apply integrable_gram <;> intro i j <;> exact kernel_entry_memLp (hA j.1) i j.2
lemma commonGram_posSemidef (A : J → Matrix n n ℂ) : (commonGram A).PosSemidef := by
apply integral_matrix_nonneg
exact Filter.Eventually.of_forall fun s => posSemidef_conjTranspose_mul_self (commonKernel A s)
omit [DecidableEq J] in
lemma commonGram_block (A : J → Matrix n n ℂ) (hA : ∀ j, (A j).PosSemidef) (j k : J) :
(commonGram A).submatrix (fun i => (j, i)) (fun i => (k, i)) =
∫ s : ℝ in Ioi 0, (purificationKernel (A j) s)ᴴ * purificationKernel (A k) s := by
ext i l
change (∫ s : ℝ in Ioi 0, (commonKernel A s)ᴴ * commonKernel A s) (j, i) (k, l) = _
rw [integral_matrix_apply (commonGram_integrable A hA),
integral_matrix_apply (kernel_cross_integrable (hA j) (hA k))]
rfl
def finitePurification (A : J → Matrix n n ℂ) (j : J) : Matrix (J × n) n ℂ :=
(CFC.sqrt (commonGram A)).submatrix id (fun i => (j, i))
lemma finitePurification_product (A : J → Matrix n n ℂ)
(hA : ∀ j, (A j).PosSemidef) (j k : J) :
(finitePurification A j)ᴴ * finitePurification A k =
∫ s : ℝ in Ioi 0, (purificationKernel (A j) s)ᴴ * purificationKernel (A k) s := by
have hh := (Matrix.nonneg_iff_posSemidef.mp (CFC.sqrt_nonneg (commonGram A))).isHermitian.eq
unfold finitePurification
rw [Matrix.conjTranspose_submatrix]
change (CFC.sqrt (commonGram A))ᴴ.submatrix (fun i => (j, i)) (Equiv.refl (J × n)) *
(CFC.sqrt (commonGram A)).submatrix (Equiv.refl (J × n)) (fun i => (k, i)) = _
rw [Matrix.submatrix_mul_equiv, hh,
CFC.sqrt_mul_sqrt_self (commonGram A) (commonGram_posSemidef A).nonneg]
exact commonGram_block A hA j k
lemma finitePurification_self (A : J → Matrix n n ℂ)
(hA : ∀ j, (A j).PosSemidef) (j : J) :
(finitePurification A j)ᴴ * finitePurification A j = A j := by
rw [finitePurification_product A hA, kernel_square_integral (hA j)]
lemma finitePurification_distance (A : J → Matrix n n ℂ)
(hA : ∀ j, (A j).PosSemidef) (j k : J) :
(finitePurification A j - finitePurification A k)ᴴ *
(finitePurification A j - finitePurification A k) =
∫ s : ℝ in Ioi 0, (purificationKernel (A j) s - purificationKernel (A k) s)ᴴ *
(purificationKernel (A j) s - purificationKernel (A k) s) := by
conv_lhs => rw [Matrix.conjTranspose_sub, Matrix.sub_mul, Matrix.mul_sub, Matrix.mul_sub]
have hd (s : ℝ) :
(purificationKernel (A j) s - purificationKernel (A k) s)ᴴ *
(purificationKernel (A j) s - purificationKernel (A k) s) =
((purificationKernel (A j) s)ᴴ * purificationKernel (A j) s -
(purificationKernel (A j) s)ᴴ * purificationKernel (A k) s) -
((purificationKernel (A k) s)ᴴ * purificationKernel (A j) s -
(purificationKernel (A k) s)ᴴ * purificationKernel (A k) s) := by
rw [Matrix.conjTranspose_sub, Matrix.sub_mul, Matrix.mul_sub, Matrix.mul_sub]
simp_rw [hd]
erw [integral_sub ((kernel_cross_integrable (hA j) (hA j)).sub
(kernel_cross_integrable (hA j) (hA k)))
((kernel_cross_integrable (hA k) (hA j)).sub (kernel_cross_integrable (hA k) (hA k))),
integral_sub (kernel_cross_integrable (hA j) (hA j)) (kernel_cross_integrable (hA j) (hA k)),
integral_sub (kernel_cross_integrable (hA k) (hA j)) (kernel_cross_integrable (hA k) (hA k))]
simp only [finitePurification_product A hA]
lemma finitePurification_jensen {R : Type u_R} [Fintype R]
(A : J → Matrix n n ℂ) (hA : ∀ j, (A j).PosSemidef) (hA1 : ∀ j, A j ≤ 1)
(e : R → J) (w : R → ℝ) (k : J)
(hw0 : ∀ r, 0 ≤ w r) (hw : ∑ r, w r = 1)
(hmean : ∑ r, w r • A (e r) = A k) :
(∑ r, w r • ((finitePurification A (e r) - finitePurification A k)ᴴ *
(finitePurification A (e r) - finitePurification A k))) ≤
(∑ r, w r • effectEntropy (A (e r))) - effectEntropy (A k) := by
simp_rw [finitePurification_distance A hA]
exact weighted_distance_jensen (A ∘ e) w (A k) (fun r => hA (e r))
(fun r => hA1 (e r)) (hA k) (hA1 k) hw0 hw hmean
theorem finite_common_purification (A : J → Matrix n n ℂ)
(hA : ∀ j, (A j).PosSemidef) :
∃ F : J → Matrix (J × n) n ℂ,
(∀ j, (F j)ᴴ * F j = A j) ∧
∀ j k, (F j)ᴴ * F k =
∫ s : ℝ in Ioi 0, (purificationKernel (A j) s)ᴴ * purificationKernel (A k) s :=
⟨finitePurification A, finitePurification_self A hA, finitePurification_product A hA⟩
end ThresholdParallelRepetition.Resolvent
end
end OAI
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