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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 | import OAI.Probability.EntangledGames.MeasurementTransport
namespace OAI
universe u_m u_n u_p u_q
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
open Matrix
namespace ThresholdParallelRepetition.QuantumSampling
variable {m : Type u_m} {n : Type u_n} {p : Type u_p} [Fintype m] [Fintype n] [Fintype p]
def hsSq (M : Matrix m n ℂ) : ℝ := ∑ i, ∑ j, Complex.normSq (M i j)
lemma hsSq_nonneg (M : Matrix m n ℂ) : 0 ≤ hsSq M :=
Finset.sum_nonneg fun _ _ => Finset.sum_nonneg fun _ _ => Complex.normSq_nonneg _
lemma hsSq_conjTranspose (M : Matrix m n ℂ) : hsSq Mᴴ = hsSq M := by
simp only [hsSq, Matrix.conjTranspose_apply, Complex.star_def, Complex.normSq_conj]
exact Finset.sum_comm
lemma hsSq_eq_trace (M : Matrix m n ℂ) : hsSq M = (Matrix.trace (Mᴴ * M)).re := by
simp only [Matrix.trace, Matrix.diag, Matrix.mul_apply, Matrix.conjTranspose_apply,
Complex.re_sum, hsSq]
rw [Finset.sum_comm]
congr 1
ext i
congr 1
ext j
simp [Complex.mul_re, Complex.normSq_apply]
lemma hsSq_eq_trace' (M : Matrix m n ℂ) : hsSq M = (Matrix.trace (M * Mᴴ)).re := by
rw [← hsSq_conjTranspose M, hsSq_eq_trace, Matrix.conjTranspose_conjTranspose]
lemma hsSq_mul_left {U : Matrix m m ℂ} [DecidableEq m] (hU : Uᴴ * U = 1)
(M : Matrix m n ℂ) : hsSq (U * M) = hsSq M := by
rw [hsSq_eq_trace, Matrix.conjTranspose_mul]
rw [show (Mᴴ * Uᴴ) * (U * M) = Mᴴ * (Uᴴ * U) * M by simp only [Matrix.mul_assoc],
hU, Matrix.mul_one, ← hsSq_eq_trace]
lemma hsSq_mul_right {U : Matrix n n ℂ} [DecidableEq n] (hU : U * Uᴴ = 1)
(M : Matrix m n ℂ) : hsSq (M * U) = hsSq M := by
rw [← hsSq_conjTranspose, Matrix.conjTranspose_mul]
rw [hsSq_mul_left (U := Uᴴ) (by simpa only [Matrix.conjTranspose_conjTranspose] using hU),
hsSq_conjTranspose]
lemma hsSq_spectral_mixed [DecidableEq n] {A B : Matrix n n ℂ}
(hA : A.IsHermitian) (hB : B.IsHermitian) (f g : ℝ → ℝ) :
hsSq (cfc f A - cfc g B) =
∑ i, ∑ j, (f (hA.eigenvalues i) - g (hB.eigenvalues j)) ^ 2 *
Complex.normSq (((hA.eigenvectorUnitary : Matrix n n ℂ)ᴴ *
(hB.eigenvectorUnitary : Matrix n n ℂ)) i j) := by
let U : Matrix n n ℂ := hA.eigenvectorUnitary
let V : Matrix n n ℂ := hB.eigenvectorUnitary
have hUU : U * Uᴴ = 1 := Unitary.mul_star_self_of_mem hA.eigenvectorUnitary.property
have hUU' : Uᴴ * U = 1 := Unitary.star_mul_self_of_mem hA.eigenvectorUnitary.property
have hVV : V * Vᴴ = 1 := Unitary.mul_star_self_of_mem hB.eigenvectorUnitary.property
have hVV' : Vᴴ * V = 1 := Unitary.star_mul_self_of_mem hB.eigenvectorUnitary.property
have hf : cfc f A = U * Matrix.diagonal (fun i => (f (hA.eigenvalues i) : ℂ)) * Uᴴ := by
erw [hA.cfc_eq]
rfl
have hg : cfc g B = V * Matrix.diagonal (fun i => (g (hB.eigenvalues i) : ℂ)) * Vᴴ := by
erw [hB.cfc_eq]
rfl
have hmixed : Uᴴ * (cfc f A - cfc g B) * V =
Matrix.diagonal (fun i => (f (hA.eigenvalues i) : ℂ)) * (Uᴴ * V) -
(Uᴴ * V) * Matrix.diagonal (fun i => (g (hB.eigenvalues i) : ℂ)) := by
rw [hf, hg, Matrix.mul_sub, Matrix.sub_mul]
simp only [← Matrix.mul_assoc, hUU', Matrix.one_mul]
simp only [Matrix.mul_assoc, hVV', Matrix.mul_one]
rw [← hsSq_mul_left (U := Uᴴ) (by simpa only [Matrix.conjTranspose_conjTranspose] using hUU)
(cfc f A - cfc g B),
← hsSq_mul_right hVV (Uᴴ * (cfc f A - cfc g B)), hmixed]
unfold hsSq
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _
simp only [Matrix.sub_apply, Matrix.diagonal_mul, Matrix.mul_diagonal]
rw [mul_comm ((Uᴴ * V) i j), ← sub_mul, ← Complex.ofReal_sub, Complex.normSq_mul,
Complex.normSq_ofReal]
simp only [pow_two]
rfl
lemma hsSq_cfc_sub_le [DecidableEq n] {A B : Matrix n n ℂ}
(hA : A.IsHermitian) (hB : B.IsHermitian) (f : ℝ → ℝ) (L : ℝ)
(hf : ∀ s t : ℝ, (f s - f t)^2 ≤ L * (s - t)^2) :
hsSq (cfc f A - cfc f B) ≤ L * hsSq (A - B) := by
have hident := hsSq_spectral_mixed hA hB (fun x => x) (fun x => x)
erw [cfc_id' ℝ A hA, cfc_id' ℝ B hB] at hident
rw [hsSq_spectral_mixed hA hB, hident, Finset.mul_sum]
apply Finset.sum_le_sum
intro i _
rw [Finset.mul_sum]
apply Finset.sum_le_sum
intro j _
simpa only [mul_assoc] using mul_le_mul_of_nonneg_right
(hf (hA.eigenvalues i) (hB.eigenvalues j)) (Complex.normSq_nonneg _)
lemma hsSq_abs_sub_le [DecidableEq n] {A B : Matrix n n ℂ}
(hA : A.IsHermitian) (hB : B.IsHermitian) :
hsSq (cfc (fun t : ℝ => |t|) A - cfc (fun t : ℝ => |t|) B) ≤ hsSq (A - B) := by
simpa using hsSq_cfc_sub_le hA hB (fun t : ℝ => |t|) 1 (fun s t => by
have h := abs_abs_sub_abs_le_abs_sub s t
nlinarith [sq_nonneg (|s-t| - |(|s| - |t|)|), abs_nonneg (s-t), abs_nonneg (|s|-|t|),
sq_abs (s-t), sq_abs (|s|-|t|)])
lemma hsSq_zero : hsSq (0 : Matrix m n ℂ) = 0 := by simp [hsSq]
lemma hsSq_blocks {q : Type u_q} [Fintype q] (A : Matrix m n ℂ) (B : Matrix m p ℂ)
(C : Matrix q n ℂ) (D : Matrix q p ℂ) :
hsSq (Matrix.fromBlocks A B C D) = hsSq A + hsSq B + hsSq C + hsSq D := by
simp only [hsSq, Fintype.sum_sum_type, Matrix.fromBlocks_apply₁₁,
Matrix.fromBlocks_apply₁₂, Matrix.fromBlocks_apply₂₁, Matrix.fromBlocks_apply₂₂,
Finset.sum_add_distrib]
ring
omit [Fintype m] [Fintype n] [Fintype p] in
lemma fromBlocks_sub {q : Type u_q} (A A' : Matrix m n ℂ) (B B' : Matrix m p ℂ)
(C C' : Matrix q n ℂ) (D D' : Matrix q p ℂ) :
Matrix.fromBlocks A B C D - Matrix.fromBlocks A' B' C' D' =
Matrix.fromBlocks (A-A') (B-B') (C-C') (D-D') := by
ext i j
cases i <;> cases j <;> rfl
lemma block_diagonal_pos [DecidableEq m] [DecidableEq n]
{A : Matrix m m ℂ} {B : Matrix n n ℂ} (hA : A.PosSemidef) (hB : B.PosSemidef) :
(Matrix.fromBlocks A 0 0 B).PosSemidef := by
have hRa := (Matrix.nonneg_iff_posSemidef.mp (CFC.sqrt_nonneg A)).isHermitian.eq
have hRb := (Matrix.nonneg_iff_posSemidef.mp (CFC.sqrt_nonneg B)).isHermitian.eq
have hR : (Matrix.fromBlocks (CFC.sqrt A) 0 0 (CFC.sqrt B))ᴴ *
(Matrix.fromBlocks (CFC.sqrt A) 0 0 (CFC.sqrt B)) = Matrix.fromBlocks A 0 0 B := by
simp only [Matrix.fromBlocks_conjTranspose, Matrix.conjTranspose_zero, hRa, hRb,
Matrix.fromBlocks_multiply, Matrix.mul_zero, Matrix.zero_mul, add_zero, zero_add,
CFC.sqrt_mul_sqrt_self A hA.nonneg, CFC.sqrt_mul_sqrt_self B hB.nonneg]
rw [← hR]
exact Matrix.posSemidef_conjTranspose_mul_self _
def dilation (M : Matrix m n ℂ) : Matrix (m ⊕ n) (m ⊕ n) ℂ :=
Matrix.fromBlocks 0 M Mᴴ 0
omit [Fintype m] [Fintype n] in
lemma dilation_isHermitian (M : Matrix m n ℂ) : (dilation M).IsHermitian := by
unfold Matrix.IsHermitian dilation
simp only [Matrix.fromBlocks_conjTranspose, Matrix.conjTranspose_zero,
Matrix.conjTranspose_conjTranspose]
lemma abs_square [DecidableEq n] {A : Matrix n n ℂ} (hA : A.IsHermitian) :
cfc (fun t : ℝ => |t|) A * cfc (fun t : ℝ => |t|) A = A * A := by
erw [← cfc_mul (fun t : ℝ => |t|) (fun t : ℝ => |t|) A
(A.finite_real_spectrum.continuousOn _) (A.finite_real_spectrum.continuousOn _)]
calc
_ = cfc (fun t : ℝ => t * t) A := by
apply cfc_congr
intro t _
nlinarith [sq_abs t]
_ = cfc (fun t : ℝ => t) A * cfc (fun t : ℝ => t) A := by
erw [cfc_mul (fun t : ℝ => t) (fun t : ℝ => t) A
(A.finite_real_spectrum.continuousOn _) (A.finite_real_spectrum.continuousOn _)]
_ = _ := by erw [cfc_id' ℝ A hA]
lemma abs_eq_sqrt_square [DecidableEq n] {A : Matrix n n ℂ} (hA : A.IsHermitian) :
cfc (fun t : ℝ => |t|) A = CFC.sqrt (A * A) := by
apply Eq.symm
exact CFC.sqrt_unique (abs_square hA) (cfc_nonneg fun t _ => abs_nonneg t)
lemma sqrt_block_diagonal [DecidableEq m] [DecidableEq n]
{A : Matrix m m ℂ} {B : Matrix n n ℂ} (hA : A.PosSemidef) (hB : B.PosSemidef) :
CFC.sqrt (Matrix.fromBlocks A 0 0 B) = Matrix.fromBlocks (CFC.sqrt A) 0 0 (CFC.sqrt B) := by
have hpos : 0 ≤ Matrix.fromBlocks (CFC.sqrt A) 0 0 (CFC.sqrt B) :=
(block_diagonal_pos (Matrix.nonneg_iff_posSemidef.mp (CFC.sqrt_nonneg A))
(Matrix.nonneg_iff_posSemidef.mp (CFC.sqrt_nonneg B))).nonneg
apply CFC.sqrt_unique _ hpos
simp only [Matrix.fromBlocks_multiply, Matrix.mul_zero, Matrix.zero_mul,
zero_add, add_zero, CFC.sqrt_mul_sqrt_self _ hA.nonneg, CFC.sqrt_mul_sqrt_self _ hB.nonneg]
lemma abs_dilation [DecidableEq m] [DecidableEq n] (M : Matrix m n ℂ) :
cfc (fun t : ℝ => |t|) (dilation M) =
Matrix.fromBlocks (CFC.sqrt (M * Mᴴ)) 0 0 (CFC.sqrt (Mᴴ * M)) := by
rw [abs_eq_sqrt_square (dilation_isHermitian M)]
unfold dilation
simp only [Matrix.fromBlocks_multiply, Matrix.mul_zero, Matrix.zero_mul, zero_add, add_zero]
exact sqrt_block_diagonal (Matrix.posSemidef_self_mul_conjTranspose M)
(Matrix.posSemidef_conjTranspose_mul_self M)
theorem hsSq_right_absolute_sub_le [DecidableEq m] [DecidableEq n] (M N : Matrix m n ℂ) :
hsSq (CFC.sqrt (Mᴴ * M) - CFC.sqrt (Nᴴ * N)) ≤ 2 * hsSq (M - N) := by
have h := hsSq_abs_sub_le (dilation_isHermitian M) (dilation_isHermitian N)
rw [abs_dilation M, abs_dilation N] at h
simp only [dilation, fromBlocks_sub, sub_self, hsSq_blocks, hsSq_zero, zero_add, add_zero] at h
rw [← Matrix.conjTranspose_sub, hsSq_conjTranspose] at h
linarith [hsSq_nonneg (CFC.sqrt (M * Mᴴ) - CFC.sqrt (N * Nᴴ))]
end ThresholdParallelRepetition.QuantumSampling
end
end OAI
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