File size: 13,747 Bytes
caec230 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 | import Mathlib
import OAI.MathematicalPhysics.PEPSFilters.LocalOperators
import OAI.MathematicalPhysics.PEPSSubvolume.FlatDirection
import OAI.MathematicalPhysics.PEPSSubvolume.TwoFactors
namespace OAI
/-! Mixed singular residual bounds from simultaneous perturbations. -/
noncomputable section
open scoped BigOperators ComplexOrder
open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator
open scoped BigOperators
open scoped Topology
open Filter
open PolynomialPEPS.PinnedEntropy
namespace PolynomialPEPS.Subvolume.ActualKernel
open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator
open PolynomialPEPS.Subvolume.SpectralCurve PolynomialPEPS.Subvolume.KernelPerturbation
open PolynomialPEPS.Subvolume.KernelNormalization PolynomialPEPS.Subvolume.OptimizerGauge
variable {L q : ℕ}
theorem real_complex_smul (t : ℝ) (v : State L q) : (t : ℂ) • v = t • v := by
ext i
simp [Complex.real_smul]
theorem two_affine_output (A : ℕ → Operator L q) (k l n : ℕ)
(hk : k<n) (hl : l<n) (hkl : k ≠ l) (B C : Operator L q)
(s t c d : ℂ) (ψ : State L q) :
asMap (L := L) (q := q) (orderedPrefix
(Function.update (Function.update A k (c • (A k+s • B)))
l (d • (A l+t • C))) n) ψ =
(c*d) • (asMap (L := L) (q := q) (orderedPrefix A n) ψ +
s • asMap (L := L) (q := q) (orderedPrefix (Function.update A k B) n) ψ +
t • asMap (L := L) (q := q) (orderedPrefix (Function.update A l C) n) ψ +
(s*t) • asMap (L := L) (q := q)
(orderedPrefix (Function.update (Function.update A k B) l C) n) ψ) := by
rw [two_affine_prefix A k l n hk hl hkl]
change Matrix.toEuclideanCLM (𝕜 := ℂ) (_ • (_ + _ + _ + _)) ψ = _
simp only [map_smul, map_add, smul_apply, add_apply]
rfl
theorem two_affine_kernel_output (A : ℕ → Operator L q) (k l n : ℕ)
(hk : k<n) (hl : l<n) (hkl : k ≠ l) (B C : Operator L q) (ψ : State L q)
(hkzero : asMap (L := L) (q := q) (orderedPrefix (Function.update A k B) n) ψ = 0)
(ζ : ℂ) (c t b d : ℝ) :
asMap (L := L) (q := q) (orderedPrefix
(Function.update (Function.update A k
((b : ℂ) • (A k + (((c*t : ℝ) : ℂ)*ζ) • B))) l
((d : ℂ) • (A l + ((t : ℂ)*Complex.I) • C))) n) ψ =
((b*d : ℝ) : ℂ) • (asMap (L := L) (q := q) (orderedPrefix A n) ψ +
t • (Complex.I • asMap (L := L) (q := q) (orderedPrefix (Function.update A l C) n) ψ) +
(c*t^2) • ((ζ*Complex.I) • asMap (L := L) (q := q)
(orderedPrefix (Function.update (Function.update A k B) l C) n) ψ)) := by
rw [two_affine_output A k l n hk hl hkl, hkzero,smul_zero,add_zero, Complex.ofReal_mul]
simp only [← real_complex_smul,smul_smul,Complex.ofReal_mul,Complex.ofReal_pow]
congr 2
ring_nf
theorem mixed_scalar_bound (N M S p d t : ℝ) (hN : 0 ≤ N) (hM : 0 ≤ M)
(hS : 1 ≤ S) (hp : 2 < p) (hd : 0 ≤ d) (hdsq : d^2*(1+t^2)=1)
(hb : scale S p*(d*M) ≤ N) : M^2 ≤ (N^2+N^2*t^2)*S := by
have H := norm_sq_bound N (d*M) S p hN (mul_nonneg hd hM) hS hp hb
calc
M^2 = (d*M)^2*(1+t^2) := by
rw [mul_pow]
calc
M^2 = (d^2*(1+t^2))*M^2 := by rw [hdsq,one_mul]
_ = _ := by ring
_ ≤ (N^2*S)*(1+t^2) := mul_le_mul_of_nonneg_right H (by positivity)
_ = _ := by ring
theorem mixed_normalized_bound (hq : 0 < q)
(X : ℕ → Finset (Vertex L)) (hX : Monotone X)
(a : ℕ → ℝ) (ψ : State L q) (n : ℕ)
(F : (j : ℕ) → LocalPositiveFilter q (X j))
(hF : IsFilterOptimizer ψ (fun j : Fin n => a j.val) (fun j : Fin n => F j.val))
(k l : ℕ) (hk : k < n) (hl : l < n) (hkl : k ≠ l) (hp : 2 < 2/a k)
(U : unitary (Matrix (RegionConfiguration q (X k)) (RegionConfiguration q (X k)) ℂ))
(e : RegionConfiguration q (X k) → ℝ) (he : ∀ i, 0 ≤ e i)
(hrep : (F k).matrix = spectralHom U (fun i => (e i : ℂ)))
(hs : ∑ i, (e i)^(2/a k) = 1)
(W : unitary (Matrix (RegionConfiguration q (X l)) (RegionConfiguration q (X l)) ℂ))
(hW : star (W : Matrix (RegionConfiguration q (X l)) (RegionConfiguration q (X l)) ℂ) = (W : Matrix (RegionConfiguration q (X l)) (RegionConfiguration q (X l)) ℂ))
(ζ : unitary ℂ) (c t : ℝ) :
let A : ℕ → Operator L q := fun j => liftLocal (X j) (F j).matrix
let B : Operator L q := liftLocal (X k) (kernel U e)
let C : Operator L q := liftLocal (X l) ((F l).matrix*(W : Matrix (RegionConfiguration q (X l)) (RegionConfiguration q (X l)) ℂ))
let v : State L q := asMap (L := L) (q := q) (orderedPrefix A n) ψ
let u : State L q := Complex.I • asMap (L := L) (q := q)
(orderedPrefix (Function.update A l C) n) ψ
let w : State L q := ((ζ : ℂ)*Complex.I) • asMap (L := L) (q := q)
(orderedPrefix (Function.update (Function.update A k B) l C) n) ψ
‖(((scale (cost e (2/a k) (c*t)) (2/a k) * ReflectionRotation.scale t : ℝ) : ℂ) •
(v + t • u + (c*t^2) • w))‖ ≤ ‖v‖ := by
dsimp only
let A : ℕ → Operator L q := fun j => liftLocal (X j) (F j).matrix
let B : Operator L q := liftLocal (X k) (kernel U e)
let C : Operator L q := liftLocal (X l) ((F l).matrix*(W : Matrix (RegionConfiguration q (X l)) (RegionConfiguration q (X l)) ℂ))
let v : State L q := asMap (L := L) (q := q) (orderedPrefix A n) ψ
let u : State L q := Complex.I • asMap (L := L) (q := q)
(orderedPrefix (Function.update A l C) n) ψ
let w : State L q := ((ζ : ℂ)*Complex.I) • asMap (L := L) (q := q)
(orderedPrefix (Function.update (Function.update A k B) l C) n) ψ
let z : State L q := v+t • u+(c*t^2) • w
let b := scale (cost e (2/a k) (c*t)) (2/a k)
let d := ReflectionRotation.scale t
let G : LocalPositiveFilter q (X k) :=
⟨normalized U e (2/a k) (c*t), normalized_posSemidef U e he (2/a k) (c*t)⟩
have hG : filterTracePower G (2/a k) = 1 :=
normalized_capacity U e he (2/a k) (by linarith) hs (c*t)
let R : unitary (Matrix (RegionConfiguration q (X l)) (RegionConfiguration q (X l)) ℂ) := ReflectionRotation.rotation W hW t
have hR : filterTracePower (rightPositive (F l) R) (2/a l) = 1 := by
rw [rightPositive_capacity]
exact hF.1 ⟨l,hl⟩
have hb := two_replacement_bound hq X hX a ψ n F hF k l hk hl hkl
(perturbUnitary U e ζ (c*t)) R G hG (rightPositive (F l) R) hR
have hmat : liftLocal (X k)
((perturbUnitary U e ζ (c*t) : Matrix _ _ ℂ)*G.matrix) =
(b : ℂ) • (A k + (((c*t : ℝ) : ℂ)*(ζ : ℂ)) • B) := by
rw [show G.matrix = normalized U e (2/a k) (c*t) from rfl, normalized_factorization, ← hrep, liftLocal_smul, liftLocal_add, liftLocal_smul]
have hmat' : liftLocal (X l) ((R : Matrix _ _ ℂ)*(rightPositive (F l) R).matrix) =
(d : ℂ) • (A l + ((t : ℂ)*Complex.I) • C) := by
rw [rightPositive_factorization]
change liftLocal (X l) ((F l).matrix*((d : ℂ) •
(1+((t : ℂ)*Complex.I) • (W : Matrix (RegionConfiguration q (X l)) (RegionConfiguration q (X l)) ℂ)))) = _
simp only [mul_smul_comm,mul_add,mul_one,liftLocal_smul,liftLocal_add]
rfl
have hkzero : asMap (L := L) (q := q) (orderedPrefix (Function.update A k B) n) ψ = 0 :=
kernel_output_zero hq X hX a ψ n F hF k hk hp U e he hrep hs
have hout : asMap (L := L) (q := q) (orderedPrefix
(Function.update (Function.update A k
(liftLocal (X k) ((perturbUnitary U e ζ (c*t) : Matrix _ _ ℂ)*G.matrix))) l
(liftLocal (X l) ((R : Matrix _ _ ℂ)*(rightPositive (F l) R).matrix))) n) ψ =
((b*d : ℝ) : ℂ) • z := by
rw [hmat,hmat']
exact two_affine_kernel_output A k l n hk hl hkl B C ψ hkzero ζ c t b d
have hfv : filteredVector (fun j : Fin n => F j.val) ψ = v := by
unfold filteredVector
rw [PhysicalCurve.orderedFilterProduct_eq_prefix]
rw [hout,hfv] at hb
exact hb
theorem mixed_envelope (hq : 0 < q)
(X : ℕ → Finset (Vertex L)) (hX : Monotone X)
(a : ℕ → ℝ) (ψ : State L q) (n : ℕ)
(F : (j : ℕ) → LocalPositiveFilter q (X j))
(hF : IsFilterOptimizer ψ (fun j : Fin n => a j.val) (fun j : Fin n => F j.val))
(k l : ℕ) (hk : k < n) (hl : l < n) (hkl : k ≠ l) (hp : 2 < 2/a k)
(U : unitary (Matrix (RegionConfiguration q (X k)) (RegionConfiguration q (X k)) ℂ))
(e : RegionConfiguration q (X k) → ℝ) (he : ∀ i, 0 ≤ e i)
(hrep : (F k).matrix = spectralHom U (fun i => (e i : ℂ)))
(hs : ∑ i, (e i)^(2/a k) = 1)
(W : unitary (Matrix (RegionConfiguration q (X l)) (RegionConfiguration q (X l)) ℂ))
(hW : star (W : Matrix (RegionConfiguration q (X l)) (RegionConfiguration q (X l)) ℂ) = (W : Matrix (RegionConfiguration q (X l)) (RegionConfiguration q (X l)) ℂ))
(ζ : unitary ℂ) (c t : ℝ) :
let A : ℕ → Operator L q := fun j => liftLocal (X j) (F j).matrix
let B : Operator L q := liftLocal (X k) (kernel U e)
let C : Operator L q := liftLocal (X l) ((F l).matrix*(W : Matrix (RegionConfiguration q (X l)) (RegionConfiguration q (X l)) ℂ))
let v : State L q := asMap (L := L) (q := q) (orderedPrefix A n) ψ
let u : State L q := Complex.I • asMap (L := L) (q := q)
(orderedPrefix (Function.update A l C) n) ψ
let w : State L q := ((ζ : ℂ)*Complex.I) • asMap (L := L) (q := q)
(orderedPrefix (Function.update (Function.update A k B) l C) n) ψ
‖v + t • u + (c*t^2) • w‖^2 ≤
(‖v‖^2+‖v‖^2*t^2)*cost e (2/a k) (c*t) := by
dsimp only
let A : ℕ → Operator L q := fun j => liftLocal (X j) (F j).matrix
let B : Operator L q := liftLocal (X k) (kernel U e)
let C : Operator L q := liftLocal (X l) ((F l).matrix*(W : Matrix (RegionConfiguration q (X l)) (RegionConfiguration q (X l)) ℂ))
let v : State L q := asMap (L := L) (q := q) (orderedPrefix A n) ψ
let u : State L q := Complex.I • asMap (L := L) (q := q)
(orderedPrefix (Function.update A l C) n) ψ
let w : State L q := ((ζ : ℂ)*Complex.I) • asMap (L := L) (q := q)
(orderedPrefix (Function.update (Function.update A k B) l C) n) ψ
let z : State L q := v+t • u+(c*t^2) • w
let b := scale (cost e (2/a k) (c*t)) (2/a k)
let d := ReflectionRotation.scale t
have hb := mixed_normalized_bound hq X hX a ψ n F hF k l hk hl hkl hp U e he hrep hs W hW ζ c t
change ‖((b*d : ℝ) : ℂ) • z‖ ≤ ‖v‖ at hb
have hbpos : 0 < b := scale_pos _ _
(lt_of_lt_of_le zero_lt_one (one_le_cost e (2/a k) (c*t)))
have hdpos : 0 < d := ReflectionRotation.scale_pos t
rw [norm_smul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (mul_pos hbpos hdpos)] at hb
have hb' : b*(d*‖z‖) ≤ ‖v‖ := by simpa [mul_assoc] using hb
exact mixed_scalar_bound ‖v‖ ‖z‖ (cost e (2/a k) (c*t)) (2/a k) d t
(norm_nonneg _) (norm_nonneg _) (one_le_cost e (2/a k) (c*t)) hp hdpos.le
(ReflectionRotation.scale_sq t) hb'
theorem mixed_transition_zero (hq : 0 < q)
(X : ℕ → Finset (Vertex L)) (hX : Monotone X)
(a : ℕ → ℝ) (ψ : State L q) (n : ℕ)
(F : (j : ℕ) → LocalPositiveFilter q (X j))
(hF : IsFilterOptimizer ψ (fun j : Fin n => a j.val) (fun j : Fin n => F j.val))
(k l : ℕ) (hk : k < n) (hl : l < n) (hkl : k ≠ l) (hp : 2 < 2/a k)
(U : unitary (Matrix (RegionConfiguration q (X k)) (RegionConfiguration q (X k)) ℂ))
(e : RegionConfiguration q (X k) → ℝ) (he : ∀ i, 0 ≤ e i)
(hrep : (F k).matrix = spectralHom U (fun i => (e i : ℂ)))
(hs : ∑ i, (e i)^(2/a k) = 1)
(W : unitary (Matrix (RegionConfiguration q (X l)) (RegionConfiguration q (X l)) ℂ))
(hW : star (W : Matrix (RegionConfiguration q (X l)) (RegionConfiguration q (X l)) ℂ) = (W : Matrix (RegionConfiguration q (X l)) (RegionConfiguration q (X l)) ℂ)) :
let A : ℕ → Operator L q := fun j => liftLocal (X j) (F j).matrix
let B : Operator L q := liftLocal (X k) (kernel U e)
let C : Operator L q := liftLocal (X l) ((F l).matrix*(W : Matrix (RegionConfiguration q (X l)) (RegionConfiguration q (X l)) ℂ))
inner ℂ (asMap (L := L) (q := q) (orderedPrefix A n) ψ)
(asMap (L := L) (q := q)
(orderedPrefix (Function.update (Function.update A k B) l C) n) ψ) = 0 := by
dsimp only
let A : ℕ → Operator L q := fun j => liftLocal (X j) (F j).matrix
let B : Operator L q := liftLocal (X k) (kernel U e)
let C : Operator L q := liftLocal (X l) ((F l).matrix*(W : Matrix (RegionConfiguration q (X l)) (RegionConfiguration q (X l)) ℂ))
let v : State L q := asMap (L := L) (q := q) (orderedPrefix A n) ψ
let u : State L q := Complex.I • asMap (L := L) (q := q)
(orderedPrefix (Function.update A l C) n) ψ
let w : State L q := asMap (L := L) (q := q)
(orderedPrefix (Function.update (Function.update A k B) l C) n) ψ
let K : ℝ := ∑ i, if e i = 0 then (1:ℝ) else 0
have hzero (ζ : unitary ℂ) :
(inner ℂ v (((ζ : ℂ)*Complex.I) • w)).re = 0 := by
let : InnerProductSpace ℝ (State L q) := InnerProductSpace.rclikeToReal ℂ (State L q)
have hb (c t : ℝ) : ‖v+t • u+(c*t^2) • (((ζ : ℂ)*Complex.I) • w)‖^2 ≤
(‖v‖^2+‖v‖^2*t^2)*(1+K*|c*t|^(2/a k)) :=
mixed_envelope hq X hX a ψ n F hF k l hk hl hkl hp U e he hrep hs W hW ζ c t
exact OptimizerFlatDirection.mixed_direction_vanishes v u
(((ζ : ℂ)*Complex.I) • w) K (2/a k) hp hb
have hi := hzero 1
let ζI : unitary ℂ := ⟨Complex.I, by
constructor <;> simp [RCLike.star_def, Complex.I_mul_I]⟩
have hr := hzero ζI
change inner ℂ v w = 0
apply Complex.ext
· simpa [ζI,inner_smul_right] using hr
· simpa [inner_smul_right] using hi
end PolynomialPEPS.Subvolume.ActualKernel
end
end OAI
|