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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 | import Mathlib
import OAI.MathematicalPhysics.PEPSFilters.LocalOperators
import OAI.MathematicalPhysics.PEPSSubvolume.SpectralPowers
namespace OAI
/-! Hermitian trace representatives and spanning by reflections. -/
noncomputable section
open scoped BigOperators ComplexOrder
open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator
open scoped BigOperators
open scoped Topology
open Filter
open PolynomialPEPS.PinnedEntropy
open scoped MatrixOrder
namespace PolynomialPEPS.Subvolume.HermitianFunctional
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
theorem map_star_of_real (l : Matrix ι ι ℂ →ₗ[ℂ] ℂ)
(h : ∀ H : Matrix ι ι ℂ, H.IsHermitian → (l H).im = 0)
(A : Matrix ι ι ℂ) : l (star A) = star (l A) := by
have hplus : (A+star A).IsHermitian := by
change star (A+star A) = A+star A
simp [add_comm]
have hminus : (Complex.I • (A-star A)).IsHermitian := by
change star (Complex.I • (A-star A)) = Complex.I • (A-star A)
simp [star_sub,star_smul,RCLike.star_def,smul_sub,neg_smul]
abel
have hp := h (A+star A) hplus
have hm := h (Complex.I • (A-star A)) hminus
simp only [map_add,Complex.add_im] at hp
simp only [map_smul,map_sub,smul_eq_mul,Complex.mul_im,Complex.I_re,
Complex.I_im,zero_mul,one_mul,zero_add,Complex.sub_re] at hm
apply Complex.ext
· simpa [RCLike.star_def] using (sub_eq_zero.mp hm).symm
· simp only [RCLike.star_def,Complex.conj_im]
linarith
def representative (l : Matrix ι ι ℂ →ₗ[ℂ] ℂ) : Matrix ι ι ℂ :=
fun i j => l (Matrix.single j i 1)
theorem apply_eq_trace (l : Matrix ι ι ℂ →ₗ[ℂ] ℂ) (A : Matrix ι ι ℂ) :
l A = (representative l * A).trace := by
rw [Matrix.matrix_eq_sum_single A]
simp only [map_sum,Matrix.mul_sum,Matrix.trace_sum]
apply Finset.sum_congr rfl
intro i hi
apply Finset.sum_congr rfl
intro j hj
rw [← show A i j • Matrix.single i j (1 : ℂ) = Matrix.single i j (A i j) from by
simp [Matrix.smul_single], map_smul]
simp [representative,smul_eq_mul,mul_comm,Matrix.trace_mul_single]
theorem isHermitian_of_trace_real (B : Matrix ι ι ℂ)
(h : ∀ H : Matrix ι ι ℂ, H.IsHermitian → (B*H).trace.im=0) : B.IsHermitian := by
let l : Matrix ι ι ℂ →ₗ[ℂ] ℂ :=
(Matrix.traceLinearMap ι ℂ ℂ).comp (LinearMap.mulLeft ℂ B)
have hs := map_star_of_real l h
apply Matrix.IsHermitian.ext
intro i j
have hij := hs (Matrix.single i j 1)
simpa [l,Matrix.star_eq_conjTranspose,Matrix.conjTranspose_single,
Matrix.trace_mul_single] using hij.symm
theorem commuting_transition (F B : Matrix ι ι ℂ) (hF : F.PosSemidef)
(hFB : (F*B).IsHermitian) (hBF : (B*F).IsHermitian) : Commute F B := by
have h1 : star B * F = F*B := by
simpa only [star_mul,hF.isHermitian.star_eq] using hFB.star_eq
have h2 : F * star B = B*F := by
simpa only [star_mul,hF.isHermitian.star_eq] using hBF.star_eq
have hc : Commute (F*F) B := by
change (F*F)*B = B*(F*F)
calc
(F*F)*B = F*(F*B) := mul_assoc _ _ _
_ = F*(star B*F) := by rw [h1]
_ = (F*star B)*F := (mul_assoc _ _ _).symm
_ = B*(F*F) := by rw [h2,mul_assoc]
have hroot := hc.cfcₙ_nnreal NNReal.sqrt
change Commute (CFC.sqrt (F*F)) B at hroot
simpa only [CFC.sqrt_mul_self F hF.nonneg] using hroot
theorem moment_commute (l : Matrix ι ι ℂ →ₗ[ℂ] ℂ) (F : Matrix ι ι ℂ)
(hF : F.PosSemidef)
(hleft : ∀ H : Matrix ι ι ℂ, H.IsHermitian → (l (H*F)).im=0)
(hright : ∀ H : Matrix ι ι ℂ, H.IsHermitian → (l (F*H)).im=0)
(A : Matrix ι ι ℂ) : l (A*F) = l (F*A) := by
let B := representative l
have hFB : (F*B).IsHermitian := isHermitian_of_trace_real _ (fun H hH => by
have hb := hleft H hH
rw [apply_eq_trace] at hb
change (F*B*H).trace.im = 0
rw [← Matrix.trace_mul_cycle B H F,mul_assoc]
exact hb)
have hBF : (B*F).IsHermitian := isHermitian_of_trace_real _ (fun H hH => by
have hb := hright H hH
rw [apply_eq_trace] at hb
simpa [B,mul_assoc] using hb)
have hc := commuting_transition F B hF hFB hBF
rw [apply_eq_trace,apply_eq_trace]
change (B*(A*F)).trace = (B*(F*A)).trace
calc
(B*(A*F)).trace = (F*B*A).trace := by rw [← mul_assoc,Matrix.trace_mul_cycle]
_ = (B*(F*A)).trace := by rw [hc.eq,mul_assoc]
end PolynomialPEPS.Subvolume.HermitianFunctional
namespace PolynomialPEPS.Subvolume.ReflectionSpan
open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator
open PolynomialPEPS.Subvolume.SpectralCurve
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
def reflection (U : unitary (Matrix ι ι ℂ)) (i : ι) : unitary (Matrix ι ι ℂ) :=
⟨spectralHom U (fun j => if j=i then -1 else 1), by
let d : ι → ℂ := fun j => if j=i then -1 else 1
have hd : star d = d := by
ext j
by_cases h : j=i <;> simp [d,h]
have hsq : d*d = 1 := by
ext j
by_cases h : j=i <;> simp [d,h]
change star (spectralHom U d)*spectralHom U d = 1 ∧
spectralHom U d*star (spectralHom U d) = 1
constructor <;> rw [← map_star,hd,← map_mul,hsq,map_one]⟩
theorem reflection_star (U : unitary (Matrix ι ι ℂ)) (i : ι) :
star (reflection U i : Matrix ι ι ℂ) = (reflection U i : Matrix ι ι ℂ) := by
change star (spectralHom U _) = spectralHom U _
rw [← map_star]
congr 1
ext j
by_cases h : j=i <;> simp [h]
theorem reflection_eq (U : unitary (Matrix ι ι ℂ)) (i : ι) :
(reflection U i : Matrix ι ι ℂ) =
1-(2:ℂ) • spectralHom U (Pi.single i 1) := by
have hd : (fun j : ι => if j=i then (-1 : ℂ) else 1) =
1-(2:ℂ) • Pi.single i 1 := by
ext j
by_cases h : j=i <;> norm_num [Pi.single_apply,h]
change spectralHom U _ = _
rw [hd,map_sub,map_smul,map_one]
theorem zero_of_hermitian_tests (l : Matrix ι ι ℂ →ₗ[ℂ] ℂ)
(h : ∀ H : Matrix ι ι ℂ, H.IsHermitian → l H = 0)
(A : Matrix ι ι ℂ) : l A = 0 := by
have hplus : (A+star A).IsHermitian := by
change star (A+star A) = A+star A
simp [add_comm]
have hminus : (Complex.I • (A-star A)).IsHermitian := by
change star (Complex.I • (A-star A)) = Complex.I • (A-star A)
simp [star_sub,star_smul,RCLike.star_def,smul_sub,neg_smul]
abel
have hp := h (A+star A) hplus
have hm := h (Complex.I • (A-star A)) hminus
simp only [map_add] at hp
have hm' : l A = l (star A) := by
apply sub_eq_zero.mp
simpa [map_smul,map_sub,smul_eq_mul] using hm
rw [← hm'] at hp
have hz : (2:ℂ)*l A = 0 := by simpa [two_mul] using hp
exact (mul_eq_zero.mp hz).resolve_left (by norm_num)
theorem zero_of_reflection_tests (l : Matrix ι ι ℂ →ₗ[ℂ] ℂ)
(h : ∀ W : unitary (Matrix ι ι ℂ), star (W : Matrix ι ι ℂ)=W → l W = 0)
(A : Matrix ι ι ℂ) : l A = 0 := by
have h1 : l 1 = 0 := h 1 (by simp)
have hP (U : unitary (Matrix ι ι ℂ)) (i : ι) :
l (spectralHom U (Pi.single i 1)) = 0 := by
have hs := h (reflection U i) (reflection_star U i)
rw [reflection_eq,map_sub,map_smul,h1] at hs
simpa [smul_eq_mul] using hs
apply zero_of_hermitian_tests l ?_ A
intro H hH
let U := hH.eigenvectorUnitary
have hdiag : (fun j => (hH.eigenvalues j : ℂ)) =
∑ i, (hH.eigenvalues i : ℂ) • Pi.single i 1 := by
ext j
simp [Pi.single_apply,Finset.sum_apply,Pi.smul_apply]
have hrep : H = spectralHom U (fun j => (hH.eigenvalues j : ℂ)) := hH.spectral_theorem
rw [hrep,hdiag,map_sum,map_sum]
simp only [map_smul,hP,smul_zero,Finset.sum_const_zero]
end PolynomialPEPS.Subvolume.ReflectionSpan
end
end OAI
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