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cdec422 | 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 Mathlib
namespace OAI
/-! Unit entries, orthogonality, characters and the cubic matrix. -/
open scoped BigOperators Matrix
namespace HadamardSix
abbrev Index := Fin 6
abbrev Matrix6 := Matrix Index Index ℂ
abbrev Charge := Index → ℤ
def Unimodular (H : Matrix6) : Prop := ∀ i j, ‖H i j‖ = 1
def IsComplexHadamard (H : Matrix6) : Prop :=
Unimodular H ∧ H.conjTranspose * H = (6 : ℂ) • (1 : Matrix6)
def HadamardEquivalent (H K : Matrix6) : Prop :=
∃ (r c : Equiv.Perm Index) (u v : Index → ℂ),
(∀ i, ‖u i‖ = 1) ∧ (∀ j, ‖v j‖ = 1) ∧
∀ i j, K i j = u i * H (r i) (c j) * v j
def Balanced (a : Charge) : Prop := ∑ i, a i = 0
noncomputable def character (x : Index → ℂ) (a : Charge) : ℂ :=
∏ i, x i ^ a i
noncomputable def g (H : Matrix6) (a : Charge) : ℂ :=
(6 : ℂ)⁻¹ * ∑ k, character (fun i => H i k) a
def permuteCharge (p : Equiv.Perm Index) (a : Charge) : Charge := fun i => a (p i)
def alpha : Charge := ![1, 1, 1, -1, -1, -1]
noncomputable def omega : ℂ := Complex.exp (2 * (Real.pi : ℂ) * Complex.I / 3)
def taoExponent (i j : Index) : ℤ :=
if i = 0 ∨ j = 0 ∨ i = j then 0
else if ((i.val : ℤ) - (j.val : ℤ)) % 5 = 1 ∨
((i.val : ℤ) - (j.val : ℤ)) % 5 = 4 then 1
else -1
noncomputable def taoMatrix : Matrix6 := fun i j => omega ^ taoExponent i j
lemma unit_ne_zero {z : ℂ} (hz : ‖z‖ = 1) : z ≠ 0 := by
intro h
simp [h] at hz
lemma unit_mul_star {z : ℂ} (hz : ‖z‖ = 1) : z * star z = 1 := by
rw [Complex.star_def, Complex.mul_conj, Complex.normSq_eq_norm_sq, hz]
norm_num
lemma unit_star_mul {z : ℂ} (hz : ‖z‖ = 1) : star z * z = 1 := by
rw [mul_comm]
exact unit_mul_star hz
lemma unit_inv {z : ℂ} (hz : ‖z‖ = 1) : z⁻¹ = star z := by
apply inv_eq_of_mul_eq_one_right
exact unit_mul_star hz
lemma Unimodular.entry_ne_zero {H : Matrix6} (hH : Unimodular H) (i j : Index) :
H i j ≠ 0 := unit_ne_zero (hH i j)
lemma IsComplexHadamard.unimodular {H : Matrix6} (hH : IsComplexHadamard H) :
Unimodular H := hH.1
lemma IsComplexHadamard.mul_conjTranspose {H : Matrix6} (hH : IsComplexHadamard H) :
H * H.conjTranspose = (6 : ℂ) • (1 : Matrix6) := by
have hleft : ((6 : ℂ)⁻¹ • H.conjTranspose) * H = 1 := by
rw [Matrix.smul_mul, hH.2, smul_smul]
norm_num
have hright := mul_eq_one_comm.mp hleft
rw [Matrix.mul_smul] at hright
have h := congrArg (fun K : Matrix6 => (6 : ℂ) • K) hright
simpa only [smul_smul, mul_inv_cancel₀ (by norm_num : (6 : ℂ) ≠ 0), one_smul] using h
lemma IsComplexHadamard.column_orthogonality {H : Matrix6} (hH : IsComplexHadamard H)
(j k : Index) :
∑ i, star (H i j) * H i k = if j = k then 6 else 0 := by
have h := congr_fun (congr_fun hH.2 j) k
simpa [Matrix.mul_apply, Matrix.one_apply, Matrix.smul_apply] using h
lemma IsComplexHadamard.row_orthogonality {H : Matrix6} (hH : IsComplexHadamard H)
(i j : Index) :
∑ k, H i k * star (H j k) = if i = j then 6 else 0 := by
have h := congr_fun (congr_fun hH.mul_conjTranspose i) j
simpa [Matrix.mul_apply, Matrix.one_apply, Matrix.smul_apply] using h
lemma isComplexHadamard_of_rows {H : Matrix6} (hu : Unimodular H)
(hr : ∀ i j, ∑ k, H i k * star (H j k) = if i = j then 6 else 0) :
IsComplexHadamard H := by
have heq : H * H.conjTranspose = (6 : ℂ) • (1 : Matrix6) := by
ext i j
simpa [Matrix.mul_apply, Matrix.one_apply, Matrix.smul_apply] using hr i j
have hleft : H * ((6 : ℂ)⁻¹ • H.conjTranspose) = 1 := by
rw [Matrix.mul_smul, heq, smul_smul]
norm_num
have hright := mul_eq_one_comm.mp hleft
rw [Matrix.smul_mul] at hright
have h := congrArg (fun K : Matrix6 => (6 : ℂ) • K) hright
refine ⟨hu, ?_⟩
simpa only [smul_smul, mul_inv_cancel₀ (by norm_num : (6 : ℂ) ≠ 0), one_smul] using h
lemma IsComplexHadamard.transpose {H : Matrix6} (hH : IsComplexHadamard H) :
IsComplexHadamard H.transpose := by
refine isComplexHadamard_of_rows (fun i j => hH.1 j i) ?_
intro i j
simpa [mul_comm, eq_comm] using hH.column_orthogonality j i
lemma IsComplexHadamard.conjugate {H : Matrix6} (hH : IsComplexHadamard H) :
IsComplexHadamard (fun i j => star (H i j)) := by
refine isComplexHadamard_of_rows (fun i j => by simpa using hH.1 i j) ?_
intro i j
have h := congrArg star (hH.row_orthogonality i j)
simpa only [star_sum, star_mul, star_star, apply_ite star, star_ofNat, star_zero, mul_comm]
using h
end HadamardSix
end OAI
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