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9eeb9c7 | 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 | import OAI.Analysis.HyperbolicCones.Model
namespace OAI
/-! Exact identities for the Choi–Lam form. -/
noncomputable section
open scoped Matrix.Norms.L2Operator
open Matrix
universe u
namespace Paper256
theorem choiLam_expand {R : Type u} [CommRing R] (z y : Fin 3 → R) :
choiLam z y =
z 0 ^ 2 * y 0 ^ 2 + z 1 ^ 2 * y 1 ^ 2 + z 2 ^ 2 * y 2 ^ 2 -
2 * (z 0 * y 0 * z 1 * y 1 + z 0 * y 0 * z 2 * y 2 +
z 1 * y 1 * z 2 * y 2) +
(z 0 ^ 2 * y 1 ^ 2 + z 1 ^ 2 * y 2 ^ 2 + z 2 ^ 2 * y 0 ^ 2) := by
simp [choiLam, Fin.sum_univ_succ]
ring
theorem qMatrix_isHermitian (y : Fin 3 → ℝ) : (qMatrix y).IsHermitian := by
show (qMatrix y)ᴴ = qMatrix y
ext i j
by_cases hij : i = j
· subst j
simp [qMatrix, Matrix.conjTranspose_apply]
· simp [qMatrix, Matrix.conjTranspose_apply, hij, Ne.symm hij, mul_comm]
theorem zMatrix_isHermitian (z : Fin 3 → ℝ) : (zMatrix z).IsHermitian := by
show (zMatrix z)ᴴ = zMatrix z
ext i j
by_cases hij : i = j
· subst j
simp [zMatrix, Matrix.conjTranspose_apply]
· simp [zMatrix, Matrix.conjTranspose_apply, hij, Ne.symm hij, mul_comm]
theorem qMatrix_quadratic (z y : Fin 3 → ℝ) :
dotProduct z (qMatrix y *ᵥ z) = choiLam z y := by
simp [dotProduct, Matrix.mulVec, qMatrix, choiLam, Fin.sum_univ_succ]
ring
theorem zMatrix_quadratic (z y : Fin 3 → ℝ) :
dotProduct y (zMatrix z *ᵥ y) = choiLam z y := by
simp [dotProduct, Matrix.mulVec, zMatrix, choiLam, Fin.sum_univ_succ]
ring
theorem zMatrix_diag_nonnegative (z : Fin 3 → ℝ) (i : Fin 3) :
0 ≤ zMatrix z i i := by
simp only [zMatrix, ↓reduceIte]
positivity
theorem zMatrix_principal_minor_two (z : Fin 3 → ℝ) (i : Fin 3) :
Matrix.det ((zMatrix z).submatrix
(fun j : Fin 2 => if j = 0 then i else i + 1)
(fun j : Fin 2 => if j = 0 then i else i + 1)) =
(z i ^ 2) ^ 2 + z (i + 2) ^ 2 * (z i ^ 2 + z (i + 1) ^ 2) := by
fin_cases i <;> simp [Matrix.det_fin_two, Matrix.submatrix, zMatrix,
Fin.reduceAdd, Fin.reduceEq] <;> ring
theorem zMatrix_principal_minor_two_nonnegative (z : Fin 3 → ℝ) (i : Fin 3) :
0 ≤ Matrix.det ((zMatrix z).submatrix
(fun j : Fin 2 => if j = 0 then i else i + 1)
(fun j : Fin 2 => if j = 0 then i else i + 1)) := by
rw [zMatrix_principal_minor_two]
positivity
theorem zMatrix_det (z : Fin 3 → ℝ) :
Matrix.det (zMatrix z) =
(z 0 ^ 2) ^ 2 * z 1 ^ 2 + (z 1 ^ 2) ^ 2 * z 2 ^ 2 +
(z 2 ^ 2) ^ 2 * z 0 ^ 2 - 3 * z 0 ^ 2 * z 1 ^ 2 * z 2 ^ 2 := by
simp [Matrix.det_fin_three, zMatrix]
ring
theorem choiLam_nonzero :
choiLam (![1, 0, 0] : Fin 3 → ℝ) (![1, 0, 0] : Fin 3 → ℝ) = 1 := by
simp [choiLam_expand, Matrix.cons_val]
theorem choiLam_cyclic_zero (i : Fin 3) :
choiLam (Pi.single i 1 : Fin 3 → ℝ) (Pi.single (i + 2) 1) = 0 := by
fin_cases i <;> simp [choiLam_expand, Fin.reduceAdd, Fin.reduceEq]
theorem choiLam_sign_zero (σ : Fin 3 → ℝ)
(hσ : ∀ i, σ i = 1 ∨ σ i = -1) : choiLam σ σ = 0 := by
rcases hσ 0 with h0 | h0 <;> rcases hσ 1 with h1 | h1 <;>
rcases hσ 2 with h2 | h2 <;> norm_num [choiLam_expand, h0, h1, h2]
theorem choiLam_curve (i : Fin 3) (s : ℝ) :
choiLam ((Pi.single i 1 : Fin 3 → ℝ) + s • Pi.single (i + 2) 1)
(Pi.single (i + 2) 1 + s • Pi.single i 1) = s ^ 4 := by
fin_cases i <;> simp [choiLam_expand, Fin.reduceAdd, Fin.reduceEq] <;> ring
theorem choiLam_smul_left (z y : Fin 3 → ℝ) (s : ℝ) :
choiLam (s • z) y = s ^ 2 * choiLam z y := by
simp only [choiLam_expand, Pi.smul_apply, smul_eq_mul]
ring
theorem choiLam_smul_right (z y : Fin 3 → ℝ) (s : ℝ) :
choiLam z (s • y) = s ^ 2 * choiLam z y := by
simp only [choiLam_expand, Pi.smul_apply, smul_eq_mul]
ring
end Paper256
end
end OAI
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