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a89d282 | 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 | import OAI.Probability.InvariantIsing.Haar.HaarPolynomialEntropyBound
namespace OAI
/-! Positive polynomial truncations of an exponential observable. -/
noncomputable section
open Matrix MvPolynomial
open scoped BigOperators
namespace InvariantIsing
def haarExpPolynomial {N : ℕ} (n : ℕ) (p : MatrixPolynomial N) : MatrixPolynomial N :=
∑ k ∈ Finset.range (n+1), ((Nat.factorial k : ℝ)⁻¹) • p^k
lemma haarExpPolynomial_eval {N : ℕ} (n : ℕ) (p : MatrixPolynomial N)
(M : Matrix (Fin N) (Fin N) ℝ) :
matrixPolynomialEval M (haarExpPolynomial n p) =
∑ k ∈ Finset.range (n+1), (matrixPolynomialEval M p)^k/(Nat.factorial k : ℝ) := by
simp only [haarExpPolynomial,map_sum,map_smul,map_pow,smul_eq_mul,div_eq_mul_inv,mul_comm]
lemma haarExpPolynomial_derivation {N : ℕ} (n : ℕ) (p : MatrixPolynomial N)
(A : Matrix (Fin N) (Fin N) ℝ) :
matrixPolynomialDerivation A (haarExpPolynomial (n+1) p) =
haarExpPolynomial n p*matrixPolynomialDerivation A p := by
unfold haarExpPolynomial
rw [Finset.sum_range_succ']
simp only [map_add,map_sum,Derivation.map_smul_of_tower,pow_zero,
Derivation.map_one_eq_zero,smul_zero,add_zero]
rw [Finset.sum_mul]
apply Finset.sum_congr rfl
intro k _
have hk : (k:ℝ)+1 ≠ 0 := by positivity
have hf : (Nat.factorial k : ℝ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero k
have hcoef : ((Nat.factorial (k+1) : ℝ)⁻¹)*((k+1:ℕ):ℝ) =
(Nat.factorial k : ℝ)⁻¹ := by
rw [Nat.factorial_succ,Nat.cast_mul,Nat.cast_add,Nat.cast_one]
field_simp
rw [Derivation.leibniz_pow]
simp only [Nat.add_sub_cancel]
rw [← Nat.cast_smul_eq_nsmul ℝ]
change (Nat.factorial (k+1) : ℝ)⁻¹ •
(((k+1:ℕ):ℝ) • (p^k*matrixPolynomialDerivation A p)) = _
rw [smul_smul,hcoef,smul_mul_assoc]
end InvariantIsing
end
end OAI
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