| import Mathlib |
| import OAI.MathematicalPhysics.PEPSFilters.LocalOperators |
| import OAI.MathematicalPhysics.PEPSSubvolume.EntropyHolder |
|
|
| namespace OAI |
|
|
| /-! Dimension bounds for finite-state entropy. -/ |
|
|
| noncomputable section |
| open scoped BigOperators ComplexOrder |
| open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator |
| open scoped BigOperators |
| open scoped Topology |
| open Filter |
| open scoped MatrixOrder |
| open scoped BigOperators Matrix.Norms.L2Operator |
| open scoped ComplexOrder BigOperators Matrix.Norms.L2Operator |
| open Matrix |
| open Filter Topology |
| open Set Filter Complex Complex.HadamardThreeLines |
| open scoped BigOperators Matrix.Norms.L2Operator MatrixOrder ComplexOrder |
| open PolynomialPEPS.PinnedEntropy |
|
|
| namespace PolynomialPEPS.Subvolume |
| open scoped BigOperators |
| variable {L q : ℕ} |
|
|
| theorem vonNeumannEntropy_nonneg (Ω : State L q) (hΩ : ‖Ω‖=1) (X : Finset (Vertex L)) : |
| 0 ≤ vonNeumannEntropy Ω X := by |
| apply Finset.sum_nonneg |
| intro i hi |
| have hp := (reducedDensity_posSemidef Ω X).eigenvalues_nonneg |
| apply Real.negMulLog_nonneg (hp i) |
| have h := Finset.single_le_sum (fun j (_ : j∈(Finset.univ : Finset (RegionConfiguration q X))) => hp j) hi |
| simpa only [marginal_eigenvalues_sum Ω hΩ X] using h |
|
|
| theorem vonNeumannEntropy_empty (Ω : State L q) (hΩ : ‖Ω‖=1) : |
| vonNeumannEntropy Ω ∅=0 := by |
| classical |
| let : IsEmpty {v : Vertex L |
| ⟨fun v => Finset.notMem_empty _ v.property⟩ |
| let i : RegionConfiguration q (∅ : Finset (Vertex L)) := fun v => isEmptyElim v |
| have hp : (reducedDensity_isHermitian Ω ∅).eigenvalues i=1 := by |
| simpa only [Fintype.sum_subsingleton _ i] using marginal_eigenvalues_sum Ω hΩ ∅ |
| simp only [vonNeumannEntropy,Fintype.sum_subsingleton _ i,hp,Real.negMulLog_one] |
|
|
| end PolynomialPEPS.Subvolume |
|
|
| namespace PolynomialPEPS.SubvolumeEntropy |
| open scoped BigOperators |
| variable {ι : Type*} [Fintype ι] [Nonempty ι] |
|
|
| theorem sum_negMulLog_le_log_card (p : ι → ℝ) (hp : ∀ i, 0≤p i) (hs : ∑ i,p i=1) : |
| ∑ i, Real.negMulLog (p i) ≤ Real.log (Fintype.card ι) := by |
| classical |
| let N : ℝ := Fintype.card ι |
| have hN : 0<N := by |
| change (0:ℝ)<(Fintype.card ι:ℝ) |
| exact_mod_cast (Fintype.card_pos (α:=ι)) |
| have hn : N≠0 := ne_of_gt hN |
| have hw : ∑ _i : ι, 1/N=1 := by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; change N*(1/N)=1; field_simp |
| have hj := Real.concaveOn_negMulLog.le_map_sum |
| (w:=fun _i : ι => 1/N) (p:=p) (t:=Finset.univ) |
| (fun i hi => le_of_lt (one_div_pos.mpr hN)) hw (fun i hi => hp i) |
| simp only [smul_eq_mul,← Finset.mul_sum,hs,mul_one] at hj |
| have hr : Real.negMulLog (1/N)=(1/N)*Real.log N := by |
| change -(1/N)*Real.log (1/N)=(1/N)*Real.log N |
| rw [Real.log_div (by norm_num) hn,Real.log_one] |
| ring |
| rw [hr] at hj |
| exact (mul_le_mul_iff_right₀ (one_div_pos.mpr hN)).mp hj |
|
|
| end PolynomialPEPS.SubvolumeEntropy |
|
|
| namespace PolynomialPEPS.Subvolume |
| open scoped BigOperators |
| variable {L q : ℕ} |
|
|
| theorem vonNeumannEntropy_le_card_log (hq : 0<q) (Ω : State L q) (hΩ : ‖Ω‖=1) |
| (X : Finset (Vertex L)) : vonNeumannEntropy Ω X ≤ (X.card:ℝ)*Real.log q := by |
| let : Nonempty (RegionConfiguration q X) := ⟨fun _ => ⟨0,hq⟩⟩ |
| have h := PolynomialPEPS.SubvolumeEntropy.sum_negMulLog_le_log_card |
| (reducedDensity_isHermitian Ω X).eigenvalues |
| (reducedDensity_posSemidef Ω X).eigenvalues_nonneg (marginal_eigenvalues_sum Ω hΩ X) |
| change vonNeumannEntropy Ω X ≤ _ at h |
| simpa only [RegionConfiguration,Fintype.card_fun,Fintype.card_fin,Fintype.card_coe, |
| Nat.cast_pow,Real.log_pow] using h |
|
|
| end PolynomialPEPS.Subvolume |
|
|
| end |
|
|
| end OAI |
|
|
