| import Mathlib |
| import OAI.MathematicalPhysics.PEPSFilters.LocalOperators |
| import OAI.MathematicalPhysics.PEPSSubvolume.Diagonal |
| import OAI.MathematicalPhysics.PEPSSubvolume.JointSpectrum |
|
|
| namespace OAI |
|
|
| /-! Exact optimizing marginal identities including singular filters. -/ |
|
|
| 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 PolynomialPEPS.PinnedEntropy |
|
|
| namespace PolynomialPEPS.Subvolume.OptimizerMarginal |
| open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator |
| open PolynomialPEPS.Subvolume.OptimizerGauge PolynomialPEPS.Subvolume.ActualTransition |
| open PolynomialPEPS.Subvolume.TraceStripping PolynomialPEPS.Subvolume.SpectralCurve |
| variable {L q : ℕ} |
|
|
| theorem optimizer_trace_moment (hq : 0 < q) |
| (X : ℕ → Finset (Vertex L)) (hX : Monotone X) |
| (a : ℕ → ℝ) (ψ : State L q) (n : ℕ) |
| (F : (j : ℕ) → LocalPositiveFilter q (X j)) |
| (hF : IsFilterOptimizer ψ (fun j : Fin n => a j.val) (fun j : Fin n => F j.val)) |
| (k : ℕ) (hk : k < n) |
| (H : Matrix (RegionConfiguration q (X k)) (RegionConfiguration q (X k)) ℂ) : |
| let A : ℕ → Operator L q := fun j => liftLocal (X j) (F j).matrix |
| let v : State L q := asMap (L := L) (q := q) (orderedPrefix A n) ψ |
| (H * reducedDensity v (X k)).trace = |
| inner ℂ v (replacementOutput A (X k) ψ k n hk (H*(F k).matrix)) := by |
| dsimp only |
| have hs := optimizer_strip_outer hq X hX a ψ n F hF (k+1) (by omega) |
| (liftLocal (X k) H) (fun j hkj hj => |
| SupportedOn.mono (hX (by omega : k≤j)) ⟨H,rfl⟩) |
| dsimp only at hs |
| rw [inner_liftLocal_trace,insertedProduct_left _ _ k n hk] at hs |
| simpa only [replacementOutput,LinearMap.coe_mk,AddHom.coe_mk,liftLocal_mul] using hs |
|
|
| theorem optimizer_marginal_commute (hq : 0 < q) |
| (X : ℕ → Finset (Vertex L)) (hX : Monotone X) |
| (a : ℕ → ℝ) (ψ : State L q) (n : ℕ) |
| (F : (j : ℕ) → LocalPositiveFilter q (X j)) |
| (hF : IsFilterOptimizer ψ (fun j : Fin n => a j.val) (fun j : Fin n => F j.val)) |
| (k : ℕ) (hk : k < n) : |
| let A : ℕ → Operator L q := fun j => liftLocal (X j) (F j).matrix |
| let v : State L q := asMap (L := L) (q := q) (orderedPrefix A n) ψ |
| Commute (F k).matrix (reducedDensity v (X k)) := by |
| dsimp only |
| let A : ℕ → Operator L q := fun j => liftLocal (X j) (F j).matrix |
| let v : State L q := asMap (L := L) (q := q) (orderedPrefix A n) ψ |
| let σ := reducedDensity v (X k) |
| have hc (H : Matrix (RegionConfiguration q (X k)) (RegionConfiguration q (X k)) ℂ) : |
| (H*((F k).matrix*σ)).trace = (H*(σ*(F k).matrix)).trace := by |
| rw [← mul_assoc,optimizer_trace_moment hq X hX a ψ n F hF k hk] |
| have hm := optimizer_moment_commute hq X hX a ψ n F hF k hk (H*(F k).matrix) |
| rw [hm] |
| have ht := optimizer_trace_moment hq X hX a ψ n F hF k hk ((F k).matrix*H) |
| rw [mul_assoc] at ht |
| rw [← ht] |
| change ((F k).matrix * H * σ).trace = (H*(σ*(F k).matrix)).trace |
| rw [Matrix.trace_mul_comm ((F k).matrix*H) σ,← mul_assoc, |
| Matrix.trace_mul_comm (σ*(F k).matrix) H] |
| change (F k).matrix*σ = σ*(F k).matrix |
| ext i j |
| have ht := hc (Matrix.single j i 1) |
| simpa [Matrix.trace_single_mul] using ht |
|
|
| theorem optimizer_marginal_spectral (hq : 0 < q) |
| (X : ℕ → Finset (Vertex L)) (hX : Monotone X) |
| (a : ℕ → ℝ) (ψ : State L q) (n : ℕ) |
| (F : (j : ℕ) → LocalPositiveFilter q (X j)) |
| (hF : IsFilterOptimizer ψ (fun j : Fin n => a j.val) (fun j : Fin n => F j.val)) |
| (k : ℕ) (hk : k < n) (hp : 2/a k ≠ 0) : |
| let A : ℕ → Operator L q := fun j => liftLocal (X j) (F j).matrix |
| let v : State L q := asMap (L := L) (q := q) (orderedPrefix A n) ψ |
| ∃ (U : unitary (Matrix (RegionConfiguration q (X k)) (RegionConfiguration q (X k)) ℂ)) |
| (e : RegionConfiguration q (X k) → ℝ), |
| (∀ i, 0 ≤ e i) ∧ |
| (F k).matrix = spectralHom U (fun i => (e i:ℂ)) ∧ |
| reducedDensity v (X k) = (‖v‖^2:ℂ) • spectralHom U (fun i => (((e i)^(2/a k):ℝ):ℂ)) := by |
| dsimp only |
| let A : ℕ → Operator L q := fun j => liftLocal (X j) (F j).matrix |
| let v : State L q := asMap (L := L) (q := q) (orderedPrefix A n) ψ |
| let σ := reducedDensity v (X k) |
| have hc := optimizer_marginal_commute hq X hX a ψ n F hF k hk |
| obtain ⟨U,e,d,he,hd,hrepF,hrepσ⟩ := JointDiagonalization.exists_positive_unitary |
| (F k).matrix σ (F k).positive (reducedDensity_posSemidef v (X k)) hc |
| change (F k).matrix = spectralHom U (fun i => (e i:ℂ)) at hrepF |
| change σ = spectralHom U (fun i => (d i:ℂ)) at hrepσ |
| have hmoment (h : RegionConfiguration q (X k) → ℝ) : |
| (∑ i, h i*d i) = ‖v‖^2 * ∑ i, (e i)^(2/a k)*h i := by |
| have ht := optimizer_trace_moment hq X hX a ψ n F hF k hk |
| (spectralHom U (fun i => (h i:ℂ))) |
| change (spectralHom U (fun i => (h i:ℂ))*σ).trace = _ at ht |
| rw [hrepσ,← map_mul,trace_spectralHom] at ht |
| rw [hrepF,← map_mul] at ht |
| have hm := ActualDiagonal.optimizer_diagonal_moment hq X hX a ψ n F hF k hk hp U e h he hrepF |
| have hprod : (fun i => (h i:ℂ))*(fun i => (e i:ℂ)) = fun i => (e i*h i:ℂ) := by |
| ext i |
| simp only [Pi.mul_apply,mul_comm] |
| rw [hprod] at ht |
| dsimp only at hm |
| rw [← ht] at hm |
| simpa only [Complex.re_sum,Pi.mul_apply,Complex.mul_re,Complex.ofReal_re, |
| Complex.ofReal_im,mul_zero,sub_zero] using hm |
| have hdval (i : RegionConfiguration q (X k)) : d i = ‖v‖^2 * (e i)^(2/a k) := by |
| simpa [Pi.single_apply] using hmoment (Pi.single i 1) |
| refine ⟨U,e,he,hrepF,?_⟩ |
| change σ = _ |
| rw [hrepσ,← map_smul] |
| apply congrArg (spectralHom U) |
| ext i |
| simp only [Pi.smul_apply,smul_eq_mul,hdval,Complex.ofReal_mul,Complex.ofReal_pow] |
| rfl |
|
|
| end PolynomialPEPS.Subvolume.OptimizerMarginal |
|
|
| end |
|
|
| end OAI |
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|
