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caec230 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 | import Mathlib
import OAI.MathematicalPhysics.PEPSMove.PhysicalCoordinates
namespace OAI
noncomputable section
open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator MatrixOrder
open Matrix
namespace PolynomialPEPS.PhysicalMove.LocalMove
open scoped BigOperators Matrix.Norms.L2Operator ComplexOrder
def spectralPower {ι : Type*} [Fintype ι] [DecidableEq ι]
(A : Matrix ι ι ℂ) (a : ℝ) : Matrix ι ι ℂ :=
cfc (fun s : ℝ => Real.rpow s a) A
def Subdensity {ι : Type*} [Fintype ι] (A : Matrix ι ι ℂ) : Prop :=
A.PosSemidef ∧ (Matrix.trace A).re ≤ 1
def moveVector {L q : ℕ} (θ : State L q) (X P Y : Finset (Vertex L))
(σ : Matrix (RegionConfiguration q (X∪P)) (RegionConfiguration q (X∪P)) ℂ)
(τ : Matrix (RegionConfiguration q (X∪Y)) (RegionConfiguration q (X∪Y)) ℂ)
(a : ℝ) : State L q :=
asMap (liftLocal (X∪P) (spectralPower σ (a/2)) *
liftLocal P (spectralPower (reducedDensity θ P) (-a/2)) *
liftLocal (X∪Y) (spectralPower τ (a/2)) *
liftLocal Y (spectralPower (reducedDensity θ Y) (-a/2))) θ
def eta {L q : ℕ} (θ : State L q) (X P Y : Finset (Vertex L)) : ℝ :=
vonNeumannEntropy θ (X∪P)-vonNeumannEntropy θ P+
vonNeumannEntropy θ (X∪Y)-vonNeumannEntropy θ Y
def ell {L : ℕ} (q : ℕ) (X : Finset (Vertex L)) : ℝ :=
Real.log (Real.exp 1*(Fintype.card (RegionConfiguration q X):ℝ))
def OneCopyMoveBound : Prop :=
∃ C : ℝ, 0<C ∧ ∃ c : ℝ, 0<c ∧
∀ (L q : ℕ), 2≤q → ∀ (θ : State L q), ‖θ‖=1 →
∀ (X P Y : Finset (Vertex L)), Disjoint X P → Disjoint X Y → Disjoint P Y →
∀ (σ : Matrix (RegionConfiguration q (X∪P)) (RegionConfiguration q (X∪P)) ℂ)
(τ : Matrix (RegionConfiguration q (X∪Y)) (RegionConfiguration q (X∪Y)) ℂ),
Subdensity σ → Subdensity τ → ∀ a : ℝ, 0<a → a*ell q X≤c →
‖moveVector θ X P Y σ τ a‖ ≤
Real.exp (-a*eta θ X P Y/2+C*Real.rpow a (5/4:ℝ)*(ell q X)^2)
end PolynomialPEPS.PhysicalMove.LocalMove
namespace PolynomialPEPS.PhysicalMove.LocalMovePhysical
open Matrix QuantumSSA SupportedCurve SpectralCurve MatrixInterpolation LocalMove
open scoped Matrix.Norms.L2Operator BigOperators ComplexOrder Kronecker
variable {L q : ℕ}
theorem asMap_mul_apply (A B : Operator L q) (ψ : State L q) :
asMap (A*B) ψ=asMap A (asMap B ψ) := by
simp only [asMap,map_mul,mul_apply_eq_comp]
theorem fourCoefficient_move (θ : State L q) (X P Y : Finset (Vertex L))
(hXP : Disjoint X P) (hXY : Disjoint X Y) (hPY : Disjoint P Y)
(σ : Matrix (RegionConfiguration q (X∪P)) (RegionConfiguration q (X∪P)) ℂ)
(τ : Matrix (RegionConfiguration q (X∪Y)) (RegionConfiguration q (X∪Y)) ℂ)
(hσ : σ.PosSemidef) (hτ : τ.PosSemidef) (a : ℝ) :
reshuffle (fourCoefficient (moveVector θ X P Y σ τ a) X Y P hXY hXP hPY.symm)=
cfc (fun t : ℝ => Real.rpow t (a/2)) (reindexHom (unionEquiv X P hXP).symm σ)*
tensorRightHom (spectralPower (reducedDensity θ P) (-a/2))*
reshuffle (cfc (fun t : ℝ => Real.rpow t (a/2)) (reindexHom (unionEquiv X Y hXY).symm τ)*
tensorRightHom (spectralPower (reducedDensity θ Y) (-a/2))*
fourCoefficient θ X Y P hXY hXP hPY.symm) := by
simp only [moveVector,asMap_mul_apply]
rw [fourCoefficient_action_XP,fourCoefficient_action_P,
fourCoefficient_action_XY,fourCoefficient_action_Y,
cfc_reindex _ _ hσ.isHermitian,cfc_reindex _ _ hτ.isHermitian]
simp only [spectralPower,Matrix.mul_assoc]
rfl
theorem fourCoefficient_eta (θ : State L q) (X P Y : Finset (Vertex L))
(hXP : Disjoint X P) (hXY : Disjoint X Y) (hPY : Disjoint P Y) :
QuantumSSA.conditionalEntropy (fourCoefficient θ X Y P hXY hXP hPY.symm*
(fourCoefficient θ X Y P hXY hXP hPY.symm).conjTranspose)+
QuantumSSA.conditionalEntropy (reshuffle (fourCoefficient θ X Y P hXY hXP hPY.symm)*
(reshuffle (fourCoefficient θ X Y P hXY hXP hPY.symm)).conjTranspose)=eta θ X P Y := by
rw [QuantumSSA.conditionalEntropy,QuantumSSA.conditionalEntropy,
fourCoefficient_gram,fourCoefficient_reshuffle_gram,
density_union_ptrL,density_union_ptrL,
entropy_reindexHom _ (reducedDensity_isHermitian θ (X∪Y)),
entropy_reindexHom _ (reducedDensity_isHermitian θ (X∪P))]
simp only [traceEntropy_spectral _ (reducedDensity_isHermitian _ _),eta,vonNeumannEntropy]
ring
theorem one_copy_move_bound : OneCopyMoveBound := by
refine ⟨200000,by norm_num,1/8,by norm_num,?_⟩
intro L q hq θ hθ X P Y hXP hXY hPY σ τ hσ hτ a ha hsmall
have hqpos : 0<q := by omega
let : NeZero q := ⟨ne_of_gt hqpos⟩
let C := fourCoefficient θ X Y P hXY hXP hPY.symm
have hC : hsEnergy C=1 := by
have hh := coefficient_hsEnergy θ (X∪Y) (unionEquiv X Y hXY)
(complementEquiv (X∪Y) P ((X∪Y)∪P)ᶜ (Finset.disjoint_union_left.mpr ⟨hXP,hPY.symm⟩) rfl)
simpa only [C,fourCoefficient,hθ,one_pow] using hh
have hdim : (1:ℝ)≤(Fintype.card (RegionConfiguration q X):ℝ) := by
exact_mod_cast (Nat.succ_le_iff.mpr (Fintype.card_pos_iff.mpr
(inferInstance : Nonempty (RegionConfiguration q X))))
have hlid : ell q X=1+Real.log (Fintype.card (RegionConfiguration q X):ℝ) := by
rw [ell,Real.log_mul (ne_of_gt (Real.exp_pos 1)) (ne_of_gt (by linarith)),Real.log_exp]
have hl : 1≤ell q X := by rw [hlid]; have := Real.log_nonneg hdim; linarith
have hl' : Real.log (Fintype.card (RegionConfiguration q X):ℝ)≤ell q X := by rw [hlid]; linarith
have hh := one_copy_move_cfc C hC
(reindexHom (unionEquiv X P hXP).symm σ) (reindexHom (unionEquiv X Y hXY).symm τ)
(reindexHom_posSemidef _ hσ.1) (reindexHom_posSemidef _ hτ.1)
(by rw [reindexHom_trace]; exact hσ.2) (by rw [reindexHom_trace]; exact hτ.2)
a (ell q X) ha hl hl' hsmall
have hY : ptrL (C*C.conjTranspose)=reducedDensity θ Y := by
dsimp only [C]
rw [fourCoefficient_gram,density_union_ptrL]
have hP : ptrL (reshuffle C*(reshuffle C).conjTranspose)=reducedDensity θ P := by
dsimp only [C]
rw [fourCoefficient_reshuffle_gram,density_union_ptrL]
have he : QuantumSSA.conditionalEntropy (C*C.conjTranspose)+
QuantumSSA.conditionalEntropy (reshuffle C*(reshuffle C).conjTranspose)=eta θ X P Y :=
fourCoefficient_eta θ X P Y hXP hXY hPY
rw [hY,hP,he] at hh
change ‖flattenCLM (_*tensorRightHom (spectralPower (reducedDensity θ P) (-a/2))*
reshuffle (_*tensorRightHom (spectralPower (reducedDensity θ Y) (-a/2))*
fourCoefficient θ X Y P hXY hXP hPY.symm))‖≤_ at hh
rw [←fourCoefficient_move θ X P Y hXP hXY hPY σ τ hσ.1 hτ.1 a,
fourCoefficient_norm] at hh
exact hh
end PolynomialPEPS.PhysicalMove.LocalMovePhysical
end
end OAI
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