File size: 4,032 Bytes
67aee1d | 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 | import OAI.Analysis.FreeEntropy.TotalSecond
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped ENNReal Topology
namespace FiniteEntropySeparation
variable {p d : ℕ}
theorem matrixHeatLaw_block_lower (hp : 2≤p) (hd : 0<d) {t v : ℝ} (ht : 0<t)
(hv : 1≤v) (htv : t≤v) (hpt : 256*t≤p)
(μ : Measure (MatrixCoordinates (p+1) d)) [IsProbabilityMeasure μ]
(hμ : ∀ᵐa ∂μ,‖a‖≤2*Real.sqrt ((p:ℝ)*d) ∧
∃f : Fin d → Fin p,balancedBlocks f ∧ ∃U : unitary (Matrix (Fin d) (Fin d) ℂ),
Real.sqrt ((p:ℝ)*d)/2≤‖coordinateMask (blockMask f) (rotateCoordinates U a)‖) :
ENNReal.ofReal (1/2:ℝ)≤ matrixHeatLaw μ t
(blockEvent p d (Real.sqrt ((p:ℝ)*d)/4) (4*Real.sqrt (v*p*d))) := by
have htd : 0<t/(d:ℝ) := div_pos ht (by exact_mod_cast hd)
let E := blockEvent p d (Real.sqrt ((p:ℝ)*d)/4) (4*Real.sqrt (v*p*d))
have hE : MeasurableSet E := (blockEvent_isClosed hd _ _).measurableSet
change _≤(volume.withDensity (fun x => ENNReal.ofReal (smoothedDensity μ (t/d) x))) E
rw [← smoothedLaw_eq μ htd,Measure.map_apply (by fun_prop) hE]
have : IsProbabilityMeasure (normalLaw (ι := MatrixIndex (p+1) d) (t/d)) := normalLaw_probability htd
rw [Measure.prod_apply (hE.preimage (show Measurable (fun z : MatrixCoordinates (p+1) d × MatrixCoordinates (p+1) d => z.1+z.2) from by fun_prop))]
calc
_ = ∫⁻a : MatrixCoordinates (p+1) d,ENNReal.ofReal (1/2:ℝ) ∂μ := by simp
_ ≤ _ := lintegral_mono_ae (hμ.mono (fun a ha => blockEvent_conditional_lower hp hd ht hv htv hpt a ha.1 ha.2))
namespace Model
variable (p : ℕ) [NeZero p]
theorem diagonalMicrostates_blocks_eventually {δ Δ : ℝ} (hδ : 0<δ) (hΔ : 0<Δ) (hδΔ : δ≤Δ/4)
(hsep : ∀i j : Fin p,i≠j → Δ≤|equidiscreteAtom p i-equidiscreteAtom p j|)
(hq : ∀l : Fin p,∀r : ℝ,|r-equidiscreteAtom p l|≤δ → (3:ℝ)/4≤((spectralBasis p l).eval r)^2)
{h : ℝ} (hh : spectralGood p δ Δ (wordMoment (τ p) (X p h))) (R : ℝ) :
∀ᶠk : ℕ in atTop,∀d : ℕ,0<d → ∀a : MatrixCoordinates (p+1) d,
a∈diagonalMicrostates (τ p) (X p h) R k d →
∃f : Fin d → Fin p,balancedBlocks f ∧ ∃U : unitary (Matrix (Fin d) (Fin d) ℂ),
Real.sqrt ((p:ℝ)*d)/2≤‖coordinateMask (blockMask f) (rotateCoordinates U a)‖ := by
have he := diagonalMicrostates_uniform_open (τ p) (X p h) R {m | spectralGood p δ Δ m}
(spectralGood_isOpen p δ Δ) hh
filter_upwards [he] with k hk d hd a ha
obtain ⟨f,hf,U,hU⟩ := spectralGood_blocks hd hδ hΔ hδΔ hsep hq a (hk d a ha)
refine ⟨f,hf,U,?_⟩
apply (sq_le_sq₀ (by positivity) (norm_nonneg _)).mp
rw [div_pow,Real.sq_sqrt (by positivity : 0≤(p:ℝ)*d)]
nlinarith
theorem matrixHeatLaw_blocks_eventually (hp : 2≤p) {δ Δ : ℝ} (hδ : 0<δ) (hΔ : 0<Δ) (hδΔ : δ≤Δ/4)
(hsep : ∀i j : Fin p,i≠j → Δ≤|equidiscreteAtom p i-equidiscreteAtom p j|)
(hq : ∀l : Fin p,∀r : ℝ,|r-equidiscreteAtom p l|≤δ → (3:ℝ)/4≤((spectralBasis p l).eval r)^2)
{h t : ℝ} (hh0 : 0<h) (hh1 : h<1) (hh : spectralGood p δ Δ (wordMoment (τ p) (X p h)))
(ht : 0<t) (hpt : 256*t≤p) (R : ℝ)
(D : ℕ → ℕ) (hD : ∀k,0<D k)
(μ : (k:ℕ) → Measure (MatrixCoordinates (p+1) (D k))) [∀k,IsProbabilityMeasure (μ k)]
(hμ : ∀k,∀ᵐa ∂μ k,a∈diagonalMicrostates (τ p) (X p h) R k (D k)) :
∀ᶠk in atTop,ENNReal.ofReal (1/2:ℝ)≤ matrixHeatLaw (μ k) t
(blockEvent p (D k) (Real.sqrt ((p:ℝ)*D k)/4) (4*Real.sqrt ((1+h+t)*p*D k))) := by
filter_upwards [diagonalMicrostates_norm_eventually p hp hh0.le hh1 R,
diagonalMicrostates_blocks_eventually p hδ hΔ hδΔ hsep hq hh R] with k hk hb
apply matrixHeatLaw_block_lower hp (hD k) ht (by linarith) (by linarith) hpt (μ k)
filter_upwards [hμ k] with a ha
exact ⟨hk (D k) (hD k) a ha,hb (D k) (hD k) a ha⟩
end Model
end FiniteEntropySeparation
end
end OAI
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