| import OAI.Probability.CubeShuffle.SchattenBounds |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators Classical |
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| namespace CubeShuffle.WeightedSweep |
| open Specht UnitaryFinite |
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| lemma finite_positive_lower_bound {ι : Type*} [Fintype ι] (f : ι → ℝ) |
| (hf : ∀ i, 0 < f i) : ∃ a : ℝ, 0 < a ∧ ∀ i, a ≤ f i := by |
| have hs (s : Finset ι) : ∃ a : ℝ, 0 < a ∧ ∀ i ∈ s, a ≤ f i := by |
| induction s using Finset.induction_on with |
| | empty => exact ⟨1, zero_lt_one, by simp⟩ |
| | @insert i s hi ih => |
| obtain ⟨a, ha, h⟩ := ih |
| refine ⟨min a (f i), lt_min ha (hf i), ?_⟩ |
| intro j hj |
| rcases Finset.mem_insert.mp hj with rfl | hj |
| · exact min_le_right _ _ |
| · exact (min_le_left _ _).trans (h j hj) |
| obtain ⟨a, ha, hs⟩ := hs Finset.univ |
| exact ⟨a, ha, fun i => hs i (Finset.mem_univ i)⟩ |
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| lemma uniformOperator_zero_of_dimension_gt_one {G V : Type*} [Group G] |
| [Fintype G] [NormedAddCommGroup V] [InnerProductSpace ℂ V] |
| [FiniteDimensional ℂ V] (ρ : Representation ℂ G V) |
| [Representation.IsIrreducible ρ] (hdim : 1 < Module.finrank ℂ V) : |
| uniformOperator ρ = 0 := by |
| apply ContinuousLinearMap.ext |
| intro v |
| change uniformOperator ρ v = 0 |
| let u := uniformOperator ρ v |
| have hfixed (g : G) : ρ g u = u := |
| congrArg (fun A : V →L[ℂ] V => A v) (uniformOperator_left ρ g) |
| let T : Representation.IntertwiningMap (Representation.trivial ℂ G ℂ) ρ := |
| { toLinearMap := LinearMap.toSpanSingleton ℂ V u |
| isIntertwining' g := by |
| apply LinearMap.ext |
| intro z |
| change z • u = ρ g (z • u) |
| rw [map_smul, hfixed] } |
| rcases Representation.IsIrreducible.surjective_or_eq_zero T with hs | hz |
| · have hd := LinearMap.finrank_le_finrank_of_surjective (f := T.toLinearMap) hs |
| simp only [Module.finrank_self] at hd |
| omega |
| · have he := congrArg (fun S : Representation.IntertwiningMap |
| (Representation.trivial ℂ G ℂ) ρ => S 1) hz |
| simpa [T] using he |
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| lemma dimension_full_row (μ : YoungDiagram) (hμ : μ.rowLen 0 = μ.card) : |
| dimension μ = 1 := by |
| let := tabloid_subsingleton_of_full_row μ hμ |
| have h := (space μ).finrank_le |
| rw [Module.finrank_fintype_fun_eq_card] at h |
| have hcard : Fintype.card (Tabloid μ) ≤ 1 := |
| Fintype.card_le_one_iff.mpr (fun _ _ => Subsingleton.elim _ _) |
| have he : Module.finrank ℂ (space μ) = 1 := by |
| have := Specht.dimension_pos μ |
| omega |
| simp [dimension, he] |
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| def localExponent (d : ℕ) (μ : Shapes (2^d)) : ℝ := |
| (1 - blockGap d) / (Real.log (dimension μ.1) + 1) |
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| lemma localExponent_pos (d : ℕ) (μ : Shapes (2^d)) : 0 < localExponent d μ := by |
| unfold localExponent |
| exact div_pos (sub_pos.mpr (blockGap_lt_one d)) |
| (by have := log_dimension_nonneg μ.1; linarith) |
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| lemma palindrome_local_bound (d : ℕ) (μ : Shapes (2^d)) : |
| ‖sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (palindromePerm d)‖ ≤ |
| Real.exp (-localExponent d μ * Real.log (dimension μ.1)) := by |
| let : NormedAddCommGroup (hilbertSpace μ.1) := inferInstance |
| let : InnerProductSpace ℂ (hilbertSpace μ.1) := inferInstance |
| let : FiniteDimensional ℂ (hilbertSpace μ.1) := inferInstance |
| by_cases hdim : dimension μ.1 = 1 |
| · rw [hdim, Real.log_one, mul_zero, Real.exp_zero] |
| exact sampleOperator_norm_le _ (cardRepresentation_unitary d μ) _ |
| have hgt : 1 < dimension μ.1 := |
| lt_of_le_of_ne (dimension_one_le μ.1) (Ne.symm hdim) |
| have hgt' : 1 < Module.finrank ℂ (hilbertSpace μ.1) := by |
| rw [finrank_hilbertSpace] |
| change 1 < (Module.finrank ℂ (space μ.1) : ℝ) at hgt |
| exact_mod_cast hgt |
| have hz := uniformOperator_zero_of_dimension_gt_one (V := hilbertSpace μ.1) |
| (cardRepresentation d μ) hgt' |
| have hgap := palindrome_off_invariants_gap d (V := hilbertSpace μ.1) (cardRepresentation d μ) |
| (cardRepresentation_unitary d μ) |
| rw [hz, sub_zero, ← sampleOperator_weighted] at hgap |
| have he : localExponent d μ * (Real.log (dimension μ.1) + 1) = 1 - blockGap d := by |
| exact div_mul_cancel₀ _ (ne_of_gt (by have := log_dimension_nonneg μ.1; linarith)) |
| have hnum : blockGap d - 1 ≤ -localExponent d μ * Real.log (dimension μ.1) := by |
| have := localExponent_pos d μ |
| nlinarith |
| have hexp : blockGap d ≤ Real.exp (blockGap d - 1) := by |
| simpa using Real.add_one_le_exp (blockGap d - 1) |
| exact hgap.trans (hexp.trans (Real.exp_le_exp.mpr hnum)) |
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| theorem uniform_palindrome_dimension_decay : |
| ∃ a : ℝ, 0 < a ∧ a ≤ 1 ∧ ∀ d : ℕ, 1 ≤ d → ∀ μ : Shapes (2^d), |
| ‖sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (palindromePerm d)‖ ≤ |
| Real.exp (-a * Real.log (dimension μ.1)) := by |
| obtain ⟨a₀, h, ha₀, hh, hevent⟩ := spectral_regimes |
| obtain ⟨d₀, hlarge⟩ := Filter.eventually_atTop.mp hevent |
| let ι := (d : Fin d₀) × Shapes (2^d.1) |
| obtain ⟨a₁, ha₁, hsmall⟩ := finite_positive_lower_bound |
| (fun i : ι => localExponent i.1.1 i.2) (fun i => localExponent_pos _ _) |
| let a := min 1 (min a₀ a₁) |
| have ha : 0 < a := lt_min zero_lt_one (lt_min ha₀ ha₁) |
| have ha₀' : a ≤ a₀ := (min_le_right _ _).trans (min_le_left _ _) |
| have ha₁' : a ≤ a₁ := (min_le_right _ _).trans (min_le_right _ _) |
| refine ⟨a, ha, min_le_left _ _, ?_⟩ |
| intro d hd μ |
| by_cases hdim : dimension μ.1 = 1 |
| · rw [hdim, Real.log_one, mul_zero, Real.exp_zero] |
| exact sampleOperator_norm_le _ (cardRepresentation_unitary d μ) _ |
| by_cases hd₀ : d₀ ≤ d |
| · by_cases hc : (μ.1.card : ℝ) / 2 < μ.1.colLen 0 |
| · have hz := palindrome_tall_zero_general d (by omega) μ.1 (cardLabels d μ) hc |
| change sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) |
| (palindromePerm d) = 0 at hz |
| rw [hz, norm_zero (E := (hilbertSpace μ.1 →L[ℂ] hilbertSpace μ.1))] |
| exact (Real.exp_pos _).le |
| · have hj : 0 < μ.1.card - μ.1.rowLen 0 := by |
| by_contra hn |
| have hrow : μ.1.rowLen 0 = μ.1.card := by |
| have := rowLen_zero_le_card μ.1 |
| omega |
| exact hdim (dimension_full_row μ.1 hrow) |
| have hspec := (hlarge d hd₀ μ.1 (cardLabels d μ) hj (not_lt.mp hc)).1 |
| apply hspec.trans (Real.exp_le_exp.mpr ?_) |
| have hmul := mul_le_mul_of_nonneg_right ha₀' (log_dimension_nonneg μ.1) |
| simpa only [neg_mul, dimension] using neg_le_neg hmul |
| · have hbase := palindrome_local_bound d μ |
| apply hbase.trans (Real.exp_le_exp.mpr ?_) |
| have hs := ha₁'.trans (hsmall ⟨⟨d, by omega⟩, μ⟩) |
| have hmul := mul_le_mul_of_nonneg_right hs (log_dimension_nonneg μ.1) |
| nlinarith |
|
|
| theorem uniform_sweep_dimension_decay : |
| ∃ b : ℝ, 0 < b ∧ b ≤ 1 ∧ ∀ d : ℕ, 1 ≤ d → ∀ μ : Shapes (2^d), |
| ‖fourierSweep d μ‖ ≤ dimension μ.1 ^ (-b) := by |
| obtain ⟨a, ha, ha1, h⟩ := uniform_palindrome_dimension_decay |
| refine ⟨a / 2, by positivity, by linarith, ?_⟩ |
| intro d hd μ |
| have hs := h d hd μ |
| rw [← physical_palindrome_norm d _ (cardRepresentation_unitary d μ)] at hs |
| have he : (Real.exp (-a * Real.log (dimension μ.1) / 2)) ^ 2 = |
| Real.exp (-a * Real.log (dimension μ.1)) := by |
| rw [pow_two, ← Real.exp_add] |
| congr 1 |
| ring |
| have hnorm : ‖fourierSweep d μ‖ ≤ Real.exp (-a * Real.log (dimension μ.1) / 2) := by |
| apply le_of_sq_le_sq ?_ (Real.exp_pos _).le |
| exact hs.trans_eq he.symm |
| convert hnorm using 1 |
| rw [Real.rpow_def_of_pos (dimension_pos μ.1)] |
| congr 1 |
| ring |
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| end CubeShuffle.WeightedSweep |
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| end |
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| end OAI |
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