| import OAI.Probability.CubeShuffle.FourierBound |
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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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| def dimension (μ : YoungDiagram) : ℝ := Module.finrank ℂ (space μ) |
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| def sweepLaw (d : ℕ) : Equiv.Perm (Card d) → ℝ := sampleLaw (run d d) |
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| lemma sweepLaw_nonneg (d : ℕ) (g : Equiv.Perm (Card d)) : 0 ≤ sweepLaw d g := |
| sampleLaw_nonneg _ _ |
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| lemma sweepLaw_sum (d : ℕ) : ∑ g, sweepLaw d g = 1 := sampleLaw_sum _ |
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| def fourierSweep (d : ℕ) (μ : Shapes (2^d)) : |
| hilbertSpace μ.1 →L[ℂ] hilbertSpace μ.1 := |
| sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d d) |
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| lemma fourierSweep_eq_group_sum (d : ℕ) (μ : Shapes (2 ^ d)) : |
| fourierSweep d μ = ∑ g : Equiv.Perm (Card d), (sweepLaw d g : ℂ) • |
| continuousRepresentation (V := hilbertSpace μ.1) (cardRepresentation d μ) g := by |
| let : NormedAddCommGroup (hilbertSpace μ.1) := inferInstance |
| let : InnerProductSpace ℂ (hilbertSpace μ.1) := inferInstance |
| let : FiniteDimensional ℂ (hilbertSpace μ.1) := inferInstance |
| exact sampleOperator_weighted (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d d) |
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| def schattenMoment {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] |
| [FiniteDimensional ℂ V] (A : V →L[ℂ] V) (p : ℝ) : ℝ := |
| ∑ i : Fin (Module.finrank ℂ V), (A.toLinearMap.singularValues i) ^ p |
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| def schattenNorm {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] |
| [FiniteDimensional ℂ V] (A : V →L[ℂ] V) (p : ℝ) : ℝ := |
| schattenMoment A p ^ (1 / p) |
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| end CubeShuffle.WeightedSweep |
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| end |
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| end OAI |
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