| import Mathlib |
| import OAI.Analysis.FreeWords.MatrixAverageTrace |
| import OAI.Analysis.FreeWords.RankOneCompression |
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|
| namespace OAI |
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| noncomputable section |
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| namespace KKMatrixQuantization |
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| open KKTraceTransfer KKWeingarten MeasureTheory |
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| open scoped BigOperators Matrix.Norms.L2Operator |
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| noncomputable section |
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| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
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| variable {M : VonNeumannAlgebra H} (τ : FaithfulNormalTrace M) |
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| variable {d : ℕ} (ρ : Matrix (Fin d) (Fin d) ℂ →⋆ₐ[ℂ] ConcreteAlgebra M) |
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| theorem traceNorm_compression_sq (q x : ConcreteAlgebra M) (hq : IsStarProjection q) : |
| (((traceNorm τ (q*x*q))^2 : ℝ) : ℂ)=τ.map (q*star x*q*x) := by |
| rw [traceNorm_sq] |
| have hq2 : q*q=q := hq.isIdempotentElem |
| have hqs : star q=q := hq.isSelfAdjoint |
| simp only [star_mul,hqs] |
| calc |
| τ.map (q*(star x*q)*(q*x*q)) = τ.map ((q*star x*(q*q)*x)*q) := by congr 1 |
| _ = τ.map (q*(q*star x*q*x)) := by rw [hq2,τ.tracial] |
| _ = _ := by simp only [← mul_assoc,hq2] |
|
|
| theorem average_traceNorm_sq [NeZero d] (x : ConcreteAlgebra M) : |
| τ.map (matrixAverage ρ (star x)*x)=(((traceNorm τ (matrixAverage ρ x))^2 : ℝ) : ℂ) := by |
| rw [traceNorm_sq, matrixAverage_star] |
| calc |
| τ.map (matrixAverage ρ (star x)*x) = τ.map (star x*matrixAverage ρ x) := matrixAverage_trace_symm ρ τ _ _ |
| _ = τ.map (matrixAverage ρ (star x)*matrixAverage ρ x) := by |
| rw [matrixAverage_trace_symm,matrixAverage_idempotent] |
|
|
| theorem continuous_compression_energy (z : Fin d) (x : ConcreteAlgebra M) : |
| Continuous (fun U => (traceNorm τ (rankOne ρ z U*x*rankOne ρ z U))^2) := by |
| have he (U : Matrix.unitaryGroup (Fin d) ℂ) : |
| (traceNorm τ (rankOne ρ z U*x*rankOne ρ z U))^2= |
| (τ.map (rankOne ρ z U*star x*rankOne ρ z U*x)).re := by |
| simpa only [Complex.ofReal_re] using congrArg Complex.re |
| (traceNorm_compression_sq τ _ x (rankOne_projection ρ z U)) |
| simp_rw [he] |
| exact Complex.continuous_re.comp (continuous_rankOne_pair τ ρ z (star x) x) |
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|
| theorem integral_compression_energy (hd : 2 ≤ d) (z : Fin d) (x : ConcreteAlgebra M) : |
| (∫ U : Matrix.unitaryGroup (Fin d) ℂ, |
| (traceNorm τ (rankOne ρ z U*x*rankOne ρ z U))^2 ∂matrixHaar d)= |
| ((traceNorm τ x)^2+(d : ℝ)*(traceNorm τ (matrixAverage ρ x))^2)/((d : ℝ)*((d : ℝ)+1)) := by |
| have : NeZero d := ⟨by omega⟩ |
| have he := integral_rankOne_pair τ ρ hd z (star x) x |
| rw [← traceNorm_sq,average_traceNorm_sq] at he |
| have hU (U : Matrix.unitaryGroup (Fin d) ℂ) : |
| τ.map (rankOne ρ z U*star x*rankOne ρ z U*x)= |
| (((traceNorm τ (rankOne ρ z U*x*rankOne ρ z U))^2 : ℝ) : ℂ) := |
| (traceNorm_compression_sq τ _ x (rankOne_projection ρ z U)).symm |
| simp_rw [hU] at he |
| rw [integral_complex_ofReal] at he |
| rw [mul_comm,← div_eq_mul_inv] at he |
| exact_mod_cast he |
|
|
| theorem integral_compression_sum {ι : Type*} [Fintype ι] |
| (hd : 2 ≤ d) (z : Fin d) (x : ι → ConcreteAlgebra M) : |
| (∫ U : Matrix.unitaryGroup (Fin d) ℂ, |
| ∑ i,(traceNorm τ (rankOne ρ z U*x i*rankOne ρ z U))^2 ∂matrixHaar d)= |
| ((∑ i,(traceNorm τ (x i))^2)+(d : ℝ)*∑ i,(traceNorm τ (matrixAverage ρ (x i)))^2)/ |
| ((d : ℝ)*((d : ℝ)+1)) := by |
| rw [integral_finsetSum _ (fun i _ => (continuous_compression_energy τ ρ z (x i)).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))] |
| simp only [integral_compression_energy τ ρ hd z,← Finset.sum_div,Finset.sum_add_distrib,Finset.mul_sum] |
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| theorem exists_rankOne_small_compression {ι : Type*} [Fintype ι] |
| (hd : 2 ≤ d) (z : Fin d) (x : ι → ConcreteAlgebra M) (ε : ℝ) |
| (hb : ((∑ i,(traceNorm τ (x i))^2)+(d : ℝ)*∑ i,(traceNorm τ (matrixAverage ρ (x i)))^2)/ |
| ((d : ℝ)*((d : ℝ)+1)) < ε/(d : ℝ)) : |
| ∃ U : Matrix.unitaryGroup (Fin d) ℂ, |
| ∑ i,(traceNorm τ (rankOne ρ z U*x i*rankOne ρ z U))^2 < ε/(d : ℝ) := by |
| by_contra hh |
| push Not at hh |
| have hi : Integrable (fun U : Matrix.unitaryGroup (Fin d) ℂ => |
| ∑ i,(traceNorm τ (rankOne ρ z U*x i*rankOne ρ z U))^2) (matrixHaar d) := by |
| apply integrable_finsetSum |
| intro i hi |
| exact (continuous_compression_energy τ ρ z (x i)).integrable_of_hasCompactSupport |
| (HasCompactSupport.of_compactSpace _) |
| have hle := integral_mono (integrable_const (ε/(d : ℝ))) hi hh |
| rw [integral_const,integral_compression_sum τ ρ hd z x] at hle |
| simp only [probReal_univ,smul_eq_mul,one_mul] at hle |
| linarith |
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| end |
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| end KKMatrixQuantization |
| end |
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| end OAI |
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