| import OAI.LinearAlgebra.MatrixState.QuarterRoot |
| import OAI.RepresentationTheory.RowColumn.Sectors |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators Classical MatrixOrder Matrix.Norms.L2Operator ComplexOrder |
| namespace RowColumn.MatrixState |
| variable {C L : Type*} [Fintype C] [DecidableEq C] [Fintype L] |
|
|
| def stateMean (p : L → ℝ) (S : L → Matrix C C ℂ) : Matrix C C ℂ := ∑ i, p i • S i |
|
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| def whitenState (A S : Matrix C C ℂ) : Matrix C C ℂ := inverseQuarter A * S * inverseQuarter A |
|
|
| omit [Fintype C] [DecidableEq C] in |
| lemma stateMean_positive (p : L → ℝ) (S : L → Matrix C C ℂ) |
| (hp : ∀ i, 0 ≤ p i) (hS : ∀ i, (S i).PosSemidef) : (stateMean p S).PosSemidef := by |
| apply Matrix.nonneg_iff_posSemidef.mp |
| exact Finset.sum_nonneg fun i _ => ((hS i).smul (hp i)).nonneg |
|
|
| omit [Fintype C] [DecidableEq C] in |
| lemma smul_le_stateMean (p : L → ℝ) (S : L → Matrix C C ℂ) |
| (hp : ∀ i, 0 ≤ p i) (hS : ∀ i, (S i).PosSemidef) (i : L) : p i • S i ≤ stateMean p S := by |
| exact Finset.single_le_sum (fun j _ => ((hS j).smul (hp j)).nonneg) (Finset.mem_univ i) |
|
|
| omit [DecidableEq C] in |
| lemma stateMean_trace_le_one (p : L → ℝ) (S : L → Matrix C C ℂ) |
| (hp : ∀ i, 0 ≤ p i) (hpsum : ∑ i, p i=1) (hS : ∀ i, (S i).trace.re ≤ 1) : |
| (stateMean p S).trace.re ≤ 1 := by |
| change (Matrix.trace (∑ i, p i • S i)).re ≤ 1 |
| rw [Matrix.trace_sum, Complex.re_sum] |
| calc |
| _ = ∑ i, p i * (S i).trace.re := by simp |
| _ ≤ ∑ i, p i := Finset.sum_le_sum fun i _ => by nlinarith [hS i,hp i] |
| _ = 1 := hpsum |
|
|
| lemma mean_support (p : L → ℝ) (S : L → Matrix C C ℂ) |
| (hp : ∀ i, 0 ≤ p i) (hS : ∀ i, (S i).PosSemidef) (i : L) (hi : 0 < p i) : |
| quarterSupport (stateMean p S) * S i = S i := by |
| have hh := quarterSupport_of_le (stateMean p S) (p i • S i) |
| (stateMean_positive p S hp hS) ((hS i).smul (hp i)) (smul_le_stateMean p S hp hS i) |
| rw [mul_smul_comm] at hh |
| exact (smul_right_injective _ (ne_of_gt hi)) hh |
|
|
| lemma whitenState_positive (A S : Matrix C C ℂ) (hS : S.PosSemidef) : |
| (whitenState A S).PosSemidef := by |
| simpa only [whitenState, (inverseQuarter_positive A).isHermitian.eq] using |
| hS.conjTranspose_mul_mul_same (inverseQuarter A) |
|
|
| lemma stateMean_whiten (p : L → ℝ) (S : L → Matrix C C ℂ) |
| (hp : ∀ i, 0 ≤ p i) (hS : ∀ i, (S i).PosSemidef) : |
| stateMean p (fun i => whitenState (stateMean p S) (S i)) = CFC.sqrt (stateMean p S) := by |
| change (∑ i, p i • (inverseQuarter (stateMean p S) * S i * inverseQuarter (stateMean p S))) = _ |
| rw [show (∑ i, p i • (inverseQuarter (stateMean p S) * S i * inverseQuarter (stateMean p S))) = |
| inverseQuarter (stateMean p S) * stateMean p S * inverseQuarter (stateMean p S) by |
| simp [stateMean, Finset.mul_sum, Finset.sum_mul]] |
| rw [quarter_whitening_mean _ (stateMean_positive p S hp hS), quarter_square] |
|
|
| lemma unwhitenState (p : L → ℝ) (S : L → Matrix C C ℂ) |
| (hp : ∀ i, 0 ≤ p i) (hS : ∀ i, (S i).PosSemidef) (i : L) (hi : 0 < p i) : |
| quarterRoot (stateMean p S) * whitenState (stateMean p S) (S i) * quarterRoot (stateMean p S) = S i := |
| undo_quarter_whitening _ _ (hS i).isHermitian (mean_support p S hp hS i hi) |
|
|
| lemma stateMean_even (odd : C → Prop) (p : L → ℝ) (S : L → Matrix C C ℂ) |
| (hS : ∀ i, S i ∈ evenAlgebra odd) : stateMean p S ∈ evenAlgebra odd := by |
| apply (evenAlgebra odd).sum_mem |
| intro i _ c d h |
| change p i • S i c d=0 |
| rw [hS i c d h, smul_zero] |
|
|
| lemma sqrt_even (odd : C → Prop) (A : Matrix C C ℂ) (hA : A.PosSemidef) |
| (he : A ∈ evenAlgebra odd) : CFC.sqrt A ∈ evenAlgebra odd := by |
| intro c d h |
| exact sqrt_block_supported odd A hA |
| (fun c d hh => he c d (fun hcd => hh (propext hcd))) c d |
| (fun he => h (Iff.of_eq he)) |
|
|
| lemma quarter_even (odd : C → Prop) (A : Matrix C C ℂ) (hA : A.PosSemidef) |
| (he : A ∈ evenAlgebra odd) : quarterRoot A ∈ evenAlgebra odd := |
| sqrt_even odd _ (Matrix.nonneg_iff_posSemidef.mp (CFC.sqrt_nonneg A)) (sqrt_even odd A hA he) |
|
|
| lemma inverseQuarter_even (odd : C → Prop) (A : Matrix C C ℂ) (hA : A.PosSemidef) |
| (he : A ∈ evenAlgebra odd) : inverseQuarter A ∈ evenAlgebra odd := by |
| let : IsClosed ((evenAlgebra odd) : Set (Matrix C C ℂ)) := |
| (evenAlgebra odd).toSubalgebra.toSubmodule.closed_of_finiteDimensional |
| exact cfc_mem (𝕜 := ℝ) (𝕜' := ℂ) (s := evenAlgebra odd) (fun x : ℝ => x⁻¹) (quarter_even odd A hA he) |
|
|
| lemma whitenState_even (odd : C → Prop) (A S : Matrix C C ℂ) (hA : A.PosSemidef) |
| (heA : A ∈ evenAlgebra odd) (heS : S ∈ evenAlgebra odd) : whitenState A S ∈ evenAlgebra odd := by |
| exact (evenAlgebra odd).mul_mem |
| ((evenAlgebra odd).mul_mem (inverseQuarter_even odd A hA heA) heS) |
| (inverseQuarter_even odd A hA heA) |
| end RowColumn.MatrixState |
|
|
| noncomputable section |
| open scoped BigOperators Classical MatrixOrder Matrix.Norms.L2Operator ComplexOrder |
| namespace RowColumn.MatrixState |
| variable {C L K : Type*} [Fintype C] [DecidableEq C] [Fintype L] [Fintype K] |
|
|
| / |
| with no invertibility assumption on either average state. -/ |
| lemma independent_overlap_mean (p : L → ℝ) (q : K → ℝ) |
| (S : L → Matrix C C ℂ) (T : K → Matrix C C ℂ) (U : Matrix C C ℂ) : |
| (∑ i, ∑ j, p i * q j * (S i * U * T j * U.conjTranspose).trace.re) = |
| (stateMean p S * U * stateMean q T * U.conjTranspose).trace.re := by |
| simp only [stateMean, Finset.sum_mul, Finset.mul_sum, smul_mul_assoc, |
| mul_smul_comm, Matrix.trace_sum, Matrix.trace_smul, Complex.re_sum] |
| simp only [Complex.real_smul, Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, |
| zero_mul, sub_zero, mul_assoc] |
| simp only [Complex.re_sum, Finset.mul_sum, Complex.mul_re, Complex.ofReal_re, |
| Complex.ofReal_im, zero_mul, sub_zero] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro j _ |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
|
|
| / |
| preserving singular substates exactly on every occupied fibre. -/ |
| theorem weighted_whitened_mean_le_one (p : L → ℝ) (q : K → ℝ) |
| (S : L → Matrix C C ℂ) (T : K → Matrix C C ℂ) |
| (hp : ∀ i, 0 ≤ p i) (hq : ∀ j, 0 ≤ q j) |
| (hpsum : ∑ i, p i = 1) (hqsum : ∑ j, q j = 1) |
| (hS : ∀ i, (S i).PosSemidef) (hT : ∀ j, (T j).PosSemidef) |
| (hStr : ∀ i, (S i).trace.re ≤ 1) (hTtr : ∀ j, (T j).trace.re ≤ 1) |
| (U : Matrix C C ℂ) (hU : U ∈ unitary (Matrix C C ℂ)) : |
| (∑ i, ∑ j, p i * q j * |
| (whitenState (stateMean p S) (S i) * U * |
| whitenState (stateMean q T) (T j) * U.conjTranspose).trace.re) ≤ 1 := by |
| rw [independent_overlap_mean, stateMean_whiten p S hp hS, stateMean_whiten q T hq hT] |
| exact trace_sqrt_conjugate_le_one _ _ U (stateMean_positive p S hp hS) |
| (stateMean_positive q T hq hT) (stateMean_trace_le_one p S hp hpsum hStr) |
| (stateMean_trace_le_one q T hq hqsum hTtr) hU |
|
|
| lemma local_overlap_nonneg (A B U : Matrix C C ℂ) (hA : A.PosSemidef) |
| (hB : B.PosSemidef) : 0 ≤ (A * U * B * U.conjTranspose).trace.re := by |
| simpa only [mul_assoc] using trace_mul_psd_nonneg A (U * B * U.conjTranspose) |
| hA (hB.mul_mul_conjTranspose_same U) |
|
|
| end RowColumn.MatrixState |
|
|
| end |
| end |
| end |
|
|
| end OAI |
|
|
