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f18e7de | 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 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 | 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
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]
/-- The independent marginal mean of the local squared Hilbert--Schmidt factors,
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
/-- Quarter-root whitening makes the independent mean at most one, while
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
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