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feat(MatrixNorm/TraceNorm): trace-norm/SVD/Hölder API #1076

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feat(MatrixNorm/TraceNorm): trace-norm/SVD/Hölder API #1076

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15 changes: 6 additions & 9 deletions QuantumInfo/Finite/Distance/TraceDistance.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -34,14 +34,11 @@ theorem ge_zero : 0 ≤ TrDistance ρ σ := by
rw [TrDistance]
simp [Matrix.traceNorm_nonneg]

theorem le_one : TrDistance ρ σ ≤ 1 :=
calc TrDistance ρ σ
_ = (1/2:ℝ) * (ρ.m - σ.m).traceNorm := by rfl
_ ≤ (1/2:ℝ) * (ρ.m.traceNorm + σ.m.traceNorm) := by
linarith [Matrix.traceNorm_triangleIneq' ρ.m σ.m]
_ = (1/2:ℝ) * (1 + 1) := by
rw [ρ.traceNorm_eq_1, σ.traceNorm_eq_1]
_ = 1 := by norm_num
theorem le_one : TrDistance ρ σ ≤ 1 := by
have htri := Matrix.traceNorm_add_le ρ.m (-σ.m)
simp [TrDistance, sub_eq_add_neg, Matrix.traceNorm_neg,
ρ.traceNorm_eq_1, σ.traceNorm_eq_1] at htri ⊢
linarith

/-- The trace distance, as a `Prob` probability with value between 0 and 1. -/
def prob : Prob :=
Expand All @@ -50,7 +47,7 @@ def prob : Prob :=
/-- The trace distance is a symmetric quantity. -/
theorem symm : TrDistance ρ σ = TrDistance σ ρ := by
dsimp [TrDistance]
rw [← Matrix.traceNorm_eq_neg_self, neg_sub]
rw [← Matrix.traceNorm_neg, neg_sub]

/-- The trace distance is equal to half the 1-norm of the eigenvalues of their difference . -/
theorem eq_abs_eigenvalues : TrDistance ρ σ = (1/2:ℝ) *
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2 changes: 1 addition & 1 deletion QuantumInfo/Finite/MState.lean
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Expand Up @@ -1238,7 +1238,7 @@ theorem traceRight_right_assoc' (ρ : MState (d₁ × d₂ × d₃)) :
@[simp]
theorem traceNorm_eq_1 (ρ : MState d) : ρ.m.traceNorm = 1 :=
have := calc (ρ.m.traceNorm : ℂ)
_ = ρ.m.trace := ρ.psd.traceNorm_PSD_eq_trace
_ = ρ.m.trace := ρ.psd.traceNorm_eq_trace
_ = 1 := ρ.tr'
Complex.ofReal_eq_one.mp this

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