Rpow mul neg rpow eq support Proj
rpow_mul_neg_rpow_eq_supportProj
Plain-language statement
For PSD A and γ ≠ 0, the product A^γ * A^{-γ} equals the support projection of A. This is because x^γ * x^{-γ} = if x = 0 then 0 else 1 for x ≥ 0.
Exact Lean statement
lemma rpow_mul_neg_rpow_eq_supportProj {A : HermitianMat d ℂ}
(hA : 0 ≤ A) (γ : ℝ) (hγ : γ ≠ 0) :
(A ^ γ).mat * (A ^ (-γ)).mat = A.supportProj.matFormal artifact
Lean source
lemma rpow_mul_neg_rpow_eq_supportProj {A : HermitianMat d ℂ} (hA : 0 ≤ A) (γ : ℝ) (hγ : γ ≠ 0) : (A ^ γ).mat * (A ^ (-γ)).mat = A.supportProj.mat := by rw [HermitianMat.supportProj_eq_cfc]; rw [HermitianMat.rpow_eq_cfc, HermitianMat.rpow_eq_cfc]; rw [ ← HermitianMat.mat_cfc_mul_apply ]; refine congr_arg _ ( HermitianMat.cfc_congr_of_nonneg hA ?_) intro x (hx : 0 ≤ x) rcases eq_or_ne x 0 with hx' | hx' · simp [hx', hγ] · simp [hx', Real.rpow_neg hx] exact mul_inv_cancel₀ (by positivity)- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entropy/DPI.lean:491-502
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Source project: quantumInfo
Person-level attribution pending.
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Source project: quantumInfo
Person-level attribution pending.
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Plain-language statement
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Source project: quantumInfo
Person-level attribution pending.