Hermitian Mat ker weighted sum le
HermitianMat.ker_weighted_sum_le
Plain-language statement
If for all i, ker(σs i) ≤ ker(ρs i), then ker(∑ w i • σs i) ≤ ker(∑ w i • ρs i), provided all weights are nonneg and all matrices are PSD.
Exact Lean statement
theorem HermitianMat.ker_weighted_sum_le {ι : Type*} [Fintype ι]
(w : ι → ℝ) (hw_nonneg : ∀ i, 0 ≤ w i)
(ρs σs : ι → HermitianMat d ℂ)
(hρs_nonneg : ∀ i, 0 ≤ ρs i)
(hσs_nonneg : ∀ i, 0 ≤ σs i)
(hker : ∀ i, (σs i).ker ≤ (ρs i).ker) :
(∑ i, w i • σs i).ker ≤ (∑ i, w i • ρs i).kerFormal artifact
Lean source
theorem HermitianMat.ker_weighted_sum_le {ι : Type*} [Fintype ι] (w : ι → ℝ) (hw_nonneg : ∀ i, 0 ≤ w i) (ρs σs : ι → HermitianMat d ℂ) (hρs_nonneg : ∀ i, 0 ≤ ρs i) (hσs_nonneg : ∀ i, 0 ≤ σs i) (hker : ∀ i, (σs i).ker ≤ (ρs i).ker) : (∑ i, w i • σs i).ker ≤ (∑ i, w i • ρs i).ker := by rw [HermitianMat.ker_sum, HermitianMat.ker_sum] · refine iInf_mono fun i ↦ ?_ by_cases hi : w i = 0 · simp [hi] · simp_all [HermitianMat.ker_pos_smul] · exact fun i => smul_nonneg (hw_nonneg i) (hρs_nonneg i) · exact fun i => smul_nonneg (hw_nonneg i) (hσs_nonneg i)- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entropy/DPI.lean:769-782
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Project documentation
The operator norm of the conjugate transpose is equal to the operator norm. -/ theorem Matrix.opNorm_conjTranspose_eq_opNorm {m n : Type*} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] (A : Matrix m n 𝕜) : Matrix.opNorm Aᴴ = Matrix.opNorm A := by unfold Matrix.opNorm rw [← ContinuousLinearMap.adjoint.norm_map (toEuclideanLin A).toContinuousLine...
Source project: quantumInfo
Person-level attribution pending.
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convex_roof_of_pure
Plain-language statement
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Source project: quantumInfo
Person-level attribution pending.
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Plain-language statement
The identity channel on D dimensional space achieves a rate of log2(D).
Source project: quantumInfo
Person-level attribution pending.