Plain-language statement
The composition of IsCompletelyPositive maps is also completely positive.
Exact Lean statement
theorem comp [DecidableEq B] {M₁ : MatrixMap A B R} {M₂ : MatrixMap B C R} (h₁ : M₁.IsCompletelyPositive)
(h₂ : M₂.IsCompletelyPositive) : IsCompletelyPositive (M₂ ∘ₗ M₁)Formal artifact
Lean source
theorem comp [DecidableEq B] {M₁ : MatrixMap A B R} {M₂ : MatrixMap B C R} (h₁ : M₁.IsCompletelyPositive) (h₂ : M₂.IsCompletelyPositive) : IsCompletelyPositive (M₂ ∘ₗ M₁) := by --sketch: (M₂ ∘ₗ M₁) ⊗ₖₘ id[n] = (M₂ ⊗ₖₘ id[n]) ∘ₗ (M₁ ⊗ₖₘ id[n]), which is a composition of positive maps. intro n x hx specialize h₁ n hx specialize h₂ n h₁ conv in LinearMap.id => change LinearMap.id ∘ₗ LinearMap.id rw [kron_comp_distrib] simpa using h₂- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/CPTPMap/Unbundled.lean:304-313
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Related declarations
Conj Transpose isometry mul isometry le one
conjTranspose_isometry_mul_isometry_le_one
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.
Convex roof of pure
convex_roof_of_pure
Plain-language statement
The convex roof extension of g : KetUpToPhase d → ℝ≥0 applied to a pure state ψ is g (KetUpToPhase.mk ψ).
Source project: quantumInfo
Person-level attribution pending.
Id achieves Rate log dim
CPTPMap.id_achievesRate_log_dim
Plain-language statement
The identity channel on D dimensional space achieves a rate of log2(D).
Source project: quantumInfo
Person-level attribution pending.