Choi of kraus R
MatrixMap.choi_of_kraus_R
Plain-language statement
The Choi matrix of a map in symmetric Kraus form is a sum of rank-1 projectors.
Exact Lean statement
theorem choi_of_kraus_R [DecidableEq A] (K : κ → Matrix B A 𝕜) :
(of_kraus K K).choi_matrix = ∑ k, Matrix.vecMulVec (fun (x : B × A) => K k x.1 x.2) (fun (x : B × A) => star (K k x.1 x.2))Formal artifact
Lean source
theorem choi_of_kraus_R [DecidableEq A] (K : κ → Matrix B A 𝕜) : (of_kraus K K).choi_matrix = ∑ k, Matrix.vecMulVec (fun (x : B × A) => K k x.1 x.2) (fun (x : B × A) => star (K k x.1 x.2)) := by unfold of_kraus choi_matrix ext i j : 2 simp only [LinearMap.coe_sum, LinearMap.coe_mk, AddHom.coe_mk, Finset.sum_apply, Matrix.sum_apply, Matrix.mul_apply, Matrix.conjTranspose_apply, RCLike.star_def, Matrix.vecMulVec, Matrix.of_apply] simp only [Matrix.single, Matrix.of_apply, mul_ite, mul_one, mul_zero] simp +contextual only [Finset.sum_ite, Finset.filter_and, Finset.filter_const, not_and, Finset.sum_const_zero, add_zero]; congr! with k simp_all only [Finset.mem_univ, Finset.filter_eq, ↓reduceIte] rw [Finset.sum_eq_single j.2] · simp · aesop · simp- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/CPTPMap/Unbundled.lean:627-642
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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.