Research proof index

The Choi matrix of a map in symmetric Kraus form is a sum of rank-1 projectors. -/ 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, LinearMa...

The linear map of conjugating a matrix by another, x → y * x * yᴴ. -/ @[simps] def root.MatrixMap.conj (y : Matrix B A R) : MatrixMap A B R where toFun x := y * x * y.conjTranspose map_add' x y := by rw [Matrix.mul_add, Matrix.add_mul] map_smul' r x := by rw [RingHom.id_apply, Matrix.mul_smul, Matrix.smul_mul] omit [DecidableEq A] in theorem conj_isPo...

The Choi matrix of a map in symmetric Kraus form is a sum of rank-1 projectors.

The composition of IsCompletelyPositive maps is also completely positive.

Definition of a CP map, but with Fintype T in the definition instead of a Fin n. -/ theorem of_Fintype {M : MatrixMap A B R} (h : IsCompletelyPositive M) (T : Type*) [Fintype T] [DecidableEq T] : (M.kron (LinearMap.id : MatrixMap T T R)).IsPositive := by obtain ⟨n, ⟨e⟩⟩ : ∃ n : ℕ, Nonempty (T ≃ Fin n) := Finite.exists_equiv_fin T convert h n using 1 h...

Definition of a CP map, but with Fintype T in the definition instead of a Fin n.