Completely positive map

In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one which satisfies a stronger, more robust condition. In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one which satisfies a stronger, more robust condition. Let A {displaystyle A} and B {displaystyle B} be C*-algebras. A linear map ϕ : A → B {displaystyle phi :A o B} is called positive map if ϕ {displaystyle phi } maps positive elements to positive elements: a ≥ 0 ⟹ ϕ ( a ) ≥ 0 {displaystyle ageq 0implies phi (a)geq 0} . Any linear map ϕ : A → B {displaystyle phi :A o B} induces another map in a natural way. If C k × k ⊗ A {displaystyle mathbb {C} ^{k imes k}otimes A} is identified with the C*-algebra A k × k {displaystyle A^{k imes k}} of k × k {displaystyle k imes k} -matrices with entries in A {displaystyle A} , then id ⊗ ϕ {displaystyle { extrm {id}}otimes phi } acts as We say that ϕ {displaystyle phi } is k-positive if id C k × k ⊗ ϕ {displaystyle { extrm {id}}_{mathbb {C} ^{k imes k}}otimes phi } is a positive map, and ϕ {displaystyle phi } is called completely positive if ϕ {displaystyle phi } is k-positive for all k. The image of this matrix under I 2 ⊗ T {displaystyle I_{2}otimes T} is