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Inmathematics apositive map is a map betweenC*-algebras that sends positive elements to positive elements. A completely positive map is one that satisfies a stronger, more robust condition.

Let${\displaystyle A}$ and${\displaystyle B}$ beC*-algebras. A linear map${\displaystyle \varphi :A\to B}$ is called apositive map if${\displaystyle \varphi }$ mapspositive elements to positive elements:${\displaystyle a\ge 0⟹\varphi (a)\ge 0}$.

Any linear map${\displaystyle \varphi :A\to B}$ induces another map

${\displaystyle \text{id}\otimes \varphi :{\mathbb{C}}^{k\times k}\otimes A\to {\mathbb{C}}^{k\times k}\otimes B}$

in a natural way. If${\displaystyle {\mathbb{C}}^{k\times k}\otimes A}$ is identified with the C*-algebra${\displaystyle A^{k\times k}}$ of${\displaystyle k\times k}$-matrices with entries in${\displaystyle A}$, then${\displaystyle \text{id}\otimes \varphi }$ acts as

We then say${\displaystyle \varphi }$ isk-positive if${\displaystyle {\text{id}}_{{\mathbb{C}}^{k\times k}}\otimes \varphi }$ is a positive map andcompletely positive if${\displaystyle \varphi }$ is k-positive for all k.

• Positive maps are monotone, i.e.${\displaystyle a_1\le a_2⟹\varphi (a_1)\le \varphi (a_2)}$ for allself-adjoint elements${\displaystyle a_1,a_2\in A_{sa}}$.
• Since${\displaystyle -\Vert a{\Vert }_A{1}_A\le a\le \Vert a{\Vert }_A{1}_A}$ for allself-adjoint elements${\displaystyle a\in A_{sa}}$, every positive map is automatically continuous with respect to theC*-norms and itsoperator norm equals${\displaystyle \Vert \varphi (1_A){\Vert }_B}$. A similar statement with approximate units holds for non-unital algebras.
• The set of positive functionals${\displaystyle \to \mathbb{C}}$ is thedual cone of the cone of positive elements of${\displaystyle A}$.

• Every *-homomorphism is completely positive.^[1]
• For every linear operator${\displaystyle V:H_1\to H_2}$ between Hilbert spaces, the map${\displaystyle L(H_1)\to L(H_2),A\mapsto VA{V}^{\ast }}$ is completely positive.^[2]Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
• Every positive functional${\displaystyle \varphi :A\to \mathbb{C}}$ (in particular everystate) is automatically completely positive.
• Given the algebras${\displaystyle C(X)}$ and${\displaystyle C(Y)}$ of complex-valued continuous functions oncompact Hausdorff spaces${\displaystyle X,Y}$, every positive map${\displaystyle C(X)\to C(Y)}$ is completely positive.
• Thetransposition of matrices is a standard example of a positive map that fails to be 2-positive. LetT denote this map on${\displaystyle {\mathbb{C}}^{n\times n}}$. The following is a positive matrix in${\displaystyle {\mathbb{C}}^{2\times 2}\otimes {\mathbb{C}}^{2\times 2}}$: The image of this matrix under${\displaystyle I_2\otimes T}$ is which is clearly not positive, having determinant −1. Moreover, theeigenvalues of this matrix are 1,1,1 and −1. (This matrix happens to be theChoi matrix ofT, in fact.)
  Incidentally, a map Φ is said to beco-positive if the composition Φ${\displaystyle \circ }$T is positive. The transposition map itself is a co-positive map.

• Choi's theorem on completely positive maps

• C*-algebras

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