Completely positive map

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In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one that 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\to B} is called a positive map if ϕ {\displaystyle \phi } maps positive elements to positive elements: a ≥ 0 ⟹ ϕ ( a ) ≥ 0 {\displaystyle a\geq 0\implies \phi (a)\geq 0} .

Any linear map ϕ : A → B {\displaystyle \phi :A\to B} induces another map

id ⊗ ϕ : C k × k ⊗ A → C k × k ⊗ B {\displaystyle {\textrm {id}}\otimes \phi :\mathbb {C} ^{k\times k}\otimes A\to \mathbb {C} ^{k\times k}\otimes B}

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

( a 11 ⋯ a 1 k ⋮ ⋱ ⋮ a k 1 ⋯ a k k ) ↦ ( ϕ ( a 11 ) ⋯ ϕ ( a 1 k ) ⋮ ⋱ ⋮ ϕ ( a k 1 ) ⋯ ϕ ( a k k ) ) . {\displaystyle {\begin{pmatrix}a_{11}&\cdots &a_{1k}\\\vdots &\ddots &\vdots \\a_{k1}&\cdots &a_{kk}\end{pmatrix}}\mapsto {\begin{pmatrix}\phi (a_{11})&\cdots &\phi (a_{1k})\\\vdots &\ddots &\vdots \\\phi (a_{k1})&\cdots &\phi (a_{kk})\end{pmatrix}}.}

We then say ϕ {\displaystyle \phi } is k-positive if id C k × k ⊗ ϕ {\displaystyle {\textrm {id}}_{\mathbb {C} ^{k\times k}}\otimes \phi } is a positive map and completely positive if ϕ {\displaystyle \phi } is k-positive for all k.

• Positive maps are monotone, i.e. a 1 ≤ a 2 ⟹ ϕ ( a 1 ) ≤ ϕ ( a 2 ) {\displaystyle a_{1}\leq a_{2}\implies \phi (a_{1})\leq \phi (a_{2})} for all self-adjoint elements a 1 , a 2 ∈ A s a {\displaystyle a_{1},a_{2}\in A_{sa}} .
• Since − ‖ a ‖ A 1 A ≤ a ≤ ‖ a ‖ A 1 A {\displaystyle -\|a\|_{A}1_{A}\leq a\leq \|a\|_{A}1_{A}} for all self-adjoint elements a ∈ A s a {\displaystyle a\in A_{sa}} , every positive map is automatically continuous with respect to the C*-norms and its operator norm equals ‖ ϕ ( 1 A ) ‖ B {\displaystyle \|\phi (1_{A})\|_{B}} . A similar statement with approximate units holds for non-unital algebras.
• The set of positive functionals → C {\displaystyle \to \mathbb {C} } is the dual cone of the cone of positive elements of A {\displaystyle A} .

• Every *-homomorphism is completely positive.^[1]
• For every linear operator V : H 1 → H 2 {\displaystyle V:H_{1}\to H_{2}} between Hilbert spaces, the map L ( H 1 ) → L ( H 2 ) ,   A ↦ V A V ∗ {\displaystyle L(H_{1})\to L(H_{2}),\ A\mapsto VAV^{\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 ϕ : A → C {\displaystyle \phi :A\to \mathbb {C} } (in particular every state) is automatically completely positive.
• Given the algebras C ( X ) {\displaystyle C(X)} and C ( Y ) {\displaystyle C(Y)} of complex-valued continuous functions on compact Hausdorff spaces X , Y {\displaystyle X,Y} , every positive map C ( X ) → C ( Y ) {\displaystyle C(X)\to C(Y)} is completely positive.
• The transposition of matrices is a standard example of a positive map that fails to be 2-positive. Let T denote this map on C n × n {\displaystyle \mathbb {C} ^{n\times n}} . The following is a positive matrix in C 2 × 2 ⊗ C 2 × 2 {\displaystyle \mathbb {C} ^{2\times 2}\otimes \mathbb {C} ^{2\times 2}} : [ ( 1 0 0 0 ) ( 0 1 0 0 ) ( 0 0 1 0 ) ( 0 0 0 1 ) ] = [ 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 ] . {\displaystyle {\begin{bmatrix}{\begin{pmatrix}1&0\\0&0\end{pmatrix}}&{\begin{pmatrix}0&1\\0&0\end{pmatrix}}\\{\begin{pmatrix}0&0\\1&0\end{pmatrix}}&{\begin{pmatrix}0&0\\0&1\end{pmatrix}}\end{bmatrix}}={\begin{bmatrix}1&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&1\\\end{bmatrix}}.} The image of this matrix under I 2 ⊗ T {\displaystyle I_{2}\otimes T} is [ ( 1 0 0 0 ) T ( 0 1 0 0 ) T ( 0 0 1 0 ) T ( 0 0 0 1 ) T ] = [ 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 ] , {\displaystyle {\begin{bmatrix}{\begin{pmatrix}1&0\\0&0\end{pmatrix}}^{T}&{\begin{pmatrix}0&1\\0&0\end{pmatrix}}^{T}\\{\begin{pmatrix}0&0\\1&0\end{pmatrix}}^{T}&{\begin{pmatrix}0&0\\0&1\end{pmatrix}}^{T}\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{bmatrix}},} which is clearly not positive, having determinant −1. Moreover, the eigenvalues of this matrix are 1,1,1 and −1. (This matrix happens to be the Choi matrix of T, in fact.)
  Incidentally, a map Φ is said to be co-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