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.

Definition

Let A and B be C*-algebras. A linear map \phi: A\to B is called a positive map if \phi maps positive elements to positive elements: a\geq 0 \implies \phi(a)\geq 0.

Any linear map \phi:A\to B induces another map

\textrm{id} \otimes \phi : \mathbb{C}^{k \times k} \otimes A \to \mathbb{C}^{k \times k} \otimes B

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

\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 \phi is k-positive if \textrm{id}_{\mathbb{C}^{k\times k}} \otimes \phi is a positive map and completely positive if \phi is k-positive for all k.

Properties

  • Positive maps are monotone, i.e. a_1\leq a_2\implies \phi(a_1)\leq\phi(a_2) for all self-adjoint elements a_1,a_2\in A_{sa}.
  • Since -\|a\|_A 1_A \leq a \leq \|a\|_A 1_A for all self-adjoint elements a\in A_{sa}, every positive map is automatically continuous with respect to the C*-norms and its operator norm equals \|\phi(1_A)\|_B. A similar statement with approximate units holds for non-unital algebras.
  • The set of positive functionals \to\mathbb{C} is the dual cone of the cone of positive elements of A.

Examples

  • Every *-homomorphism is completely positive.[1]
  • For every linear operator V:H_1\to H_2 between Hilbert spaces, the map L(H_1)\to L(H_2), \ A \mapsto V A 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 \phi:A \to \mathbb{C} (in particular every state) is automatically completely positive.
  • Given the algebras C(X) and C(Y) of complex-valued continuous functions on compact Hausdorff spaces X, Y, every positive map 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 \mathbb{C}^{n \times n}. The following is a positive matrix in \mathbb{C}^{2\times 2} \otimes \mathbb{C}^{2\times 2}:
\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 \otimes T is

\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 \circ T is positive. The transposition map itself is a co-positive map.

See also

References

  1. ^ K. R. Davidson: C*-Algebras by Example, American Mathematical Society (1996), ISBN 0-821-80599-1, Thm. IX.4.1
  2. ^ R.V. Kadison, J. R. Ringrose: Fundamentals of the Theory of Operator Algebras II, Academic Press (1983), ISBN 0-1239-3302-1, Sect. 11.5.21