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 a natural way. If is identified with the C*-algebra of -matrices with entries in , then acts as
is called k-positive if is a positive map and completely positive if is k-positive for all k.
Properties
Positive maps are monotone, i.e. for all
self-adjoint elements .
Since for all
self-adjoint elements , every positive map is automatically continuous with respect to the
C*-norms and its
operator norm equals . A similar statement with approximate units holds for non-unital algebras.
The set of positive functionals is the
dual cone of the cone of positive elements of .
For every linear operator between Hilbert spaces, the map is completely positive.[2]Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
Every positive functional (in particular every
state) is automatically completely positive.
Given the algebras and of complex-valued continuous functions on
compact Hausdorff spaces, every positive map 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 . The following is a positive matrix in : The image of this matrix under is 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 T is positive. The transposition map itself is a co-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 a natural way. If is identified with the C*-algebra of -matrices with entries in , then acts as
is called k-positive if is a positive map and completely positive if is k-positive for all k.
Properties
Positive maps are monotone, i.e. for all
self-adjoint elements .
Since for all
self-adjoint elements , every positive map is automatically continuous with respect to the
C*-norms and its
operator norm equals . A similar statement with approximate units holds for non-unital algebras.
The set of positive functionals is the
dual cone of the cone of positive elements of .
For every linear operator between Hilbert spaces, the map is completely positive.[2]Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
Every positive functional (in particular every
state) is automatically completely positive.
Given the algebras and of complex-valued continuous functions on
compact Hausdorff spaces, every positive map 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 . The following is a positive matrix in : The image of this matrix under is 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 T is positive. The transposition map itself is a co-positive map.