Which states can be reached from a given state by unital completely positive maps?

Abstract

For a state $\omega$ on a C${}^{\ast }$-algebra $A$ we characterize all states $\rho$ in the weak* closure of the set of all states of the form $\omega \circ \phi$, where $\phi$ is a map on $A$ of the form $\phi (x)={\sum }_{i=1}^n{a}_i^{\ast }x{a}_i,$ ${\sum }_{i=1}^n{a}_i^{\ast }{a}_i=1$ ($a_i\in A$, $n=1,2,...$). These are precisely the states $\rho$ that satisfy $\Vert \rho |J\Vert \le \Vert \omega |J\Vert$ for each ideal $J$ of $A$. The corresponding question for normal states on a von Neumann algebra $R$ (with the weak* closure replaced by the norm closure) is also considered. All normal states of the form $\omega \circ \psi$, where $\psi$ is a quantum channel on $R$ (that is, a map of the form $\psi (x)={\sum }_j{a}_j^{\ast }x{a}_j$, where $a_j\in R$ are such that the sum ${\sum }_j{a}_j^{\ast }{a}_j$ converge to $1$ in the weak operator topology) are characterized. A variant of this topic for hermitian functionals instead of states is investigated. Maximally mixed states are shown to vanish on the strong radical of a C${}^{\ast }$-algebra and for properly infinite von Neumann algebras the converse also holds.

18 pages, to appear in PEMS