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

Abstract

For a state $\omega$ on a C$^*$-algebra $A$ we characterize all states $\rho$ in the weak* closure of the set of all states of the form $\omega\circ\varphi$, where $\varphi$ is a map on $A$ of the form $\varphi(x)=\sum_{i=1}^na_i^*xa_i,$ $\sum_{i=1}^na_i^*a_i=1$ ($a_i\in A$, $n=1,2,...$). These are precisely the states $\rho$ that satisfy $\|\rho|J\|\leq\|\omega|J\|$ 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 $\psi(x)=\sum_ja_j^*xa_j$, where $a_j\in R$ are such that the sum $\sum_ja_j^*a_j$ 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$^*$-algebra and for properly infinite von Neumann algebras the converse also holds.