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

TL;DR

This paper characterizes the states reachable from a given state via unital completely positive maps on C*-algebras and von Neumann algebras, linking state transformations to ideal and radical properties.

Contribution

It provides a complete characterization of states obtained through unital completely positive maps, extending to von Neumann algebras and exploring the role of radicals.

Findings

States reachable by unital CP maps satisfy specific ideal norm inequalities.

Normal states of the form ω∘ψ are characterized where ψ is a quantum channel.

Maximally mixed states vanish on the strong radical of a C*-algebra.

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.