Recoverable states on von-Neumann algebras

Let $(\mathcal{M},\tau)$ and $(\mathcal{N},\tau^{\prime})$ be tracial von-Neumann algebras and let $\phi:\mathcal{M}\to\mathcal{N}$ be a strictly completely positive, trace preserving map. Given a positive, invertible $B\in\mathcal{M}$ with $\tau(B)=1$, a state on $\mathcal{M}$ given by a positive $A\in L^1(\mathcal{M}, \tau)$ is said to be recoverable if $\mathcal{R}(\phi(A))=A$ where $\mathcal{R}$ is the Petz recovery map corresponding to $B$ and $\phi$. In this paper, we study recoverable states and show how an arbitrary state can be made close to a recoverable state via iterates of $\mathcal{R}\circ\phi$. We show that there exists a completely positive, trace preserving map $\psi:\mathcal{M}\to\mathcal{M}$ such that $\psi(A)$ is recoverable for all $A$ and $(\mathcal{R}\circ\phi)^n\to\psi$ in norm as operators on $L^p(\mathcal{M},\tau)$ for all $1\,\textless p\,\textless\infty$, and discuss potential applications to quantum information theory. We also show that this convergence holds strongly in $L^1$. Finally, we prove an interesting decomposition theorem for normal states on $\mathcal{M}$.