Recoverable states on von-Neumann algebras

Let $(\mathcal{M},\tau )$ and $(\mathcal{N},{\tau }^{\mathrm{\prime }})$ 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 $1\text{textless}p\text{textless}\mathrm{\infty }$, 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 $\mathcal{M}$.