Recoverable states on von-Neumann algebras

TL;DR

This paper investigates the properties of recoverable states in von Neumann algebras, demonstrating how arbitrary states can be approximated by recoverable ones through iterative Petz recovery maps, with implications for quantum information theory.

Contribution

It introduces a method to approximate any state by a recoverable state using iterates of the Petz recovery map and establishes convergence results in various operator norms.

Findings

Existence of a completely positive, trace-preserving map making all states recoverable.

Iterates of the Petz recovery map converge in norm to a map with recoverable outputs.

Convergence also holds strongly in the $L^1$ norm, with a decomposition theorem for normal states.

Abstract

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 $\mathcal{M}$.