Let $(\mathcal{M},τ)$ and $(\mathcal{N},τ^{\prime})$ be tracial von-Neumann algebras and let $φ:\mathcal{M}\to\mathcal{N}$ be a strictly completely positive, trace preserving map. Given a positive, invertible $B\in\mathcal{M}$ with $τ(B)=1$ , a state on $\mathcal{M}$ given by a positive $A\in L^1(\mathcal{M}, τ)$ is said to be recoverable if $\mathcal{R}(φ(A))=A$ where $\mathcal{R}$ is the Petz recovery map corresponding to $B$ and $φ$ . In this paper, we study recoverable states and show how an arbitrary state can be made close to a recoverable state via iterates of $\mathcal{R}\circφ$ . We show that there exists a completely positive, trace preserving map $ψ:\mathcal{M}\to\mathcal{M}$ such that $ψ(A)$ is recoverable for all $A$ and $(\mathcal{R}\circφ)^n\toψ$ in norm as operators on $L^p(\mathcal{M},τ)$ 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 $\mathcal{M}$ .Let $(\mathcal{M},τ)$ and $(\mathcal{N},τ^{\prime})$ be tracial von-Neumann algebras and let $φ:\mathcal{M}\to\mathcal{N}$ be a strictly completely positive, trace preserving map. Given a positive, invertible $B\in\mathcal{M}$ with $τ(B)=1$, a state on $\mathcal{M}$ given by a positive $A\in L^1(\mathcal{M}, τ)$ is said to be recoverable if $\mathcal{R}(φ(A))=A$ where $\mathcal{R}$ is the Petz recovery map corresponding to $B$ and $φ$. In this paper, we study recoverable states and show how an arbitrary state can be made close to a recoverable state via iterates of $\mathcal{R}\circφ$. We show that there exists a completely positive, trace preserving map $ψ:\mathcal{M}\to\mathcal{M}$ such that $ψ(A)$ is recoverable for all $A$ and $(\mathcal{R}\circφ)^n\toψ$ in norm as operators on $L^p(\mathcal{M},τ)$ 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 $\mathcal{M}$.

quant-pharXiv:2605.08829May 12, 20260 citations

Pure Completely Positive Maps As A Dual Object Of

c^{*}

-Algebras

Let

(\mathcal{M},τ)

and

(\mathcal{N},τ^{\prime})

be tracial von-Neumann algebras and let

φ:\mathcal{M}\to\mathcal{N}

be a strictly completely positive, trace preserving map. Given a positive, invertible

B\in\mathcal{M}

with

τ(B)=1

, a state on

\mathcal{M}

given by a positive

A\in L^1(\mathcal{M}, τ)

is said to be recoverable if

\mathcal{R}(φ(A))=A

where

\mathcal{R}

is the Petz recovery map corresponding to

B

and

φ

. 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}\circφ

. We show that there exists a completely positive, trace preserving map

ψ:\mathcal{M}\to\mathcal{M}

such that

ψ(A)

is recoverable for all

A

and

(\mathcal{R}\circφ)^n\toψ

in norm as operators on

L^p(\mathcal{M},τ)

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}