Contractivity of positive and trace-preserving maps under Lmp norms

Mary Beth Ruskai, Stanisław J. Szarek, Elisabeth M. Werner

Masahito Hayashi

Saptak Bhattacharya

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 $\mathcal{M}$.

Frederik vom Ende, Dariusz Chruściński, Gen Kimura +1 more

David Pérez-Garcı́a, Michael M. Wolf, Dėnes Petz +1 more

We provide a complete picture of contractivity of trace preserving positive maps with respect to p-norms. We show that for p>1 contractivity holds in general if and only if the map is unital. When the domain is restricted to the traceless subspace of Hermitian matrices, then contractivity is shown to hold in the case of qubits for arbitrary p⩾1 and in the case of qutrits if and only if p=1, ∞. In all noncontractive cases best possible bounds on the p-norms are derived.