Contractivity of positive and trace preserving maps under $L_p$ norms

TL;DR

This paper characterizes when trace-preserving positive maps are contractive under $L_p$ norms, showing unitality is necessary for $p>1$, with specific results for qubits and qutrits.

Contribution

It provides a complete characterization of contractivity of positive trace-preserving maps under $L_p$ norms, including new bounds and conditions for qubits and qutrits.

Findings

Contractivity holds for $p>1$ iff the map is unital.

For qubits, contractivity holds for all $p \\geq 1$ on traceless subspace.

For qutrits, contractivity holds only for $p=1$ and $p=\\infty$.

Abstract

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 $p\geq 1$ and in the case of qutrits if and only if $p=1,\infty$. In all non-contractive cases best possible bounds on the $p$-norms are derived.