# Divisibility of qubit channels and dynamical maps

## TL;DR

This paper investigates the divisibility properties of quantum channels and dynamical maps, characterizing various divisibility classes for qubit channels, and explores their implications for entanglement breaking and quantum dynamics.

## Contribution

It introduces a detailed analysis of L-, CP-, and P-divisibility for quantum channels, extending previous results and visualizing these properties for qubit channels.

## Key findings

- Characterization of divisibility subsets for qubit channels
- Transitions between divisibility classes are possible
- Divisible but not infinitesimal divisible qubit channels are entanglement breaking

## Abstract

The concept of divisibility of dynamical maps is used to introduce an analogous concept for quantum channels by analyzing the \textit{simulability} of channels by means of dynamical maps. In particular, this is addressed for Lindblad divisible, completely positive divisible and positive divisible dynamical maps. The corresponding L-divisible, CP-divisible and P-divisible subsets of channels are characterized (exploiting the results by Wolf et al., Comm. Math. Phys., 279(1):147-168, 2008) and visualized for the case of qubit channels. We discuss the general inclusions among divisibility sets and show several equivalences for qubit channels. To this end we study the conditions of L-divisibility for finite dimensional channels, especially the cases with negative eigenvalues, extending and completing the results of Phys. Rev. Lett., 101(15):150402, 2008. Furthermore we show that transitions between every two of the defined divisibility sets are allowed. We explore particular examples of dynamical maps to compare these concepts. Finally, we show that every divisible but not infinitesimal divisible qubit channel (in positive maps) is entanglement breaking, and open the question if something similar occurs for higher dimensions.