# Information flow versus divisibility for qubit evolution

## TL;DR

This paper proves the universal equivalence between lack of information backflow and CP divisibility in qubit dynamical maps, extending previous results to all qubit evolutions and providing new insights into their structure.

## Contribution

It demonstrates that the equivalence between information backflow and CP divisibility holds for all qubit dynamical maps, not just image non-increasing ones.

## Key findings

- Equivalence between information backflow and CP divisibility is universal for qubit maps.
- No CPTP projector exists onto a 3D subspace spanned by qubit density operators.
- Includes examples of non-image non-increasing qubit evolutions.

## Abstract

We study the relation between lack of Information Backflow and completely positive divisibility (CP divisibility) for non-invertible qubit dynamical maps. Recently, these two concepts were shown to be fully equivalent for the so called image non-increasing dynamical maps. Here we show that this equivalence is universal for any qubit dynamical map. A key ingredient in our proof is the observation that there does not exist CPTP projector onto a 3-dimensional subspace spanned by qubit density operators. Our analysis is illustrated by several examples of qubit evolution including also dynamical maps which are not image non-increasing.