# When the assignment map is completely positive

## TL;DR

This paper clarifies and generalizes the conditions under which the reduced dynamics of a quantum system remains completely positive, building on recent theoretical advances in the characterization of system-environment states.

## Contribution

It restates and clarifies a previous result on CP assignment maps within a new framework, leading to a straightforward generalization.

## Key findings

- Restates the CP assignment map result using a new framework
- Provides a clearer understanding of system-environment state conditions
- Generalizes the previous CP map result

## Abstract

Finding the general set of system-environment states for which the reduced dynamics of the system is completely positive (CP) is the subject of some recent works. An advance in this context appeared in [X. Lu, Phys. Rev. A 93, 042332 (2016)], where the problem has been solved for the case of CP assignment map. Here, we restate this result using the framework introduced in [J. M. Dominy et al., Quantum Inf. Process. 15, 465 (2016)]. This, we think, clarifies the mentioned result better and so leads to a generalization of it, straightforwardly.