Sufficient conditions placed on initial system-environment states for positive maps

TL;DR

This paper establishes a sufficient condition for the positivity of quantum maps based on initial system-environment states, expanding understanding beyond complete positivity and aiding in identifying positive but not completely positive maps.

Contribution

It introduces a new sufficient condition for positivity of quantum maps considering initial correlations, extending previous work on complete positivity.

Findings

Provides a criterion for positive maps with initial correlations

Offers examples illustrating the application of the condition

Suggests procedures to identify positive but not completely positive maps

Abstract

A system interacting with its environment will give rise to a quantum evolution. After tracing over the environment the net evolution of the system can be described by a linear Hermitian map. It has recently been shown that a necessary and sufficient condition for this evolution to be completely positive is for the initial state to have vanishing quantum discord. In this paper, we provide a sufficient condition for the map to be positive with respect to the initial system-environment correlation. This could lead to ways in which to identify positive but not completely positive maps. Illustrative examples and suggestive procedures are also provided.