On complete positivity, Markovianity, and the quantum data-processing inequality, in the presence of initial system-environment correlations

TL;DR

This paper establishes that complete positivity is both necessary and sufficient for the quantum data-processing inequality to hold, even with initial system-environment correlations, linking dynamics and information flow.

Contribution

It demonstrates that the reduced dynamics remain completely positive if and only if initial correlations do not permit backward information flow, unifying previous results.

Findings

Complete positivity is necessary and sufficient for the data-processing inequality.

Initial correlations without backward information flow preserve complete positivity.

Provides an information-theoretic framework unifying previous results.

Abstract

We show that complete positivity is not only sufficient but also necessary for the validity of the quantum data-processing inequality. As a consequence, the reduced dynamics of a quantum system are completely positive, even in the presence of initial correlations with its surrounding environment, if and only if such correlations do not allow any anomalous backward flow of information from the environment to the system. Our approach provides an intuitive information-theoretic framework to unify and extend a number of previous results.