# Markovianity of the reference state, complete positivity of the reduced dynamics, and monotonicity of the relative entropy

## TL;DR

This paper investigates the conditions under which the reduced dynamics of a quantum system remain completely positive, emphasizing the importance of fixed system-environment dimensions and exploring the limitations of monotonicity of quantum relative entropy under Hermitian maps.

## Contribution

It demonstrates that the Markovianity condition is necessary for complete positivity when system-environment dimensions are fixed and shows the non-generalizability of quantum relative entropy monotonicity to Hermitian maps.

## Key findings

- Complete positivity can occur without Markovianity if dimensions are fixed.
- Counterexample provided for non-Markovian states with CP dynamics.
- Monotonicity of quantum relative entropy does not extend to Hermitian maps.

## Abstract

Consider the set $\mathcal{S}=\lbrace\rho_{SE}\rbrace$ of possible initial states of the system-environment, steered from a tripartite reference state $\omega_{RSE}$. Buscemi [F. Buscemi, Phys. Rev. Lett. 113, 140502 (2014)] showed that the reduced dynamics of the system, for each $\rho_{S}\in \mathrm{Tr}_{E}\mathcal{S}$, is always completely positive if and only if $\omega_{RSE}$ is a Markov state. There, during the proof, it has been assumed that the dimensions of the system and the environment can vary through the evolution. Here, we show that this assumption is necessary: we give an example for which, though $\omega_{RSE}$ is not a Markov state, the reduced dynamics of the system is completely positive, for any evolution of the system-environment during which the dimensions of the system and the environment remain unchanged. As our next result, we show that the result of Muller-Hermes and Reeb [A. Muller-Hermes and D. Reeb, Ann. Henri Poincare 18, 1777 (2017)], of monotonicity of the quantum relative entropy under positive maps, cannot be generalized to the Hermitian maps, even within their physical domains.