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

Abstract

Consider the set $\mathcal{S}=\{{\rho }_{SE}\}$ 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.

8 pages, 1 figure

References in corpus (5)

• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Vanishing quantum discord is necessary and sufficient for completely positive maps
• Completely Positive Maps and Classical Correlations
• Vanishing quantum discord is not necessary for completely-positive maps
• Completely positive maps within the framework of direct-sum decomposition of state space

Cited by in corpus (1)

• Necessary and sufficient condition for the reduced dynamics of an open quantum system interacting with an environment to be linear