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

@article{Sargolzahi2019MarkovianityOT,
  title={Markovianity of the reference state, complete positivity of the reduced dynamics, and monotonicity of the relative entropy},
  author={Iman Sargolzahi and Sayyed Yahya Mirafzali},
  journal={Physical Review A},
  year={2019}
}
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… 
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