Markovianity of the reference state, complete positivity of the reduced dynamics, and monotonicity of the relative entropy
Abstract
Consider the set S ={ρ_{S E}} of possible initial states of the systemenvironment, steered from a tripartite reference state ω_{R S E}. Buscemi [F. Buscemi, Phys. Rev. Lett. 113, 140502 (2014), 10.1103/PhysRevLett.113.140502] showed that the reduced dynamics of the system, for each ρ_{S}∈Tr_{E}S , is always completely positive if and only if ω_{R S E} 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 ω_{R S E} is not a Markov state, the reduced dynamics of the system is completely positive, for any evolution of the systemenvironment during which the dimensions of the system and the environment remain unchanged. As our next result, we show that the result of MullerHermes and Reeb [A. MullerHermes and D. Reeb, Ann. Henri Poincare 18, 1777 (2017), 10.1007/s0002301705509], of monotonicity of the quantum relative entropy under positive maps, cannot be generalized to the Hermitian maps, even within their physical domains.
 Publication:

Physical Review A
 Pub Date:
 October 2019
 DOI:
 10.1103/PhysRevA.100.042121
 arXiv:
 arXiv:1908.01203
 Bibcode:
 2019PhRvA.100d2121S
 Keywords:

 Quantum Physics
 EPrint:
 8 pages, 1 figure