# 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}
}
• Published 4 August 2019
• Mathematics
• Physical Review A
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…
3 Citations

## Figures from this paper

Positivity of the assignment map implies complete positivity of the reduced dynamics
If there exists a positive assignment map, then the so-called reference state is a Markov state, which implies that there exists another assignment map which is completely positive, which means the reduced dynamics of the system is also completely positive.
Necessary and sufficient condition for the reduced dynamics of an open quantum system interacting with an environment to be linear
The dynamics of a closed quantum system, under a unitary time evolution $U$, is, obviously, linear. But, the reduced dynamics of an open quantum system $S$, interacting with an environment $E$, is
When the reduced dynamics is linear
The dynamics of a closed quantum system, under a unitary time evolution $U$, is, obviously, linear. But, the reduced dynamics of an open quantum system $S$, interacting with an environment $E$, is

## References

SHOWING 1-10 OF 25 REFERENCES
Positivity of the assignment map implies complete positivity of the reduced dynamics
If there exists a positive assignment map, then the so-called reference state is a Markov state, which implies that there exists another assignment map which is completely positive, which means the reduced dynamics of the system is also completely positive.
A general framework for complete positivity
• Mathematics
Quantum Inf. Process.
• 2016
A complete and consistent mathematical framework for the discussion and analysis of completePositivity for correlated initial states of open quantum systems is described and it is shown that the constrained nature of the problem gives rise to not one but three inequivalent types of complete positivity.
When the Assignment Map Is Completely Positive
• Mathematics
Open Syst. Inf. Dyn.
• 2018
This work restates the result of [7], where the problem was solved for the case of CP assignment map, using the framework introduced in [8], and leads to a generalization of it, straightforwardly.
Complete positivity, Markovianity, and the quantum data-processing inequality, in the presence of initial system-environment correlations.
This work shows that complete positivity is not only sufficient but also necessary for the validity of the quantum data-processing inequality, and provides an intuitive information-theoretic framework to unify and extend a number of previous results.
Monotonicity of the Quantum Relative Entropy Under Positive Maps
• Mathematics
• 2015
We prove that the quantum relative entropy decreases monotonically under the application of any positive trace-preserving linear map, for underlying separable Hilbert spaces. This answers in the
Erratum: Vanishing Quantum Discord is Necessary and Sufficient for Completely Positive Maps [Phys. Rev. Lett. 102, 100402 (2009)].
• Mathematics
Physical review letters
• 2016
This work developed a complete and consistent mathematical framework for the discussion and analysis of linear subsystem dynamics, including the question of complete positivity for correlated initial states, and developed a general method for constructing examples yielding completely positive subdynamics, even though they may feature highly entangled states.
Structure of correlated initial states that guarantee completely positive reduced dynamics
We use the Koashi-Imoto decomposition of the degrees of freedom of joint system-environment initial states to investigate the reduced dynamics. We show that a subset of joint system-environment
Dynamics beyond completely positive maps : some properties and applications
• Mathematics
• 2008
Maps that are not completely positive (CP) are often useful to describe the dynamics of open systems. An apparent violation of complete positivity can occur because there are prior correlations of
Completely positive maps and classical correlations
• Mathematics, Physics
• 2008
We expand the set of initial states of a system and its environment that are known to guarantee completely positive reduced dynamics for the system when the combined state evolves unitarily. We
Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality
• Mathematics
• 2004
We give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von Neumann entropy. By combining a result of Petz characterising the equality