The fact that, in the course of open evolution, the creation of system-environment entanglement out of the initial coherence occurs in the vast majority of cases has been known for many years. However, how much entanglement can be created under general assumptions has not been analyzed yet. In this work, we investigate system-environment entanglement generation under a broad and important class of interactions known as pure-dephasing interactions for arbitrary initial conditions. Our main results are the following bounds on the relative entropy of entanglement $E_r\left({\sigma }_{SE}\right)$ after evolution: $C_r\left({\rho }_S\right)-H\left(\mathcal{I}\right|\mathcal{M})\le E_r\left({\sigma }_{SE}\right)\le C_r\left({\rho }_S\right)-C_r\left({\sigma }_S\right)+H\left(\mathcal{I}\right)$, connecting it on one side to the initial relative entropy of coherence $C_r\left({\rho }_S\right)$ and information extractable from the environment $H\left(\mathcal{I}\right|\mathcal{M})$ and on the other side to the decoherence strength, $C_r\left({\rho }_S\right)-C_r\left({\sigma }_S\right)$, making the statement that larger entanglement leads to more decoherence quantitatively precise. As an illustration, we apply our results to the spin-boson model, where a qubit interacts with a bosonic bath, showing the time evolution of the entanglement and the crossover between Markovian and non-Markovian dynamics. Our results help us better understand and quantify the complicated correlations produced during open evolution, which will hopefully lead to better and new decoherence-mitigation techniques.

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• Open quantum systems & decoherence
• Resource theories