Interplay between coherence and mixedness as well as its geometry for arbitrary two-qubit X-states

Recently, a relationship between coherence and mixedness for an arbitrary d-dimensional quantum states has been built by Singh et al. (Phys Rev A 91:052115, 2015). However, whether this relationship holds under the action of decoherence remains to be further investigated. In this paper, we firstly study the geometry of coherence and mixedness for a class of two-qubit X-states and demonstrate new pictures and structures of trade-off between coherence and mixedness. At the same time, we also examine the dynamical behaviors of coherence, mixedness and their trade-off for both qubit-qubit system and environment-environment system where a two-qubit composite system is interacting with their own environmental channels. Different types of channels, such as amplitude-damping, phase-damping and bit-phase-flip channels with (non-)Markovian effects, are taken into consideration. Our results show that (i) Coherence can be completely transferred from the qubit-qubit system to the environment-environment system in the Markovian environment while in non-Markovian scenarios, the coherence between the qubit-qubit system and environment-environment system can be transferred each other. (ii) The decrease in the coherence is not always accompanied by an increase in the mixedness. (iii) The trade-off between coherence and mixedness still holds in the open system.