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.