Geometry of Quantum Coherence for Two Qubit X States

The geometry of the structure of entanglement and discord for Bell-diagonal states is depicted by Lang and Caves (Phys. Rev. Lett. 105 , 150501, 2010 ). In this paper, we investigate the geometry with respect to several distance-based quantifiers of coherence for Bell-diagonal states. We find that as both l 1 norm and relative entropy of coherence vary continuously from zero to one, their related geometric surfaces move from the region of separable states to the region of entangled states, a fact illustrating intuitively that quantum states with nonzero coherence can be used for entanglement creation. We find the necessary and sufficient conditions that quantum discord of Bell-diagonal states equals to its relative entropy of coherence, and depict the surfaces related to the equality. We give surfaces of relative entropy of coherence for X states. We show the surfaces of dynamics of relative entropy of coherence for Bell-diagonal states under local nondissipative channels and find that all coherences under local nondissipative channels decrease.