Entanglement wedge cross section inequalities in AdS/BCFT

The entanglement wedge cross section in holographic picture provides a geometrical description of the entanglement for mixed states. In this paper we study the inequalities for the entanglement wedge cross section in AdS/BCFT duality. In the presence of the boundary in conformal field theory (CFT), the dual entanglement wedge cross section exhibits abundant phase structures since the extremal surface may end on the brane. We present a universal treatment which is applicable for all the possible phases such that the inequalities for the entanglement wedge cross section can be proved in an algebraic manner rather than a diagrammatic manner. We show that the entanglement wedge cross section in AdS/BCFT satisfies the same inequalities as in AdS/CFT.